Surrogate based sensitivity analysis and uncertainty quantification of floating wind turbine mooring systems

A Floater module containing several empirical parameters has been added to the TNO’s Cost model in order to include the analysis of floating wind turbine support structures and mooring systems. It is of our interest to know which model parameters within the Floater module contribute most significantly to the mooring system costs and ultimately to the levelized cost of energy (LCOE). The strategy employed relies on constructing a surrogate model (based on Kriging), which is then used to perform global sensitivity analysis. For the scenarios studied here, it was found that the model parameter related to the mooring line breaking load coefficient remained the most sensitive to the capital expenditure (CapEx) cost, while the model parameter related to the failure event cost for mooring line repair remained most sensitive to the operational expenditure (OpEx) cost. Additionally, the study aimed at expanding the deterministic Cost model to systematically account for stochastic model parameter inputs in order to reduce modelling uncertainties and contribute towards more reliable mooring line designs.


Introduction
The global wind industry will focus on scaling up of wind energy projects contributing towards reaching the net zero targets (carbon neutrality) by 2050 [1].To this aim, the European Commission plans to install 30GW of new offshore wind farms every year, between now and 2030 [2].In recent years, a considerable effort towards the development of floating offshore wind turbine technology is seen [4].These floating wind turbines are generally secured by mooring systems that consists of the mooring lines and the anchors.The stability of a floating wind turbine depends on the design of the mooring system.Moreover, the mooring system constitute an important element in the cost breakdown of the floating wind turbine system, typically about 18-30% [8] of the CapEx cost, and therefore have a large impact on the LCOE.Offshore mooring systems have very high failure rates.Brindley and Comley [6] analyzed the mean times to failure (MTTF) of mooring systems during the years 1996-2005 and concluded that approximately 3 mooring line failures every 2 years, occur, based on an average population of 34 units.According to Kvitrud [7], 16 mooring lines in the Norwegian Continental Shelf failed between 2010 and 2014.The failures were due to fatigue (four cases), overload (six cases), mechanical damage (four cases), and manufacturing errors (two cases).In other words, more than 60% of the mooring line failures were attributed to design errors.Due to the high costs associated with the mooring Cost modeling is a systematic method for calculating a realistic baseline cost for the offshore wind project and can be used to assess the project's viability to carry out due diligence [5].TNO's Wind Farm Cost model (hereinafter referred to as Cost model) is an in-house developed parametric modeling tool for predicting the LCOE of offshore wind farms [9].A Floater module containing several empirical coefficients has been added to the existing Cost model in order to include the analysis of floating wind turbine support structures and mooring systems.To reduce the cost and optimise the design of the mooring systems, it is of our interest to know which model parameters within the Floater module contribute most significantly to the mooring system costs and ultimately to the LCOE.Moreover, we want to understand how uncertainty in these model parameters propagate through the Cost model's output related to mooring line design.Based on our earlier work [10], we will employ the UQ4WIND framework to perform Sobol' variance-based global sensitivity analysis, which will quantify the uncertainties of the model parameters onto the Cost model's output.To estimate the Sobol' indices, we will resort to the traditional Monte Carlo (MC) sampling technique.The main downside associated to the MC sampling step is its high computational expense due to large number of model evaluations required.We will alleviate this issue by constructing a Kriging-based surrogate (metamodel) of the Cost model to evaluate Sobol' indices.
The paper is organized as follows.Section 2 describes the Cost model and its uncertain model parameters related to mooring line design.Global sensitivity analysis using Sobol' indices, which is accelerated by Kriging-based surrogate model, is described in section 3. Finally, the results of global sensitivity analysis and uncertainty quantification are detailed in section 4 followed by conclusions drawn in section 5.

Cost model
The Cost model employs a range of engineering models and their interactions, empirical relationships, and statistical data for wind sites; an overview of the Cost model is shown in Figure 1.In addition to that, a number of user-input model parameters to provide a detailed cost breakdown of an offshore wind farm are present in these engineering models.The model parameters are related to both Capital Expenditure (CapEx) and the Operational Expenditure (OpEx) costs.CapEx, which comprises all investment costs paid before the start of commercial operation, is the major component of the life cycle cost of wind farms whereas the OpEx component includes all expenses incurred following the commercial operation data.The Floater module within the UpWind model is responsible for floating support structure and mooring Event cost N(394800, 10) system design.The integrated analysis using the model parameters of the Floater module contribute to the overall model output: LCOE.A detailed description of the Cost model is beyond the scope of the current discussion, and can be found in [9].Important for the current discussion is to distinguish between model parameters (θ), model scenario (S) and model output (Y).θ consists of user-input parameters, such as design parameters, empirical coefficients etc. S consists of values that are pre-determined such as wind farm site, turbine type, operation etc. Lastly, Y contains the model quantity of interest such as mooring line costs, cable installation costs etc.The goal of this research was to expand the deterministic Cost model to include uncertain model parameters in order to quantify the uncertainty onto the model output.Towards this direction, the model parameters that were considered as uncertain were the ones that are within the Floater module and are listed in Table 1.In mathematical notation, these uncertainties will be captured in a vector of model parameters θ = (θ 1 , ..., θ 7 ) ∈ R. A Cost model evaluation, denoted by µ, returns a vector of outputs Y, depending on the (uncertain) model parameters θ and on the (given) value of the model scenario S i , where subscript i indicates Y that corresponds to S. Each of the model parameters in , where values for the model parameters and their uncertainty margins are obtained from combination of references [3,11,12] and experts elicitation.

Kriging-based global sensitivity analysis
The propagation of uncertainties from θ = θ 1 , ..., θ 7 onto the model output Y i in equation 1 using the traditional Monte Carlo sampling technique [13] is infeasible because of the high Consequently, a surrogate model Y i (θ) that approximates the behavior of Y i (θ) is built.Y i (θ) possesses similar statistical properties with Y i (θ), while maintaining an easy-to-evaluate form.In a non-intrusive approach, which is of interest herein, the surrogate model is developed using a set of realizations x∈X called the experimental design (ED) points for X = (x (1) ,...,x (n) ) and the corresponding model response Kriging, also called a Gaussian process (GP), is a surrogate modelling technique [14], which is described by the following equation: µ K (x) is the Kriging model response, β T f (x) is the mean value of the Gaussian process (i.e.trend) and Z(x) is the covariance, which is given by: where σ 2 ϵ is the constant variance of the GP, θ is a vector of unknown parameters and R() is the correlation function which describes the spatial correlation between observations (x i ) and new points (x j ) that are part of the ED points.
In our investigation, ordinary Kriging is selected [15], which means that β T f (x) is a scalar value β 0 to be determined.A valid correlation function needs to satisfy two conditions, saying that the R(., .) in equation 3 should be symmetric and positive semi-definite.There are many functions which have been proven effective for these two conditions such as the constant, Gaussian and Matérn functions [16].For our investigation, Gaussian correlation function is selected.For the construction of a Kriging surrogate model, randomly chosen X are used for the estimation of the hyper-parameters (β 0 , σ 2 ϵ , θ).The input in the ED is generated by Latin hypercube sampling method [17] based on the input distribution shown in Table 1.Covariance matrix adaptation-evolution strategy (CMA-ES) is chosen for the optimization method [18] and Leave-one-out (LOO) approach [19] for evaluating the cross-validation error: where Y (x−1) denotes the Kriging surrogate model output trained by leaving the x − 1-th sample out.The surrogate model with the smallest ϵ LOO is then chosen as the surrogate model to perform sensitivity analysis.
The idea of a variance-based global sensitivity analysis is to relate the variance in the model inputs to the variance in the model output.The total order Sobol' indices are defined as a ratio of variances: which measures the contribution to the output variance of θ, including all variance caused by its interactions, of any order.For the sake of conciseness, we describe this technique briefly; a more detailed description is available elsewhere [10].S T otal i can be interpreted as an importance measure for the parameter θ: a large S T i implies, roughly speaking, that θ n has a strong influence on Y.It should be noted that S T otal i can be greater than one because the higher order model interactions are accounted for in the total Sobol' indices.

Results and discussion
In this section, the results of sensitivity analysis of the Cost model parameters will be presented and discussed.Kriging surrogate model is used in place of Cost model to perform sensitivity analysis.In order to build the Kriging surrogate model as described in equation ( 2), Cost model is evaluated using Monte Carlo sampling following the distribution for model parameters indicated in Table 1.In total 500 model evaluations were sufficient to achieve the Kriging surrogate models, separate for the CapEx and the OpEx scenarios, with LOO-errors (equation ( 4)) smaller than 10 −3 .Sensitivity analysis takes less than 2 hours with the surrogate model, whereas the computational cost would be one order of magnitude higher with the full Cost model (when running on a desktop computer).
Sobol' indices S T otal i are computed expressing the sensitivity of Y, following equation ( 5), towards the perturbation in θ.The results are shown in Figure 3.For S i = CapEx, θ 2 by far has the largest influence among the five model parameters studied.Line breaking load coefficient (θ 2 ) is the ratio between the breaking strength of the mooring line and the maximum design load that the mooring system is expected to experience.Factors such as the type of mooring line material, the manufacturing process, and quality control measures determine the value of line breaking load coefficient.Thus, higher variation of mooring line breaking load coefficient reflects the higher mooring cost needed during the CapEx phase.Additionally, the mooring line cost rate coefficient (θ 5 ) also shows a noticeable influence.The type of material used for the mooring line design, length, and diameter determines the mooring line cost rate coefficient, and therefore shows a noticeable impact on the mooring cost.Other model parameters that are related to the mooring line tension (θ 1 ; the ratio of the maximum surge force on a mooring line to the force exerted by the waves in the horizontal direction), density (θ 3 ; the ratio of density of the mooring line material to the density of water) and stiffness (θ 4 ; the ratio of the axial force required to produce a unit deformation in the mooring line material to the cross-sectional area of the mooring line) show a small influence on the mooring cost.For S i = OpEx, the variation in the Y, i.e. annual costs (operations and maintenance costs), can be primarily attributed to the variation in θ 7 , while θ 6 has almost negligible influence.The event cost θ 7 that refers to the cost of repairing and restoring the wind turbine or its component in the event of failure.The event cost represents a large expense that must be factored while calculating the O&M annual costs during the OpEx phase.The mooring line failure rate (θ 6 ) is a parameter used in mooring analysis and design to estimate the expected time (in hours) until a mooring line fails due to excessive loading.For this case study, the failure rate for mooring lines is subject to operational  constraints during its design life, and thus the changes in the frequency of maintenance shows little effect on the overall OpEx cost.In order to quantify the uncertainty resulting due to the distribution of the model parameters (θ) onto the model outputs (Y), the sampled model output data (gathered from Monte Carlo sampling) were plotted in the form of histograms shown in Figure 4.The histograms are overlaid with the normal (Gaussian) distribution curve that best fits the data.For S i = CapEx, the resulting distribution of the Mooring Cost shows a mean (µ) = 254.02k€and standard deviation (σ) = 21.41k€.For S i = OpEx, the resulting distribution of the Annual cost shows a mean (µ) = 42.69k€ and standard deviation (σ) = 5.29k€.The applied uncertainty quantification allows for a more informed decision-making process by providing a range of potential costs and probabilities associated with each cost.This helps stakeholders to understand the risks and make better decisions based on the project constraints.
Overall, the input parameter distribution (Table 1) plays a critical role on the results for both sensitivity analysis and uncertainty quantification.Therefore, it is important to note that the results of sensitivity analysis and uncertainty quantification may vary for different values for input parameter distribution.

Conclusions
In this study, the previously developed UQ4WIND framework is applied to perform sensitivity analysis and uncertainty quantification study on TNO's Wind Farm Cost model.The model parameters investigated here are related to mooring line design for floating wind turbines.In an effort to reduce the computational cost from the required repeated Cost model evaluations for performing sensitivity analysis, a surrogate model was constructed based on Kriging (Gaussian process) method.Two model scenarios are studied here: (i) CapEx scenario with mooring line cost as the model output and (ii) OpEx scenario with annual maintenance cost as the model output.The results of Kriging-based sensitivity analysis using the Sobol' indices highlights the influential parameters affecting the model output.It was found that the model parameter tied to the mooring line breaking load coefficient remained the most influential to the total model output variance for the CapEx cost, while the model parameter related to the failure event cost for mooring line repair primarily influenced the total model output variance for the OpEx cost.Uncertainty quantification study of the sampled model outputs using distribution allowed probabilistic statements on the costs to be deduced.By considering uncertainties and their effects, stakeholders can make informed decisions and mitigate risks while analyzing mooring systems, and ultimately contributing to improved and more reliable mooring system designs.

Figure 2 .
Figure 2. Flowchart of the Kriging-based global sensitivity analysis framework.

Figure 3 .
Figure 3. Sobol' indices indicating the sensitivity of mooring cost with respect to 5 uncertain model parameters (left) and the sensitivity of annual cost with respect to 2 uncertain model parameters (right).

Figure 4 .
Figure 4. Distributions obtained for the mooring cost (left) and annual cost (right) due to input model parameter uncertainty.

Table 1 .
Description of uncertain model parameters used within the Floater module of the TNO's Cost model chosen for sensitivity analysis and uncertainty quantification.