4.1.2. Wind Turbine

The substructure carries a representative 7 MW turbine, which is modeled in LACflex aero-elastic code [65]. The turbine includes a 90 m tubular tower with a diameter ranging between 4 and 6 m. Along the tower, three concentrated masses are assumed to emulate the effect of secondary-structures. The aero-elastic code employs a modal-based representation of the turbine (including the tower, rotor and blades). An aerodynamic damping contribution is included through the fluid-structure interaction when calculating aero-elastic forces. The wind turbine model was originally developed for industrial purposes, where it was applied in commercial projects. A rather similar model (albeit a 5 MW turbine instead of 7 MW), which adheres to the same modeling principles, has been applied in other studies on structural dynamics of wind turbines [34,66].

#### 4.1.3. Load Cases

In this study, we consider the fatigue failure mode in the normal operating condition (design load case (DLC) 1.2 [40]). For a typical offshore wind jacket substructure, this DLC accounts for most of the fatigue damage [67].

The met-ocean parameters applied in this study are derived based on measurements from a representative North Sea site [68] and are summarized in Table 1. The wind speed ranges between 4 and 31 ms<sup>−</sup>1, resulting in *nb* = 15 wind speed bins. For each wind speed bin, representative wave parameters, i.e., the significant wave height and peak period, are assigned. The significant wave height ranges from 0.1 to 7.9 m while the peak period ranges from 3.0 to 9.6 s. The met-ocean parameters along with their yearly probability of occurrence are derived from a site-specific joint probability distribution function, which is a common design practice [40]. A total of *nd* = 12 wind directions are analyzed (wind and waves are assumed fully aligned). For each wind speed, a total of *nT I* = 5 turbulence intensity quantiles, namely, *q* ∈ [*q*10, *q*30, *q*50, *q*70, *q*90], are considered. The quantiles for each wind speed are calculated based on the Weibull distribution according to the IEC standard [40] for turbulence class B. The turbulence intensities for the given site ranges from 0.09 to 0.31. The fatigue damage is scaled with the corresponding turbulence intensity quantile probability, hence representing the target Weibull distribution. Every load case (wind speed, wave height, peak period and turbulence intensity) is simulated with *ns* = 6 seeds. The total number of load cases analyzed is *nt* = *nbndnT Ins* = 5400.

**Table 1.** Load case definitions according to IEC [40] and representative site-specific parameters.


#### *4.2. Nominal Results*

The structural reliability of selected joints of the jacket substructure is evaluated based on model (1) and the variables are summarized in Table 2. The stress ranges, Δ*σ*, and number of cycles, *N*, were obtained from simulations. The SN curves for tubular joints in air and in seawater with cathodic protection are used according to [48]. The SN curve for the air environment are applied to the joint at level 50. For the remaining joints, the SN curve for seawater with cathodic protection is applied. For tubular joints exposed to seawater with cathodic protection, negative inverse slopes of *m*<sup>1</sup> = 3 and *m*<sup>2</sup> = 5 and intercepts of log *Kc*<sup>1</sup> = 12.18 and log *Kc*<sup>2</sup> = 16.13 are assumed to calculate the characteristic SN curve. For tubular joints in air environment, the following values can be used: log *Ka*<sup>1</sup> = 12.48 and log *Ka*<sup>2</sup> = 16.13, while assuming the same *m* values as for seawater environment. The mean SN curve for the probabilistic analysis was calculated from the characteristic SN curve's intercepts assuming a standard deviation of 0.20 [48].


**Table 2.** Variables used in the probabilistic model to estimate fatigue damage accumulation in the nominal case [32].

Distribution: N-normal, LN-logNormal, D-deterministic.

#### 4.2.1. Annual Reliability

The annual reliability index as a function of time, Δ*β*(*t*), is calculated based on the state-of-the-art probabilistic methods described in Section 2. The limit state Equation (1) was applied using the standard-based variables provided in Table 2. The reliability indices are presented in Figure 4 and Table 3 and are denoted as the nominal results. The results represent the situation where no additional knowledge from a digital twin is available. The results are provided for 10 selected joints, which are typically critical for a jacket design.

The structure is designed to have a fatigue lifetime of 25 years. The fatigue lifetime ends when the annual reliability index reaches the target value Δ*β* = 3.3, which serves as the basis for reliability-based calibration of safety factors in recognized design codes [31,40]. For the considered case study, the design driving joints are 13BU and 40CU with a lifetime of 25 and 27 years. Joint 13BU is located close to the mudline, while joint 40CU is located slightly below the splash zone. Joints 40CL, 40BL, 25BU and 25BL have a lifetime between 50 and 100 years, while the remaining joints have a lifetime above 100 years.

**Table 3.** Fatigue lifetime derived based on probabilistic model (1) and stochastic variables presented in Table 2.


**Figure 4.** Structural reliability as function of time for the nominal model.

#### **5. Case Study Results**

In this section, we exemplify how new information from digital twins can be included in the proposed framework to quantify uncertainty and subsequently update structural reliability for the particular case study. We use information from previously established digital twins [33,34]. The effect of structural dynamics uncertainty, *Xd*, is investigated based on a model updating study presented in [33], where the soil stiffness, *ks*, was calibrated based on in situ measurements. The effect of loading uncertainty, *Xl*, is investigated based on a virtual sensing study [34], where modal expansion was used to estimate unmeasured field quantities. The results are presented and discussed based on two design driving joints, namely, 13CU and 40BU.
