*5.2. Loading Uncertainty Update*

In this subsection, we investigate the effect of updating loading uncertainty on the structural reliability. First, we present a sensitivity study on wave loading calibration, followed by updating the reliability based on load calibration using two virtual sensing configurations. The virtual sensing study is presented based on uncertainty quantified in [34]. In this subsection, the *Xl* uncertainty is updated based on an updated *Cm* parameter. It is assumed, similarly as in Section 5.1, that only one uncertain parameter affects the uncertainty modeling, i.e., *α* = *Cm*.

#### 5.2.1. Wave Loading Sensitivity

The effect of updating the wave loading coefficient, *Cm*, on the structural reliability of joint 13CU is presented in Figure 8 and in Table 6. The mean value of the wave loading coefficient is modified by a factor of 0.8–1.2, which results in modifications of the loading uncertainty. It is assumed that new information from the digital twin is obtained; in this particular case, the mean value of uncertainty related to loading uncertainty, *μXl* , is updated. The results are derived by using the limit state Equation (1) with the standardbased variables provided in Table 2 and updated values for *μXl* .

The wave loading modification has a medium impact on the fatigue lifetime. Updating the wave loading by a factor of 0.8 (reducing the inertia-induced wave loading by 20%) results in an increased lifetime by a factor of 1.6. Increasing the wave loading by a factor of 1.2 results in reducing the lifetime by a factor of 0.7. The effect of updating wave loading on joint 40BU is more pronounced, as indicated in Figure 9. For this joint, reducing the wave loading by 20% results in a lifetime increase by more than fourfold (>100 years), while a wave loading increase by 20% results in a lifetime reduction by a factor of 0.3.

**Figure 8.** Impact of updating wave loading on structural reliability-joint 13CU.


**Table 6.** Fatigue lifetime derived for different distributions of *Xl*.

**Figure 9.** Fatigue lifetime derived for different distributions of *Xl*-joint 40BU.

5.2.2. Reliability Update-Virtual Sensing Uncertainty

The virtual sensing uncertainty quantified for two virtual sensing configurations are considered based on results presented in [34]. The following virtual sensing uncertainty configurations are used: (1) basic setup: CoV = 0.10 and (2) extended setup: CoV = 0.05, while the mean value for both setups is assumed to be 1.00. The basic setup includes only acceleration sensors above the water level, while the extended one, in addition, includes sub-sea acceleration sensors and a wave radar sensor. It is assumed that the virtual sensing uncertainty are combined with the nominal *Xl* uncertainty. Furthermore, it is assumed, for illustrative purposes, that the mean value of *Xl* equals 0.9. The *Xl* distribution parameters used in this study are summarized in Table 7 for joints 40CU and 13BU.

The results for joint 40BU are presented in Figure 10. As each model update configuration results in the same mean value update so the only difference in the stochastic model is the CoV, the higher the CoV, the shorter lifetime we should derive. This is confirmed in the results as the direct sensing method (measuring directly), with CoV = 0.00 resulting in a lifetime of 60 years, followed by the extended virtual sensing method (lifetime of 50 years and CoV = 0.05), while the most uncertain method (basic virtual sensing with CoV = 0.10) results in a fatigue lifetime of 40 years. In this case, each configuration derives a fatigue lifetime larger than the nominal one, i.e., 25 years. However, this is not the case for joint 13CU, where the fatigue lifetime using the basic virtual sensing configuration is 22 years, as depicted in Figure 11. Even though the mean value of the update results in reduced fatigue damage (*μXl* = 0.97 for this case), the negative effect of increased uncertainty (CoV *Xl* = 0.14) results in a fatigue lifetime reduction of 3 years.

**Table 7.** Fatigue lifetime updated based on various uncertain wave loading calibration methods. *Xl* distribution is updated (mean and CoV).


**Figure 10.** Impact of updating wave loading based on uncertain virtual sensing methods-40BU.

**Figure 11.** Impact of updating wave loading based on uncertain virtual. sensing methods-13CU.

#### *5.3. Uncertainty Correlation*

In the previous subsections, the *Xd* and *Xl* uncertainties were investigated separately, hence neglecting a potential correlation. In this subsection, we consider updating both *Xd* and *Xl* with varying correlation coefficients. The correlation can stem from interaction between the structural dynamics and loading parameters. For example, the loading parameters can be calibrated based on responses from a previously updated structural model.

We assume the structural and loading uncertainties are quantified based on new information from a digital twin, resulting in updated mean values of structural and load uncertainties: *μXd* = 0.80 and *μXl* = 0.97 and using the reference uncertainty level CoV = 0.10. The updated uncertainty value corresponds to increasing the soil stiffness by 50%, *ks* = 1.5, and reducing the wave loading coefficient by 10%, *Cm* = 0.9. The results are presented for joint 13CU.

Three scenarios of correlation between *Xd* and *Xl* are investigated: (1) *ρ* = 0 (no correlation), which can be the case if the load calibration was performed without using information from the updated structural model, (2) *ρ* = 1 (full correlation), when, for example, load calibration using mode shapes from an updated structural model and (3) an intermediate case with *ρ* = 0.5, where both analytical and measured mode shapes were used for load calibration.

The structural reliability calculated for various scenarios is presented in Figure 12. The nominal setup yields a fatigue lifetime of 25 years, while the updated uncertainty results in a fatigue lifetime ranging between 23 and 48 years, where the difference stems solely from varying correlations. The largest fatigue lifetime is obtained when assuming no correlation, while the lowest lifetime is derived for full correlation. Note that despite reducing the mean values of *Xd* and *Xl*, the fatigue lifetime is reduced compared to the nominal result for the full correlation case. The results are in line with expectations, because positive correlation increases the combined *XdXl* uncertainty.

**Figure 12.** Impact of *Xd* and *Xl* correlation on structural reliability.

#### *5.4. Application for New Structures*

Assuming a number of digital twins for similar structures have been established in the past, we can, by applying the proposed framework, obtain a distribution function of *XdXl*, which indicates what is the expected outcome of updating the structural and load model. This knowledge can be used at the design stage, resulting in an optimized design given the expected model update is realized. However, the updated information may be at a preliminary stage of validation and therefore subject to some degree of uncertainty, i.e., the expected model update outcome only represents our (best) knowledge. Hence, we must confirm our expectation by performing model updates during the structural lifetime and consider all potential outcomes of the experiment (model update) in the design stage. This is accounted for by preparing a decision rule, which for any outcome introduces an action that guarantees that the wind turbine has a sufficient reliability level until the intended lifetime is reached. The proposed application is based on Bayesian pre-posterior decision theory [54] and has, in the offshore wind industry, been applied in, for example, optimization of operation and maintenance of wind turbines [55].

In the following, an illustrative example is presented for this application to new structures. Assume that, based on previous digital twins, we obtain a prior distribution function of quantified uncertainties, *Xf* = *XdXl*. This prior distribution can be regarded as the future (yet to be realized) distribution of the updated uncertainties and can be used already at the design stage.

For the sake of illustration, we assume that the future outcome of model updates can be modeled as *Xf* ∼ N 0.9,(0.9 × 0.05)<sup>2</sup> , as depicted in Figure 13. The prior distribution is used together with model (1) to design the optimized structure. This is obtained by assuming that the generic structural dynamics and loading uncertainty are substituted with the expected uncertainty quantified based on the future experiment, *XdXl* = *Xf* . The decision models are derived based on (1), where, depending on the outcome of the model update, different values of *Xf* are assumed. The *Xf* values are summarized in Table 8. As a result, we derive an optimized structure, which has sufficient reliability until the intended lifetime is reached. This is indicated in Figure 14 by the blue curve. In the design, we have used the prior distribution of the updated uncertainty and assumed that updating the model is performed during the operation of the structure to confirm our expectation (obtain the posterior distribution). The point in time when updating the model must be performed can be derived by applying model (1) with the nominal uncertainty from Table 2, as shown in Figure 13 with the orange curve. Finally, we derive a point when the structure reaches the target reliability level and some action is needed to confirm its structural reliability. This is indicated by the orange curve in Figure 14.

When the model update time is reached, updating of the model is performed. As a result of model update, we can obtain one of the three outcomes for *Xf* , which will have an impact on the decision models, as depicted in Figure 15. In particular, we have the following potential outcomes:


Given the expected or positive outcome of updating the model is realized, no further action is required. However, if the outcome of updating the model is unexpectedly negative, the following mitigation actions can be considered to ensure the required level of reliability during the intended lifetime: (1) strengthening or (2) curtailing of the wind turbine (thereby reducing fatigue damage) and operating until the end of the intended lifetime. If it is economically infeasible to continue the operation of a particular turbine given the model updating outcome, one can consider premature decommissioning. The reliability level after the mitigation action is performed as indicated by the solid red line in Figure 15.


**Table 8.** Pre-posterior stochastic model.

**Figure 13.** Stochastic model for pre-posterior design.

**Figure 14.** Benefit of including pre-posterior design (including prior knowledge on *Xf* ).

**Figure 15.** Pre-posterior design at inspection time.
