Mooring system fatigue analysis of a floating offshore wind turbine

Ocean Engineering xxx (xxxx) xxx

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Mooring system fatigue analysis of a floating offshore wind turbine ~igo J. Losada Carlos Barrera, Tommaso Battistella, Raúl Guanche *, In Environmental Hydraulics Institute, Universidad de Cantabria (IHCantabria), Avda. Isabel Torres, 15, Parque Científico y Tecnol� ogico de Cantabria, 39011, Santander, Spain

A R T I C L E I N F O

A B S T R A C T

Keywords: Fatigue Mooring system Floating offshore wind turbine Non-linear interpolation techniques

Mooring systems are under a cyclic loading process caused by the randomness of metocean conditions, which could lead to a fatigue failure of the station keeping system. The present paper presents an innovative meth­ odology for the assessment of floating offshore wind turbine mooring system fatigue considering the full lifetime of the structure. The method integrates the impact of the life cycle metocean conditions over the dynamic performance of the platform thanks to coupled numerical models, selection and non-linear data interpolation techniques and commonly accepted fatigue approaches. One of the benefits of using this methodology is that there are no uncertainties due to the selection of a reduced set of sea states. The methodology is applied to a set of moorings with different properties in the DeepCwind platform to evaluate the solution which offers the best compromise between size and fatigue damage. Results show that the best long-term mooring behaviour is achieved with a weight of approximately 300 kg/m. A comparison is conducted between the fatigue damage obtained through the life-cycle method and conventional methods. The mean differences observed between the standard and the new method proposed are between 13% and 49% depending on the use of the S–N or T-N curves.

1. Introduction Wind industry has experienced a huge growth in recent years moti­ vated by the need for energy sources alternative to fossil fuels. In particular, offshore wind energy technology evidences important po­ tential in the coming years. In fact, this trend is being led by the Euro­ pean Union with a total installed offshore wind capacity of 15,780 MW in 2017 (Wind Europe, 2018a). The dominant substructures in offshore wind farms are fixed foundations including monopiles, jackets, and gravity base foundations (Wind Europe, 2018a). However, new tech­ nological advances point to offshore floating wind farms in intermediate and deep waters. These new solutions are an opportunity for countries with important wind resources but with narrow continental shelves. Hywind Scotland was the first floating offshore wind farm in the world with a total of five floating spar buoys installed in 2017. After this success and according to European policies, floating offshore wind farms could provide between 4 and 5 GW by 2030 (Wind Europe, 2018b). Floating offshore wind turbines (FOWTs), in comparison with fixed foundations, have a higher level of design complexity. Despite the sys­ tem stability is mainly driven by the floating platform characteristics, the mooring system plays a critical role in the design of structure

motions and natural periods. Hence, station keeping systems based on mooring lines and anchors are crucial to guarantee structure surviv­ ability and its components (e.g. power cable) under different metocean conditions. Traditionally, a successful mooring design considers several limit states (LS) (DNVGL-OS-E301, 2018) as follows: ultimate (ULS), accidental (ALS), fatigue (FLS) and service (SLS). These limit states contribute to properly ensuring the resistance of the mooring and its service criteria. This article is focused on the analysis of fatigue loads (FLS) on the mooring lines of a FOWT. Moorings are under continuous cyclic meto­ cean loads, therefore fatigue damage is a potential failure mechanism. Fatigue damage can be evaluated by means of either an S–N or a T-N curve. These curves relate a constant stress (S) or tension (T) range with the maximum number of cycles until component failure (N). Numerous investigations have been conducted since the 1980s to understand the fatigue failure mechanism of offshore mooring chains. Building on prior research on this topic, van Helvoirt (Van Helvoirt, 1982) described an experimental test campaign related to the static and fatigue strength of stud-link chains and connecting links under high load cycles in the marine environment. Lereim (1985) presented a complete study of chain reliability including experimental and numerical assessments. He

* Corresponding author. E-mail address: [email protected] (R. Guanche). https://doi.org/10.1016/j.oceaneng.2019.106670 Received 5 April 2019; Received in revised form 27 September 2019; Accepted 31 October 2019 0029-8018/© 2019 Elsevier Ltd. All rights reserved.

Please cite this article as: Carlos Barrera, Ocean Engineering, https://doi.org/10.1016/j.oceaneng.2019.106670

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proposed an S–N curve for chain links based on a fatigue crack propa­ gation analysis for a corrosive environment assuming different initial surface crack depths. The American Petroleum Institute (API) (API RP 2FP1, 1993) proposed a standard based on a normalised tension range to define the fatigue lifetime for each mooring component. Different T-N curves are defined according to floating offshore structure experiments. Later, Det Norske Veritas (DNV) (DNV-OS-E301, 2001) published different design S–N curves to estimate the fatigue life. These two standards are widely used as references for fatigue design by industry and researchers (Lassen and Syvertsen, 1997) (Xue et al., 2018) (Thies et al., 2014). As has been shown, early investigations have been focused on building fatigue damage curves to allow a safe mooring design. Currently, the mooring integrity management is a current topic among researchers and engineers. Issues such as residual stress, anomalous loading modes or corrosion are receiving increased awareness by the offshore industry. Martinez et al. (2017) estimated how residual stresses generated during the manufacturing process contribute to the fatigue life of mooring chains depending on the loading mode. Rampi et al. (2015) described a new fatigue mechanism based on a combination of high pretension levels and motions generating out-of-plane bending fatigue loading and proposed a new S–N fatigue curve diagram. Gabri­ elsen et al. (2018) conducted mooring fatigue tests considering surface roughness, corrosion pits and mean loads using chain segments recov­ ered from a floating structure in the North Sea. Their results concluded that the degradation of the chains reduces the fatigue capacity although the fatigue design capacity is still above the S–N design curve given by (DNVGL-OS-E301, 2018). As seen in the previous literature review, a complete analysis of the different mechanisms that induce mooring fatigue already has been carried out. However, there are significantly less investigations of longterm fatigue performance evaluation. Traditionally, a set of environ­ mental states is chosen to discretise the long-term environmental con­ ditions (occurrence matrix) (DNVGL-OS-E301, 2018) (API, 2008) but this selection may affect the long-term fatigue life of mooring chains of FOWTs. The present paper proposes a new methodology to estimate the long-term fatigue analysis involving long-term metocean databases, advanced selection methods, FOWT numerical models and non-linear interpolation techniques. All of these combined allow us to recreate the long-term damage to a mooring system with a low numerical cost. The new methodology is applied to different mooring systems in order to select the most appropriate mooring on a FOWT located in the BiMEP test site (north of Spain) considering the influence of all metocean conditions on the floating structure life-cycle. The reference platform used in this work is the DeepCwind semisubmersible platform (Rob­ ertson, Jonkman, Wendt, Goupee, Dagher). This paper is organised as follows. In section 2, the new methodol­ ogy, the metocean databases, the advanced selection techniques, the FOWT numerical model, the methods to evaluate the fatigue damage and the non-linear interpolation techniques are described. Section 3 describes the case study involving the site and the definition of FOWT, the selected sea states and the long-term results. Finally, a discussion of the obtained results and the main conclusions of this investigation are presented in sections 4 and 5, respectively.

the FOWT lifetime. A maximum dissimilarity selection technique is then applied to select the most representative sea state subset from this database. FOWT dynamics and mooring fatigue are evaluated by means of a numerical model for the full subset of sea states. Finally, the FOWT dynamics and mooring fatigue results are rebuilt for the full lifetime using a radial basis function (RBF) interpolation technique. 2.1. Site assessment and metocean database Long-term analysis requires the use of databases including time se­ ries of the relevant environmental parameters to assess wind and wave conditions at a given offshore location. They are usually built upon metocean reanalysis techniques and provide metocean parameters extended over several decades. The metocean database used in this work is based on the reanalysis developed by IHCantabria and BiMEP in the framework of the TRLþ project (Metocean Analysis of BiMEP for Offshore Design, 2017). Wind data are obtained from the Seawind database (Menendez et al., 2014). Wind data are modelled with the Weather Research & Fore­ casting model and the Advanced Research dynamical solver module, developed by the National Center for Atmospheric Research (Skamarock et al., 2008). Wind speed (W) and direction (β) for the 1985–2015 period at 10 m above the sea surface are provided with a resolution of 1 h. Wind speed at the nacelle height is obtained using the empirical expression for the wind power law (Jonkman and Kilcher, 2012) (Emeis, 2013). Wave data are taken from the Global Ocean Waves database (Reguero et al., 2012). Significant wave height (Hs), wave peak period (Tp) and direc­ tion (α), among other wave parameters, for the 1985–2015 period are provided with a 1-h resolution. The numerical simulation of the dynamic response of the FOWT and its mooring system over several decades using hourly metocean time series would require a huge computational effort. As a consequence, the number of sea states to be simulated must be reduced by using a reliable selection technique. 2.2. Maximum dissimilarity selection technique In general, the objective of the selection techniques is to reduce the large data amounts provided by metocean databases at given locations into a representative subset maintaining the variability of the original

2. Methodology to predict mooring fatigue damage and FOWT dynamics The most accurate method to estimate the mooring fatigue response is a dynamic analysis in the time domain (API, 2008) where all non­ linearities and dynamics are captured. The main disadvantage of this method is the excessive computational cost associated with the evalu­ ation of all observed sea states at the target location. The proposed methodology attempts to provide a more efficient approach by following a different set of steps, as shown in Fig. 1. The first step is to collect from a metocean database all historical sea states for a time period equal to

Fig. 1. Methodology to predict the lifetime mooring fatigue damage and FOWT dynamics. 2

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data. The maximum dissimilarity algorithm (MDA) (Camus et al., 2011) is used to select this representative subset. The samples included in this subset are hourly sea states of dimension 5 defined by W, β, Hs, Tp and α. From a database P including I sea states, pii ¼ {p1, p2, …, pI}, a representative data subset O with J sea states, ojj ¼ {o1, o2, …, oJ}, is selected, with J < I. The selection starts by choosing an initial sea state from the full database P. In this case, the sea state containing the maximum significant wave height is chosen as the initial sea state. The rest of the sea states are iteratively selected, calculating the dissimilarity between of the remaining sea states in the database and the sea states added to the subset by choosing those with the largest dissimilarity at each iteration. During the application of the algorithm, the subset O is composed of R sea states, orr ¼ {o1, o2, …, oR}, with R < J. The selection finishes when the number of required sea states, J, is reached. The dissimilarity (di) is evaluated by means of the Euclidean-Circular norm ( k k) between the vectors p and o. The dissimilarity value is taken ac­ cording to Polinsky (Polinsky et al., 1996) as follows: � pii oR 1 di ¼ minimum (1) minimum ​ pii orr ; rr ¼ 1; …; R 2

phase and Qj;α ðωl Þ ¼ ql;α ðωl Þ eiφlj;α , where ql;α is the amplitude of the firstorder force per wave amplitude unit and φlj;α is its phase. The symbol * denotes the complex conjugate. Qj;α ðωl ; ωm Þ represents the quadratic

transfer function (QTF) associated with the frequency difference be­ tween pairs of wave components. The mooring system is simulated using a dynamic model that allows the estimation of mooring loads with higher accuracy than a quasi-static model (Barrera et al., 2019a) (Robertson et al., 2017) although it is computationally more demanding. The dynamic formulation is based on Newton’s second law (5) and is solved by a finite element method as follows: � � ∂2 ! r ∂ Ts ∂! r ! ρ0 2 ¼ þ f ð1 þ eÞ (5) ∂s 1 þ e ∂s ∂t where ρ0 is the linear weight, ! r is the position vector, s is the longitu­ ! dinal coordinate, e is the deformation, Ts is the tension and f is the sum ! of external forces acting on the cable. External forces ð f Þ result from the ! ! ! sum of the buoyancy force ð f hg Þ, the normal ð f dn Þ and tangential, ð f dt Þ ! components of the drag forces, the inertial force ð f i Þ and the seabed ! ! contact force in the normal ð f sb;n Þ and horizontal ð f sb;t Þ directions. Their formulations are given by the following equations:

2.3. Numerical model description The numerical model used this work was presented and validated in (Barrera et al., 2019a). However, a brief description is provided next because some differences with respect to the original model have been implemented. The numerical model is built by coupling a hydrody­ namic, an aerodynamic and a mooring model. The hydrodynamic component models the behaviour of the FOWT in the frequency domain using potential flow theory based on a boundary element method (BEM), which is transformed to the time domain using the relationship proposed by Ogilvie (1964). The BEM model used is ANSYS AQWA (ANSYS AQWA, 2013). Hydrodynamic results are computed together with the aerodynamic and mooring results by means of the Cummins equation (Cummins, 1962), a second-order ordinary differential equation with a convolution integral applied to solve the radiation problem (2) as follows: Z t _ τÞdτ þ GkðtÞ ¼ Fe ðtÞ þ Fw ðtÞ þ Fm ðtÞ ðM þ A∞ Þ€kðtÞ þ (2) Kðt τÞkð

! ! ! ! ! ! f ¼ f hg þ f dn þ f dt þ f i þ f sb ! f dn ¼

! f sb; ​ n ¼ ​ d½ðzbot

ation, velocity and displacement, respectively. External forces are rep­ resented by wave excitation forces (Fe ), wind forces (Fw ) and mooring system forces (Fm ). Wave excitation forces include both the first and second order difference-frequency components, expressed in the time domain by the following equations:

(3)

l¼1

Feð2

Þ;j;α

L L L �X X X ¼ Re Al A*l Qj;α ðωl ; ωl Þ þ 2 Al A*m Qj;α ðωl ; ωm Þeiðωl l¼1

ωm Þt

​ rz ÞKG

r_z KB �rz

1 CDT dρw j! v t j! vt 2 2 πd ! CI ρ an 4 w

! f i

� xy � r_ r_xy ! f sb; ​ t ¼ ​ fhg Kμ min ;1 vμ k r_xy k

(6)

where ! g is gravitational acceleration, ρc is the cable density and ρw is the density of water. CDN and CDT are the normal and tangential drag coefficients, respectively; d is the mooring diameter; CI is a hydrody­ namic mass coefficient; ! v and ! a are the velocity and acceleration denoted by the subscripts n and t, which are the decompositions into the normal and tangential directions, respectively; zbot is the vertical coor­ dinate of the seabed; and rz is the vertical projection of the position vector and r_z its velocity. Constants KG and KB represent the stiffness and viscous coefficients, respectively. In the horizontal term, Kμ is the co­ efficient of kinetic friction corresponding to a maximum velocity, vμ , and r_xy represents the velocity of the horizontal projection of the position vector. The main difference with respect to the numerical model used in (Barrera et al., 2019a) is the aerodynamic model. Normally, aero­ dynamic forces are estimated through blade element momentum theory (BEMT) (Hansen, 2015). However, fatigue analysis requires a huge number of simulations to characterise properly the mooring damage. A simplification of the aerodynamic model is implemented here in order to reduce the computational cost but retaining a sufficient level of accuracy in the estimation of wind forces, evidenced by the validations shown in section 3.4. The aerodynamic model calculates the thrust force by means of a thrust coefficient defined for different relative wind speeds seen by the rotor (Martini et al., 2016) (Karimirad and Moan, 2012). It is assumed that only the normal component of the rotor is generating a force and that the nacelle is always aligned with the wind direction. The thrust and thrust coefficients are obtained from simulations made with FAST (Jonkman and Buhl, 2005) considering a rigid tower and constant and turbulent winds defined across the rotor following the well-known power law with the exponent equal to 0.14. Ten iterations per wind

where M is the inertia matrix of the floating structure, A∞ is the added mass matrix at infinite frequency, K is the retardation matrix, G is the hydrostatic stiffness matrix, t is time, τ is the integration variable of the € k; _ and ​ k are the floating platform acceler­ convolution integral and k;

j ¼ 1; 2; 3; 4; 5; 6

! f dt ¼ ¼

0

L � �X Feð1Þ;j;α ¼ Re Al Qj;α ðωl Þeiðωl tÞ ;

1 CDN dρw j! v n j! vn 2

ρ ρw ! ! f hg ¼ ρ0 c g ð1 þ eÞ ρc

�

l¼1 m¼lþ1

(4) where Feð1Þ is the first-order wave excitation force and Feð2 Þ the secondorder wave excitation force with j representing the degrees of freedom. L is the number of wave components, Al eiðωl tÞ the complex wave component, Al the complex wave amplitude, ωl the wave frequency and i represents the imaginary number. Qj;α ðωl Þ stands for the first-order complex excitation transfer function associated with ωl , j and α. Al can be written as Al ¼ al eiεl ; where al is the wave amplitude, εl is the wave 3

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speed are simulated in the turbulent cases and the mean values of thrust and the thrust coefficient are adopted. Fig. 2 shows the thrust force for different wind speeds considering constant (static thrust) and turbulent (dynamic thrust) wind. Comparing both approximations, it should be noted that an important discrepancy in the thrust estimation is found between 9.5 m/s and 12.5 m/s that is higher for constant speeds. No substantial discrepancies appear outside this range of simulated speeds. The differences are mainly related to the thrust variability resulting in turbulent wind simulations, which reaches the rated thrust value only when the incoming wind speed approaches the rated wind speed or in case sharp gusts occur. Provided that wind fluctuations prevent the development of the aerodynamic loads generated under static condi­ tions, the thrust coefficient law calculated for turbulent wind will be used. The thrust curves have been calculated using a conventional pitch controller. These curves have a negative slope for wind speed above the rated value and may introduce negative damping at the pitch natural frequency (Larsen and Hanson, 2007). A filter is implemented to remove the contribution of the pitch natural frequency band from the calculated relative wind speed to avoid this effect before evaluating the thrust force (Martini et al., 2016) (Karimirad and Moan, 2012). A smooth transition of the trust curve is added from the thrust curve value at 25 m/s to 0 at 26 m/s, avoiding unrealistic thrust jumps when the wind speed fluctu­ ates around 25 m/s. The relative wind speed seen by the rotor and the thrust force can be formulated following: ! v rotor ¼ ! v

��! �����! ðv��! SWL þ wSWL � rSWL rotor Þ

sea state. The tension at the fairlead is mainly induced by the platform translational movements (surge, sway and heave) (Barrera et al., 2019b) and, therefore, their assessment has an important impact on the fatigue evaluation. For this reason, floating platform rotations are not a primary focus of investigation in this work. Additionally, the nacelle acceleration is analysed as a potential parameter that influences the wind turbine production as it is assumed that excessive tower top accelerations would trigger the turbine shut down. Different theories can be considered to estimate the fatigue damage (Fatemi and Yang, 1998). However, two approaches have been estab­ lished as the most trustworthy: the crack growth approach and the S–N approach. The crack growth approach is based on fracture mechanics and as­ sumes that the strength of a component fails when an initial crack grows to a critical crack size. In spite of the fact that this method considers the load sequence in the crack growth, different previous works have evi­ denced that most mooring chain fatigue lifetime is spent on crack initiation (P�erez-Mora et al., 2015). Therefore, it seems appropriate to adopt a fatigue criterion based on crack initiation. The present work evaluates the cumulative fatigue damage through S–N and T-N ap­ proaches. The S–N approach assumes that fatigue failure occurs when a number of cycles, N, is reached. N is a function of the constant cyclic stress range, S, applied to the specimen. This approach provides different S–N curves according to the type of material. These curves are modelled from a linear regression of normalised experimental test re­ sults. The number of fatigue cycles (N) for a particular range of constant cyclic stress (S) is formulated in (9) as follows:

(7)

where ! v rotor is the relative wind speed seen by the rotor, ! v is the un­ disturbed wind speed, v��! SWL is the platform velocity at the sea water level, ���! is the platform angular velocity, and r����� �! w SWL SWL rotor is the position vector between the sea water level and the rotor axis at the tower centreline. Also: Trotor ¼

1 Arotor ρa CT v2rotor 2

N ¼ aS m log N ¼ log a m log S

(9)

where a is the intercept parameter of the S–N curve and m is the slope of the S–N curve. It should be noted that there is no endurance limit in mooring S–N curves. The fatigue damage accumulation during the mooring life-cycle is evaluated according to Palmgren-Miner’s rule (Palmgrem, 1924) (Miner, 1945). This rule assumes linear accumulation damage without considering the load sequence in the life-cycle. The parameters (a and m) (Table 1) for a chain mooring with the corrosive influence of seawater are found in DNVGL–OS–E301 (DNVGL-OS-E301, 2018). This standard provides the parameters as a function of a stud chain or studless chain regardless of the steel grade. Recent studies have revealed the importance of the steel grade on the S–N curve estimation (Arredondo et al., 2016) and the possible con­ servative results of the curve proposed by the standard. It should be

(8)

where Trotor is the thrust force, Arotor is the rotor area, ρa is the air density and CT is the thrust coefficient. 2.4. Dynamics and fatigue evaluation The main objective of this work is to evaluate the mooring fatigue damage considering different mooring properties. Fatigue analysis re­ quires the determination of the tension of each mooring line for every

Fig. 2. Thrust and thrust coefficient for different wind speeds. 4

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effective cycles is required to evaluate the fatigue damage through the S–N and T-N curves. Different time-domain cycle counting methods are used to obtain the equivalent response between regular and irregular tension series. The most popular are the peak counting, the range counting, the level-crossing counting and the rainflow counting methods (ASTM E1049-1985, 1985). The rainflow cycle counting method is the most accurate and widely used method to estimate the fatigue damage according to (API, 2008) (Dowling, 1972) (Watson and Dabell, 1975). It was proposed by (Matsuishi and Endo, 1968) and a new equivalent definition was performed by (Rychlik, 1987). Fatigue damage in this work is determined making use of PalmgrenMiner’s rule (12) (Palmgrem, 1924) (Miner, 1945) in connection with the rainflow counting method proposed by (Rychlik, 1987) and imple­ mented by (The WAFO group, 2017). The fatigue damage of a particular sea state is assessed as the sum of the individual tension/stress ranges indicated by the rainflow algorithm during the sea state duration. Finally, the total fatigue damage is obtained by adding all the sea states during the life-cycle causing the failure if the damage is higher than 1 as follows:

Table 1 S–N fatigue curve parameters according to DNVGL–OS–E301. S–N FATIGUE CURVE PARAMETERS COMPONENT STUD CHAIN STUDLESS CHAIN

a

m 11

1.2 10 6.0 1010

3 3

noted that the measured nominal tension (Tn ) on moorings must be transformed to a nominal stress (σ n ) considering the chain nominal cross sectional ðAc Þ as the representative area (10). The cross-sectional area to be considered is twice the chain link (dc denotes the link diameter) area as follows: � � � σn ½MPa� ¼ Tn ½N� Ac mm2 �� (10) Ac ¼ πd2c 2 Another standard widely used in the mooring fatigue design is APIRP 2SK (API, 2008). This standard proposes the use of a T-N curve, similar to S–N but considering the tension range and not the stress range. The T-N curves define the number of cycles to failure, N, when mooring is repeatedly cycled by means of a given effective tension range (11). The effective tension range is defined as a relation between the tension range (T) and the reference breaking strength (RBS). The parameters a and m for a chain mooring considering a T-N curve are collected in Table 2. � � m N ¼ a T= (11) RBS

Damage ðsea stateÞ ¼

2.5. Reconstruction of fatigue and dynamics: radial basis function interpolation technique Once the fatigue damage and dynamics have been evaluated for the selected subset of metocean data according to 2.2, it is possible to interpolate results in the original set of metocean conditions by means of a non-linear interpolation technique called Radial Basis Function (RBF) (Rippa, 1999). This method aims at finding an objective function (c) through an approximation function (~c) built as a weighted sum of basic symmetric radial functions and a linear polynomial as follows: J X

cðpÞ ffi ~cðpÞ ¼ uðpÞ þ

1000 316

3 3

�

(13)

A Gaussian expression (15) is used as the radial basis function in this work. The shape of the radial basis function is dominated by parameter q0. The optimal q0 can be estimated by (Rippa, 1999) or by means of a sensitivity analysis. Values of 0.1 and 0.175 are used to estimate the response of fatigue damage and tension, respectively. � �2

T-N FATIGUE CURVE PARAMETERS

STUD CHAIN STUDLESS CHAIN

ojj

where pðhÞ is a linear polynomial equal to the multivariate data dimension (mv) (W, β, Hs, Tp and α), ajj are the RBF adjustment co­ efficients, ∅ð Þ is the basic radial function and ojj are the approximation centres. The linear polynomial is defined on a monomial basis fu0 ; u1 ; …; umv g, including a number of monomials of degree 1 and a monomial of degree 0, where b ¼ fb0 ; b1 ; …; bmv g are the coefficients of these mo­ nomials. Coefficients a and b are calculated by enforcing the interpola­ tion constraints as follows: � � ~c ojj ¼ c ojj ; jj ¼ 1; …; J (14)

ϕ p m

ajj ϕ p jj¼1

Table 2 T-N fatigue curve parameters according to API-RP 2SK a

(12)

where n is the number of cycles in the sea state with the stress/tension range interval Sk , N is the number of cycles to failure at the normalised stress/tension range Sk , provided by the appropriate S–N or T-N curve. Sk is the succession of stress/tension ranges obtained by rainflow counting.

The reference breaking strength is usually provided by mooring manufacturers (Vicinay Cadenas brochure., 2018). An R4S studless chain mooring is considered for the purpose of this work. The R4S me­ chanical properties are shown in Table 3. Frequency domain or time domain approaches can be considered to predict the fatigue damage due to low frequency and wave frequency tensions (DNVGL-OS-E301, 2018) (API, 2008). Frequency domain methods have been discussed by several re­ searchers. A bimodal theoretical models for predicting fatigue damage under stationary and non-stationary Gaussian processes was presented by (Jiao and Moan, 1990). Later, the previous theoretical models to estimate fatigue damage were imporved incorporating a trimodal spectral formulation to account for other processes such as vortex-induced vibrations (VIV) or wind loads at the low and wave frequencies (Gao and Moan, 2008). Previous studies have been incor­ porated into design codes (DNVGL-OS-E301, 2018) (API, 2008). The standards admit three possible frequency domain methods to evaluate fatigue damage: a simple summation of low frequency and wave fre­ quency fatigue damage independently, the combined spectrum of low and wave frequencies and the combined spectrum with a dual narrow-banded correction factor. Despite the considerable computational cost, the time domain approach is the most accurate methodology for predicting mooring fa­ tigue response (API, 2008) because all nonlinearities related to mooring stiffness, seabed friction, drag and damping are taken into account. The S–N and T-N curves are presented for regular stress/tension ranges. However, the mooring response is irregular due to the randomness of metocean loads. Hence, a conversion of tension/stress time histories to

COMPONENT

Xn ðSk Þ 1 X n ðSk Þ*ðSk Þm ¼ NðSk Þ a

5

� ojj ¼ e

p ojj q0

(15)

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Table 3 R4S mechanical properties (Vicinay Cadenas brochure., 2018). Ultimate strength (MPa min.)

Yield strength (MPa min.)

Reduction of area (% min)

Elongation (% min)

960

700

50

12

a

Design temperature (� C) 20

Proof loada (kN min.) Stud chain

Studless chain

0.0240aZl

0.0213aZl

Break loada (kN min.) 0.0304aZl

Zl ¼ d2l (44–0.08dl); dl: link diameter. Chain weight per metre: stud chain ¼ 0.0219 d2l ; studless chain ¼ 0.02 d2l .

3. Case study: description and results

year long BiMEP metocean hourly time series (Metocean Analysis of BiMEP for Offshore Design, 2017). Fig. 6 shows the full dataset and the selected subset. Each database sample contains five variables (wind speed, wind direction, significant wave height, wave direction and peak period). The number of database samples (sea states) is 271,728 and they are represented by small black circles in this figure. Based on a preliminary sensitivity analysis, the number of selected samples in the subset is 1,000, which are represented by means of large blue circles in the figure. Additionally, 225 samples represented as red squares are selected to validate the interpolation technique of multivariate data based on the radial basis function (RBF) approach presented in section 2.5. As a result of this selection, a data subset with high variability is chosen including operational and extreme sea states. The selected 1,225 sea states are used to generate wind and wave synthetic time series, and by means of a wind turbine numerical model to predict the floating structure dynamics. Finally, the RBF is used to transfer all dynamics in terms of movements, tensions and fatigue damage to the original database.

3.1. Site assessment The location selected for this study is the BiMEP test site ( 2.894� , 43.563� ), an area offshore the town of Armintza on the Basque Coast (north of Spain). The site water depth ranges between 50 and 90 m. The selected period is between 1985 and 2015. Therefore, a period of 30 years is taken as the floating structure life-cycle which results in a total of 271,728 1-h sea states (DNV-OS-J101, 2014). Each sea state contains data of significant wave height, wave period, wave direction, wind speed at 90 m above the sea water level and wind direction. Wind and wave roses for this location are presented in Fig. 3. Waves mainly come from the north-west and wind has three predominant directions: east, south and west. Wind and wave directions, unless otherwise stated, considers the north direction at 0� with positive angles clockwise. 3.2. FOWT definition The DeepCwind semisubmersible platform (Robertson, Jonkman, Wendt, Goupee, Dagher) with a 5 MW wind turbine (Jonkman et al., 2009) is considered in this work. Station keeping is provided by three equal chain mooring lines in catenary configuration separated by angles of 120� with a length of 835.5 m. The design weight of each mooring is 125.6 kg/m in 200 m water depth. Despite the water depth at BiMEP is lower than 200 m, it has been chosen to maintain the original configu­ ration of the DeepCwind mooring system. However, five other types of mooring systems (MS) with a different weight and pretension level are investigated. The most important features of these mooring systems are shown in Table 4. A schematic side view is presented in Fig. 4 for the six mooring systems evaluated. As shown in Fig. 5, the DeepCwind platform has been oriented with the main wind direction (west). The coordinate system used in this work is set such that the x-axis is directed from west to east and the y-axis from south to north.

3.4. Wind turbine numerical model validation A complete validation of the numerical model was performed in (Barrera et al., 2019a) against laboratory tests published in the context of the OC5 Project (Robertson et al., 2017) based on results pertaining to static tension, decay tests and regular and irregular waves with and without wind. However, a new validation of experimental tests of irregular waves with wind is incorporated in this work because the aerodynamic module is now based on a look-up table of thrust co­ efficients and not on the blade element momentum theory (BEMT) used in (Barrera et al., 2019a). Results are validated for all wind approaches comparing the move­ ments and the tensions at the fairlead of the mooring lines. However, the focus of this investigation is only on the tension validation. It should be noted that the experimental tests were conducted in a single direction with waves and wind oriented from the negative to the positive x-axis. The rotor torque was not considered in the experiments. Therefore, the mooring line responses are equal in M1 and M3. The results of this new validation for wind and irregular waves are presented in Fig. 7 through

3.3. Metocean conditions The maximum dissimilarity selection technique is applied to the 30

Fig. 3. Wind and wave roses at the BiMEP test site. 6

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Table 4 Mooring system parameters. MOORING SYSTEM

WEIGHT (kg/ m)

PRETENSION (kN)

LINK DIAMETER (mm)

EQUIVALENT DIAMETER (m)

AXIAL STIFFNESS (N)

MS MS MS MS MS MS

92 125.6 200 308 462 634

811 1112 1770 2735 4117 5675

68 79 100 124 152 178

0.1199 0.1398 0.1763 0.2186 0.2679 0.3138

5.43Eþ08 7.45Eþ08 1.16Eþ09 1.77Eþ09 2.62Eþ09 3.55Eþ09

MINIMUM BREAKING STRENGTH R4S (N)

1 2 3 4 5 6

5,420,364 7,200,284 10,944,000 15,930,028 22,363,193 28,664,642

Fig. 4. Mooring schematic side view.

theoretical distribution function. The input of the validated numerical model are wind and wave time series generated synthetically from the selected 1,000 sea states using the maximum dissimilarity algorithm as explained in section 3.3. The numerical model outputs are the time series of the movements (surge, sway and heave), tensions (M1, M2 and M3) and the nacelle accelera­ tion. These time series are fitted to a generalised extreme value (GEV) distribution function because the sea states include both operational and extreme cases. The MPM of the distribution function is selected next as representative of each time series. The correlation coefficients between the empirical and GEV distribution for movements, tensions and nacelle acceleration are presented in Table 5. The correlation coefficients are obtained as the average of the 1,000 sea states for each variable and type of mooring. A good correlation is found between the empirical and theoretical data with values between 0.98 and 1. The long-term values are obtained through a non-linear interpolation technique based on the radial basis function (RBF) method previously introduced in section 2.5. The non-linear interpolation technique re­ quires two development steps. First, the construction of the interpola­ tion function, based on the numerical simulations using the 1,000 sea states selected by the maximum dissimilarity algorithm. Second, the accuracy verification provided by the non-linear interpolation. This verification is conducted through the selection of 225 additional sea states shown in Fig. 6. The response to these additional sea states is predicted by means of the non-linear interpolation (RBF) and the nu­ merical simulation. Comparisons between the responses of the RBF and the simulation are presented in Fig. 10 through Fig. 13 for mooring system 3. The graphs represent on the x-axis the response given by the RBF and on the y-axis the numerical model response (simulation). If the RBF and simulation responses are coincident, they will be represented by a point in the bisector of the graph. In contrast, if they are not coincident, they will be far off the bisector. A linear fit line is built with the RBF and the simulation data to analyse its position with respect to the bisector. In general, RBF presents a good agreement with the simulation due to the proximity between the fit line and the bisector (See. Figs. 11 and 12). From the 1,000 simulated cases for each type of mooring, 30 years of data are rebuilt using the RBF technique for the 13 variables. Each variable can be represented as a function of wind characteristics (speed

Fig. 5. Coordinate system and location of the different mooring lines (M1, M2, M3) in DeepCwind.

Fig. 9 considering three approaches: the BEMT, the thrust coefficients for constant wind speed and the thrust coefficients for turbulent wind speed. In the coming validation figures, these approaches are called the control (dashed blue line), quasi-static (dotted red line) and quasidynamic (dashed-dotted green line), respectively. In general, the agreement between the different numerical approaches and the exper­ imental test results is very good. In this work, the quasi-dynamic approach is selected to estimate wind forces for the reasons explained in section 2.3(See. Fig. 8). 3.5. FOWT simulation for fatigue analysis The long-term FOWT analysed variables are platform movements, which are the most probable maximum (MPM) of surge, sway and heave; moorings, which are the MPM of tension and fatigue damage for each mooring line; and nacelle acceleration, which is the MPM and percentiles 90%, 95% and 99%. A total of 13 variables are considered. The MPM is obtained fitting the empirical data of each variable to a 7

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Fig. 6. Selection of a representative metocean database by means of the maximum dissimilarity algorithm.

Fig. 7. Tensions: irregular waves and steady wind (Irregular waves: Hs ¼ 7.1 m, Tp ¼ 12.1 s, γJONSWAP ¼ 2.2 (γJONSWAP is defined as the peak enhancement factor in the JONSWAP spectrum; Wind: RPM (revolutions per minute) ¼ 12.1, W ¼ 12.91 m/s).

and direction) or wave characteristics (significant height, direction and peak period). Fig. 14 shows an example of this reconstruction consid­ ering the fatigue damage on M2 of mooring system 3. Mooring line M2 is located in the west direction (270� ) and, therefore, the damage will be a maximum at that direction. As the main direction of wave propagation is at 315� , M2 is the mooring line most exposed to wave action. The higher the significant wave height, the more fatigue-accumulated damage. In addition, according to Fig. 14, peak periods between 13 s and 21 s generate high fatigue damage. As far as the wind is concerned, there is a range of directions between 225� and 315� that generate high fatigue damage on M2. Unlike waves, higher wind speeds do not generate the higher fatigue damage. The reason is that the turbine thrust force is maximal at 12 m/s according to Fig. 2. A significant fatigue damage is found starting at wind speed of 10 m/s. However, it should be noted that high wind speeds may be associated with high wave heights and therefore the fatigue damage will be dominated by the significant wave height rather than the wind speed. Further results will only be shown as a function of wind characteristics due to the important variability of

wind direction (Fig. 3) at the target location and the dependence of thrust force on wind speed. The influence of the different mooring systems proposed in Table 4 on the tension and fatigue damage is displayed in Fig. 15 through Fig. 18. Platform movements determine the tension at the mooring system. Thus, higher platform movement produces lower tension while a major restriction to the movement induces higher tension in the mooring system. A wide range of platform movements can be found depending on the mooring system from movements of up to 14 m in MS 1, with ten­ sions of 500 kN to restricted movements to 2.5 m in MS 6, with tensions of approximately 6,200 kN. According to the mooring system configuration, a predominant directional sector can be assigned to each mooring line. Consequently, the higher fatigue damages come from the sector between northwest by north (NWbN) and east (E) in M1 (330� - 90� ), between southwest by south (SWbS) and northwest by north in M2 (210� - 330� ) and between east and southwest by south in M3 (90� - 210� ). As shown in Fig. 3, directions coming from NWbN and E are less likely and for this reason 8

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Fig. 8. Tensions: irregular waves and dynamic wind (Irregular waves: Hs ¼ 7.1 m, Tp ¼ 12.1 s, γJONSWAP ¼ 2.2; Wind (NPD spectrum): RPM ¼ 12.1, W ¼ 13.05 m/s).

Fig. 9. Tensions: white noise wave and steady wind (White noise: Hs ¼ 10.5 m, Tp ¼ 6 s–26 s, γJONSWAP ¼ 2.2; Wind: RPM ¼ 12.1, W ¼ 12.91 m/s). Table 5 Correlation coefficients between the empirical and GEV distribution for movements, tensions and acceleration of the 1,000 sea states selected by the maximum dissimilarity algorithm. MOORING SYSTEM

SURGE

SWAY

HEAVE

TENSION M1

TENSION M2

TENSION M3

ACCELERATION

MS MS MS MS MS MS

0.9913 0.9921 0.9926 0.9931 0.9938 0.9936

0.9912 0.9922 0.9932 0.9942 0.9944 0.9945

0.9879 0.9880 0.9884 0.9888 0.9892 0.9895

0.9944 0.9950 0.9956 0.9960 0.9964 0.9967

0.9935 0.9941 0.9947 0.9955 0.9962 0.9966

0.9952 0.9958 0.9964 0.9969 0.9970 0.9972

0.9962 0.9958 0.9959 0.9959 0.9958 0.9965

1 2 3 4 5 6

M1 will have minor fatigue damage. In general, fatigue damage de­ creases with higher mooring weights. By comparing the different mooring systems, the high fatigue damage achieved in MS 1 (92 kg/m) is remarkable, as is how it decreases with the increase in mooring weight until reaching the lowest damage in MS 6 (634 kg/m). In addition, it should be noted that although the weight increment reduces the fatigue damage, all mooring lines show a high fatigue damage coinciding with the sector between 250� and 330� . According to the wind and wave

roses, this direction corresponds to the extreme events of this location, with significant wave heights between 7 m and 10 m and wind speeds between 25 m/s and 40 m/s. The developed method allows capturing the relevance and contri­ bution of directionality to fatigue damage and selection of a suitable mooring system to the prevailing metocean conditions in the target location. In consonance with the obtained results, the most appropriate mooring system for the proposed FOWT and location could be MS 4 9

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Fig. 10. Comparison of movements (MPM) in mooring system 3: simulation & RBF.

Fig. 11. Comparison of tensions (MPM) in mooring system 3: simulation & RBF.

Fig. 12. Comparison of fatigue damage in mooring system 3: simulation & RBF.

(FLS) stated in the DNVGL–OS–E301 (DNVGL-OS-E301, 2018) is 5 for mooring lines which are not regularly inspected, in tension-tension processes and when the accumulated damage (dF) is lower than 0.8. If dF is higher than 0.8, the safety factor is defined by the expression 5 þ 3 ((dF-0.8)/0.2). Conversely, the API (API, 2008) recommends a value of 3 as safety factor. The proposed methodology may lead to a reduction in the safety factors by considering all the climate variability in the target location. In addition to the uncertainty associated with the selection of design sea states, there are other elements that may affect an accurate fatigue definition such as the design fatigue curve selection; the randomness of wind and wave time series; the mooring line pretension and the corrosion.

Fig. 13. Comparison of accelerations (MPM and percentiles 95% and 99%) in mooring system 3: simulation & RBF.

because it drastically reduces the fatigue damage found in MS 1, MS 2 and MS 3. No relevant improvement is provided by MS 5 and MS 6(See. Figs. 16 and 17).

4.1. Selection of design sea states and fatigue curves

4. Discussion

In this section, the proposed methodology, which is based on the estimation of fatigue damage through the simulation of all sea states in the FOWT life-cycle, is compared with the fatigue damage provided by assuming that the long-term behaviour can be represented by a limited number of sea states as proposed by standards (DNVGL-OS-E301, 2018) and (API, 2008). In general, these standards recommend setting a discrete number of sea states between 10 and 50. To select the discrete sea states, the metocean database is divided into four ranges of significant wave height: 0–1.5 m, 1.5–3 m, 3–4.5 m

The reported methodology allows estimation of the dynamics and fatigue damage considering all metocean conditions throughout the FOWT life-cycle. In general, the standards (DNVGL-OS-E301, 2018) (Section 6.3) (API, 2008) (Section 6.3) propose the selection of a limited number of long-term representative sea states and perhaps this is the reason for the extremely conservative safety factors recommended by these standards. In particular, the safety factor for the fatigue limit state 10

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Fig. 14. Reconstruction of fatigue damage (T-N approach) on M2 of mooring system 3 as function of wind (W and β) and waves (Hs, Tp and α).

Fig. 15. Reconstruction of tensions (MPM) on M2 for all mooring systems as a function of wind characteristics.

and higher than 4.5 m. The average wave height of each range is set as the representative height of each subset. In turn, for each range of sig­ nificant wave height three ranges of wind speed are set: 0–7 m/s, 7–14 m/s and higher than 14 m/s. For each range, wind speed is estimated as the average of the speeds in each range. Finally, four ranges of peak periods are proposed for each speed range: 0–6 s, 6–9 s, 9–12 s and higher than 12 s. As with the significant wave height and the wind speed, the average is the representative value for each peak period range. In all, 48 sea states result from this discretisation. However, a total of 35 cases are set as a consequence of the fact that the metocean database does not contain data in some of the previously set ranges. Wave and wind di­ rections are obtained as the most likely value corresponding to each

range. The percentage of occurrence of each sea state is obtained from the presentation probability in the metocean database. The selected sea states are presented in Table 6. Fig. 19 shows the distribution of the sea states selected to reproduce the long-term environmental conditions and estimate the mooring fatigue damage. The fatigue curves used in this work are the T-N curve and the S–N curve corresponding to studless chains proposed by API (API, 2008) and DNV (DNVGL-OS-E301, 2018), respectively. The fatigue damage results using both approaches for each of the six mooring systems (Table 4) with three mooring lines considering the matrix of 35 cases and the life-cycle simulation are presented in Fig. 20. According to the obtained results, the following conclusions can be drawn. 11

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Fig. 16. Reconstruction of fatigue damage on M1 for all mooring systems as a function of wind characteristics.

Fig. 17. Reconstruction of fatigue damage on M2 for all mooring systems as a function of wind characteristics.

First, the S–N curve provides estimations of more conservative damages than does the T-N. The reason for this is that the S–N curve provides the same fatigue damage regardless of the grade of the chain steel, while the T-N curve takes into account the breaking strength of the steel and therefore a higher steel quality generates less fatigue damage. Different investigations have shown the importance of steel quality on the evaluation of fatigue damage (P� erez-Mora et al., 2015) (Arredondo et al., 2016). Second, the fatigue damage obtained using the life-cycle method presented in this work is higher than that presented using the discrete sea states proposed by the standards. Consequently, simulating the FOWT life-cycle can reduce the uncertainty in the selection of sea states and thus can estimate a more accurate fatigue damage. Table 7 shows

the percentage difference between the life-cycle method and the discrete sea states matrix according to both the S–N and T-N curves. Mean dif­ ferences between 13% and 49% are obtained in the fatigue damage estimation between both approaches. Finally, MS 1 and MS 2 present an important fatigue damage. In both mooring systems, M2 is the mooring line that suffers the most fatigue damage. The reason is because these mooring systems have large ranges of movement and therefore large stress/tension ranges. In contrast, M2 and M3 are the mooring lines with the most important damage in MS 3, MS 4, MS 5 and MS 6 due to the presence of wind in the region between east and west according to Fig. 3. These results show that the mooring line characteristics can influence the distribution of fatigue damage for all lines included in the mooring system. 12

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Fig. 18. Reconstruction of fatigue damage on M3 for all mooring systems as a function of wind characteristics.

statistical uncertainty into the results due to the randomness of waves and wind. The generation of wind and wave time series by wind and wave spectra uses random phases for transforming spectra into the time domain. Therefore, multiple time series meet the criteria characterising in the spectra. This fact causes an uncertainty in the estimation of fatigue damage due to the presence of random phases. To highlight the importance of random phases in the fatigue damage assessment, twenty wave and wind time series are generated for the same sea state. MS 2 and the sea state defined by Hs ¼ 3.35 m, Tp ¼ 9.21 s, α ¼ 316.09� , W ¼ 12.10 m/s, β ¼ 328.09� are considered in these simulations. Fig. 21 shows the total lifetime of each mooring line if the previous sea state is repeated cyclically. As seen, lines M1 and M2 are those most exposed to this sea state and, therefore, they have a shorter lifetime until failure. It is noteworthy to highlight that with the same sea state, there are lifetime differences of up to 4.16 and 3.42 years, with average values of 6.81 and 8.09 years considering the mooring lines M1 and M2, respectively. These results show the importance of randomness on the estimation of fatigue damage.

Table 6 Selection of sea states. Cases

Hs (m)

Tp (s)

α

W (m/s)

В

%

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35

1 1 1 1 1 1 1 1 1 1 1 1 2.1 2.1 2.1 2.1 2.1 2.1 2.1 2.1 2.1 2.1 2.1 2.1 3.6 3.6 3.6 3.6 3.6 3.6 3.6 5.4 5.4 5.4 5.4

5 8 10 13 5 8 10 13 5 8 10.5 13 5.6 8 10.5 14 5.5 7.5 10.5 13.5 5.5 7.5 10.5 13.5 11 15 11 14.5 8 10.5 14 17 16 11 15

0 320 310 310 40 310 310 310 290 290 300 310 10 310 10 340 10 310 310 310 290 310 310 310 310 310 310 310 0 310 310 310 310 310 310

4 4 4 4 9.5 9.5 9.5 9.5 16 16 16 16 4.2 4.2 4.2 4.2 10.5 10.5 10.5 10.5 16.5 16.5 16.5 16.5 4.5 4.5 10.8 10.8 17.5 17.5 17.5 5 11.1 21 21

0 320 90 90 90 90 90 90 270 190 190 190 30 220 30 330 40 270 270 190 270 270 270 190 240 190 250 270 300 270 270 190 200 290 270

3.22 7.67 17.15 6.20 3.23 4.71 8.40 3.15 0.39 0.26 0.68 0.38 0.03 0.38 4.08 8.06 0.27 2.27 6.38 8.03 0.24 1.16 1.51 1.79 0.02 1.15 0.35 3.22 0.16 1.33 1.87 0.08 0.49 0.15 1.54

4.3. Effect of corrosion on fatigue life Another important process in mooring fatigue life is corrosion. A steel chain permanently in contact with sea water suffers degradation of its physical and mechanical characteristics. The standards assess this process as a section loss (DNVGL-OS-E301, 2018). The section loss ratio depends on the type of water (polar, temperate or tropical), the type of inspection and the part of the mooring involved (bottom, catenary, splash zone). To assess the impact of corrosion on fatigue damage, a fatigue analysis over M1 is performed considering MS 2, a corrosion rate of 0.2 mm/year and the sea state proposed in 4.2. Fatigue damage evolution for M1 is presented in Fig. 22. Fatigue damage is calculated for the initial state and every five elapsed years. According to the obtained results, the importance of considering corrosion in the fatigue design is relevant because damage may double the initial values at the end of the lifetime of the structure.

4.2. Influence of the randomness of waves and wind time series on fatigue damage evaluation

4.4. Influence of mooring line pretension on fatigue and power production

The most accurate method to estimate fatigue damage is by means of a time domain method (API, 2008). However, this method introduces a

The fatigue curves proposed by the standards do not consider the mean stress effect when the fatigue cycle number is estimated for a 13

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Fig. 19. Summary of selected sea states to reproduce the long-term environmental conditions.

Fig. 20. Fatigue damage for different mooring systems and fatigue curves. Table 7 Percentage difference between the life-cycle method and the sea states matrix according to S–N and T-N curves. MOORING SYSTEM

MS1

MOORING LINE

M1

M2

M3

MS2 M1

M2

M3

MS3 M1

M2

M3

MS4 M1

M2

M3

M1

M2

M3

M1

M2

M3

M1

M2

M3

T-N (%) S–N (%)

51 58

11 15

5 2

59 62

12 15

12 12

50 50

13 16

24 16

37 42

14 16

18 17

41 44

8 10

26 31

35 38

23 7

30 36

46 49

13 13

19 19

particular stress range. Goodman, Gerber or Soderberg corrections are commonly used to take into account the mean stress effect on the ma­ terial fatigue behaviour (Rodríguez et al., 2005) (Bannantine et al., 1990). These corrections relate the stress amplitude (Δσ) for a mean stress (σm), with the stress that would provide the same fatigue life with a mean stress equal to zero (Δσ0), by means of the following expression: � � �n � σm Δσ ¼ Δ σ 0 1 (16)

MS5

MS6

MEAN VALUES

are presented in Table 8. Two configurations are considered: the FOWT static position and a second one considering a sea state. MS 2 and the sea state proposed in 4.2 are chosen to estimate the correction factors. The steel quality is set to be R4S with an ultimate strength of 960 MPa and a yield strength of 700 MPa (Vicinay Cadenas brochure., 2018). The re­ sults show that the mean stress reduces fatigue life. The Soderberg correction is the most conservative while the Gerber correction presents values close to 1. However, the results clearly show the importance of considering the mean stress in the evaluation of fatigue damage.

σR

where n ¼ 1 for the Goodman and Soderberg corrections, n ¼ 2 for the Gerber correction, σR is the yield strength for the Soderberg correction and σR is the ultimate strength for the Goodman and Gerber corrections. Mean stress correction factors through the use of the ratio Δσ= Δσ 0

5. Conclusions This paper presents an innovative methodology to assess the fatigue 14

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Fig. 21. Elapsed time until fatigue failure. MS 2. Sea state: Hs ¼ 3.35 m, Tp ¼ 9.21 s, α ¼ 316.09� , W ¼ 12.10 m/s, β ¼ 328.09�

Fig. 22. Evolution of fatigue damage due to corrosion. MS 2, M1 Sea state: Hs ¼ 3.35 m, Tp ¼ 9.21 s, α ¼ 316.09� , W ¼ 12.10 m/s, β ¼ 328.09�

life-cycle of FOWT moorings considering the time series of metocean conditions during the full lifetime. It is shown that the present method reduces the uncertainty related to the discrete selection of representa­ tive long-term sea states proposed by the standards. This approach in­ tegrates multivariate metocean data selection techniques, FOWT numerical models, fatigue modelling methods and non-linear interpo­ lation techniques. From a 30-year metocean reanalysis hourly dataset, a subset of 1,000 sea states is selected using a maximum dissimilarity algorithm. These sea states are simulated using hydrodynamic, aerodynamic and mooring coupled numerical models to obtain the FOWT dynamic response and

the mooring fatigue damage. This numerical model is first validated with experimental tests to demonstrate the model reliability in the prediction of dynamics. Finally, the results are propagated for all the sea states contained in the metocean database, a total of 271,728 sea states, making use of non-linear interpolation techniques based on radial basis functions. This non-linear interpolation is validated with 225 additional numerical cases for the following variables: platform movements (surge, sway and heave), mooring tension, fatigue damage and nacelle accel­ erations. A correlation coefficient between 0.96 and 0.99 is obtained for all variables. This approach allows the evaluation of the FOWT dy­ namics throughout its life-cycle at a reduced computational cost and 15

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corrosion, mean tension effects or residual stress from chain manufacturing. A sensitivity analysis to some key fatigue parameters is also conducted in this work. In particular, the wave and wind random­ ness, corrosion and mean tension effects are evaluated in this research. The wave and wind randomness is analysed for an operational sea state with the turbine working close to the rated wind speed. Twenty cases are proposed to evaluate the time series statistical uncertainty. The lifetime until fatigue failure evidences a difference among all cases of around four years for the mooring lines with the greatest loading. Corrosion is a complex degradation process of the physical and mechanical properties of moorings. Fatigue damage could increase up to a factor of two be­ tween the initial state and the elapsed life-cycle considering a degra­ dation ratio of 0.2 mm/year according to the results presented. Finally, mooring pretension is another important source of uncertainty in the evaluation of fatigue damage. Its effect can be evaluated by means of different formulations proposed in the fatigue theory. The different approaches show an increase of up to 20% in the stress amplitude used to estimate the fatigue damage.

Table 8 Mean stress correction factors. Ratio.Δσ =Δσ0 FOWT STATIC POSITION: M1, M2, M3 MOORING SYSTEM

MEAN STRESS (MPa)

MEAN STRESS CORRECTIONS GOODMAN

SODERBERG

GERBER

MS MS MS MS MS MS

111.6563 112.5471 112.6817 113.2385 113.4420 114.0265

0.8837 0.8828 0.8826 0.8820 0.8818 0.8812

0.8405 0.8392 0.8390 0.8382 0.8379 0.8371

0.9865 0.9863 0.9862 0.9861 0.9860 0.9859

MOORING SYSTEM

MEAN STRESS (MPa)

MEAN STRESS CORRECTIONS GOODMAN

SODERBERG

GERBER

MS MS MS MS MS MS

146.5869 137.5512 128.5214 123.7932 121.0556 120.1624

0.8473 0.8567 0.8661 0.8710 0.8739 0.8748

0.7906 0.8035 0.8164 0.8232 0.8271 0.8283

0.9767 0.9795 0.9821 0.9834 0.9841 0.9843

MOORING SYSTEM

MEAN STRESS (MPa)

MEAN STRESS CORRECTIONS GOODMAN

SODERBERG

GERBER

MS MS MS MS MS MS

153.0608 142.6110 131.8447 125.8337 122.4716 121.2557

0.8406 0.8514 0.8627 0.8689 0.8724 0.8737

0.7813 0.7963 0.8117 0.8202 0.8250 0.8268

0.9746 0.9779 0.9811 0.9828 0.9837 0.9840

MOORING SYSTEM

MEAN STRESS (MPa)

MEAN STRESS CORRECTIONS GOODMAN

SODERBERG

GERBER

MS MS MS MS MS MS

71.9353 79.2807 88.3341 95.6605 100.1367 102.8488

0.9251 0.9174 0.9080 0.9004 0.8957 0.8929

0.8972 0.8867 0.8738 0.8633 0.8569 0.8531

0.9944 0.9932 0.9915 0.9901 0.9891 0.9885

1 2 3 4 5 6

SEA STATE: M1

1 2 3 4 5 6

Acknowledgements

SEA STATE: M2

1 2 3 4 5 6

The authors acknowledge financial support from the Spanish Min­ istry of Science, Innovation and Universities to PhD candidate Carlos Barrera S� anchez through his research training scholarship under Grant Agreement No. BES-2014-070381. Raúl Guanche also acknowledges financial support from the Ramon y Cajal Program (RYC-2017-23260) of the Spanish Ministry of Science, Innovation and Universities. This work �lisis del COmportamiento dina �mico de is part of the ACOPLE (Ana �licas flotantes para la optimizacio �n del disen ~ o de aguas PLataformas Eo profundas) research project (Grant Agreement No. ENE2017-89716-R within the National Programme for Research, Development and Inno­ vation Aimed at the Challenges of Society, call 2017. Spanish Ministry of Science, Innovation and Universities).

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achieves an accurate and detailed definition of mooring fatigue damage assessment. A set of mooring systems with different properties for the same FOWT configuration is proposed to estimate the most appropriated mooring against fatigue damage. The fatigue damage evaluation is performed by means of a time domain method using Palmgren-Miner’s rule in connection with the rainflow counting method making use of the S–N curve and T-N curve proposed by the standards. Despite the sig­ nificant differences between the two fatigue curves, both approaches estimate a mooring of a weight approximately 300 kg/m is the most suitable. The proposed methodology evaluates the fatigue damage taking into account all life-cycle sea states and eliminates possible uncertainty or bias due to the discrete selection of sea states proposed by the standards. The comparison between both approaches evidences that the discrete selection underestimates the fatigue damage. Mean percentage differ­ ences between 13% and 49% are obtained comparing both approaches using the S–N and T-N curves. However, it should be noted that the discrete selection results can change depending on selected sea states. Here, a reasonable selection is applied taking into account that fatigue is a long-term process. The consideration of all sea states during the FOWT life-cycle could contribute to a reduction in the fatigue safety factors proposed by standards, although with a certain conservatism due to the fatigue process complexity in mooring made up of chains where other phenomena should be contemplated such as the randomness of wave and wind time series, anomalous loading modes (out-of-plane bending), 16

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