# Quantifying and clustering the wake-induced perturbations within a wind farm for load analysis

A. Lovera, E. Fekhari, B. Jézéquel, Marine Dupoiron, Martin Guiton, E. Ardillon

► To cite this version:

A. Lovera, E. Fekhari, B. Jézéquel, Marine Dupoiron, Martin Guiton, et al.. Quantifying and clustering the wake-induced perturbations within a wind farm for load analysis. Journal of Physics: Conference Series, 2023, 2505 (1), pp.012011. 10.1088/1742-6596/2505/1/012011. [https://doi.org/10.1088/1742-6596/2505/1/012011](https://doi.org/10.1088/1742-6596/2505/1/012011). [arXiv:2208.09636](https://arxiv.org/abs/2208.09636)

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# Quantifying and clustering the wake-induced perturbations within a wind farm for load analysis

To cite this article: A. Lovera et al 2023 J. Phys.: Conf. Ser. 2505 012011

View the article online for updates and enhancements.

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# Quantifying and clustering the wake-induced perturbations within a wind farm for load analysis

A. Lovera¹, E. Fekhari², B. Jézéquel¹, M. Dupoiron³, M. Guiton³ and E. Ardillon²

¹ EDF R&D, 7 bd Gaspard Monge, Palaiseau, France

² EDF R&D, 6 Quai Watier, Chatou, France

³ IFPEN, Rond-point de l'échangeur de Solaize, Solaize, France

E-mail: anais.lovera@edf.fr

**Abstract.** The main objective of the HIPERWIND project is to reduce the Levelized Cost Of Energy by controlling the uncertainties in a chain of numerical models simulating an offshore wind farm. This paper studies the wake-induced perturbations of the ambient wind conditions inside a wind farm. A theoretical floating offshore wind farm off the coast of South Brittany in France is considered as a test case, and a numerical model simulating the wake of this farm is exploited. This model gives an analytical prediction of wake deficit and added turbulence, also considering the influence of the floaters' position due to mean wind forces. An uncertainty propagation on the wake model is performed, considering the multivariate joint distribution of ambient wind conditions as input. This results, for each turbine, in a wake-modified probabilistic distribution on the wind speed and turbulence intensity averaged over the rotor. A new approach for clustering these wake-modified distributions is proposed in this paper to regroup the wind turbines exposed to similar environmental conditions, leading to a similar structural response. This analysis is essential to ensure the practical feasibility of the reliability analysis of an overall project with the computational cost of aero-servo-hydro-elastic simulations of offshore wind turbines. Several methods classically used for clustering are compared. Finally, four clusters only are sufficient to represent the whole set of turbines in the wind farm.

## 1. Introduction

In the offshore wind industry, the wake effect is considered crucial for electricity production and structural fatigue of turbine components. For instance, the standards developed by the International Electrotechnical Commission (see appendix E in [1]) review some analytical [2], and numerical models (e.g., the dynamic wake meandering approach by [3]) to simulate the wind speed deficit and the wake added turbulence. Since the pioneering work of [4], several wake models were developed and compared in a benchmark by [5]. This work takes advantage of the low computational cost of steady-state wake models to propagate the uncertainty from ambient to wake-induced wind conditions seen on a farm. It is worth noting that the wake creates a heterogeneous wind field in a wind farm, resulting in different loading conditions which should be considered at the stage of reliability based design (RBD). Such heterogeneity is not taken into account by the design load cases of international standards (see e.g., [1, 6]) where wind and wave conditions are derived from scenarios occurring over long-term periods. Wake models allow us to simulate the wind conditions' distribution at each turbine, however, the RBD step is too costly to be performed for each turbine. For further details regarding the RBD, one may refer to the

<figure><img src="image_2.png" /></figure>
work of [7, 8, 9, 10, 11], proposing several approaches to reduce the computational cost of this step. In order to speed up the RBD at the scale of a wind farm, the present work aims at building clusters of WT similarly affected by the wake. Then, the RBD over a wind farm can be computed only on a few WT, each representing a cluster of turbines enduring similar wake-modified wind loading. This clustering is done on two wind parameters, following the conclusions of the global sensitivity analysis of [12]. In order to discriminate the wake-perturbed distributions of wind parameters, the maximum mean discrepancy (MMD) is used as a statistical metric between multivariate distributions (as reviewed by [13]). To illustrate this novel approach, a theoretical wind farm for the ongoing tenders off the coast of South Brittany in France is studied, with a modified version of the floating offshore wind turbine (FOWT) IEA-15MW (initially proposed in [14, 15]). Figure 1 illustrates the layout of the 25 FOWT considered in the following. This layout is regular with an inter-turbine distance of seven times the rotor diameter in the dominant wind direction and five times the rotor diameter in the orthogonal (crosswind) direction. More details regarding the FOWT modifications and theoretical wind farm can be found in [16, 17].

<figure><img src="image_3.png" /><figcaption>Figure 1: South Brittany wind farm layout (left). South Brittany wind rose from the ANEMOC data (right, source: [18]).</figcaption></figure>

Based on the results of previous numerical studies, with either dynamic wake meandering [19] or LES [20], we retain the time-averaged floater position (translation and rotation) as the main effect of the floater motion on the far wake. It was shown that this effect is small, both on the wake-added turbulence and on the wind speed deficit. However, noticeable uplift of the wake may influence the fatigue design.

The key idea in this paper to reduce the number of cases for RBD is to employ clustering techniques on the wake-induced wind parameters, to constitute groups of WT that are exposed to similar design load conditions. To do so, the present work is divided into four sections. First, the wake model used in this work is presented in section 2. Then, section 3 describes the uncertainty propagation through the wake model, from the probabilistic distribution of ambient wind parameters to the wake-modified wind parameters distribution within the farm. In order to regroup the similar wake-modified distributions, an adapted statistical metric on multivariate distributions is introduced in section 4. Finally, the application of several clustering methods to the South Brittany case study is compared in section 5. The main conclusion suggests four clusters among the 25 FOWT which can then be used for RBD analysis of the farm.

## 2. Wake simulation model description

The uncertainty propagation and subsequent clustering require a great number of farm scale simulations. Each simulation predicts per WT, the wind velocity deficit and added turbulence
due to the wake of the upstream machines. As the WT are supported by a floater, their positions are affected by the environmental conditions. We coupled Farmshadow™, an engineering farm wake software developed by IFPEN with a hydro-static computation to predict the average floater position, wind speed and turbulence intensity. Initially, all floaters are oriented to face the dominant wind, which is also the farm orientation (see Figure 1). The floaters are considered to be rigid, and all degrees of freedom are modelled in this computation (surge, sway and heave for the three translations and roll, pitch and yaw for the three rotations). The floater surge due to the aerodynamic forces was shown to have a negligible impact on the wake seen by the subsequent WT, as the relative motion between two WT due to surge is around 20m, less than 2% of the smallest inter-turbine distance. Therefore, the hydrodynamic loads due to the current and the drift associated to second order wave loads [21] were not considered here. These load contributions could have a second order effect by altering the stiffness of the mooring system, but this effect is also neglected. The forces applied on the floater by the mooring system are linearized (see [18] for details). For the aerodynamic load computations, the wind is considered constant and uniform, meaning that no turbulence or vertical shear was taken into account, as this would have required an active control of the blade pitch and rotation speed. The vertical shear and turbulence were however used in the farm-scale wake computation. Finally, the floater yaw was not considered as a relevant parameter in this study, as the rotors wind alignment is assumed to be faster than the floater motion.

<figure><img src="image_4.png" /><figcaption>Figure 2: Representation of the two nested iterative loops solving the wind field in the farm</figcaption></figure>

The hydro-static computation (inner loop in Figure 2) is first done by computing the floater equilibrium position, and second by averaging the aerodynamic load on the rotor over two rotations. This load is used to compute the new equilibrium position. The aerodynamic calculation uses a Blade Element Momentum (BEM) method, with rigid blades. The rotation speed and blade pitch are obtained from a look-up table depending on the averaged wind speed over the rotor, following the reference cases computed for this FOWT [16]. The process of “computing equilibrium position then calculate aerodynamic loads” is iterated until the floater position converges (variation of the position vector norm less than 1%). Since the floater pitch is defined in the floater referential, and the rotor tilt is defined according to the wind direction, the rotor tilt $\beta$ is the sum of the shaft tilt $\beta_0$ and the tower tilt derived from the floater roll $\alpha$ and pitch $\beta$, and the nacelle yaw $\gamma$. The floater yaw is very small and neglected in this calculation. The total rotor tilt is eventually given by the following expression:
$$ \hat{\beta} = \tan^{-1} \left( \frac{\cos(\gamma) \sin(\beta) - \sin(\gamma) \cos(\beta) \sin(\alpha)}{\sqrt{1 - (\cos(\gamma) \sin(\beta) - \sin(\gamma) \cos(\beta) \sin(\alpha))^2}} \right) + \hat{\beta}_0. \quad (1) $$

The second loop (in Figure 2) consists in computing the incoming wind on each WT in the farm and deducing their new position. FarmShadow™ uses engineering wake models to predict the wind field throughout the whole farm, starting from the most upstream WT and working downwards. Turbulence and wind vertical shear are considered in this computation, and modelling of the velocity deficit, added turbulence and wake superposition is done through published analytical models. Here, the models used are super-gaussian [22] for the wake speed deficit, Qian [23] for the wake added turbulence and local linear sum [24] for the superposition. No blockage model is used. The loop is initialized with a wake computation taking all WT in their initial position (no pitch or surge). Then the average incoming wind is computed for each rotor and the hydro-static loop is run to determine the new floater position. The process is repeated until the average speed at each rotor has converged. The convergence usually takes 1 or 2 iterations.

The main impact of the floater position on the wake comes from the additional tilt applied to the rotor (obtained from Eq. 1) on top of shaft tilt. Figure 3 shows the farm-averaged total rotor tilt as a function of the wind direction for different incoming wind speeds. The maximal total tilt is around 13°, more than twice the shaft tilt of 6°, and is obtained for 10.75 m/s wind speed with 67.5° orientation (wind coming from the floaters stern or back end). The irregularities in the curves, particularly for 8.75 and 10.75 m/s are linked to the WTs alignments in the farm.

<figure><img src="image_5.png" /><figcaption>Figure 3: Farm averaged rotor tilt for different wind conditions, as a function of compass azimuth (0° means a wind coming from the north). Head wind (wind coming from the floaters bow, or front end) is represented with a grey line and cross wind (orthogonal to the farm and floaters) with a dashed line.</figcaption></figure>

The impact of the floater position on the average velocity and turbulence intensity is relatively small (around 1% at most). It is maximal for tailwind conditions when the induced rotor tilt is maximal, or crosswind conditions when the spacing between WTs is the smallest. For example, with a rated tailwind ($u = 11$ m/s, $TI = 14\%$), the turbulence intensity decreases by around 1% when taking the WTs position into account, while the wind speed increases by 1%. The elevation of the wake centre is more significant (10 to 20%) and affects the loads applied on the WT encountering this wake. Figure 4 shows the difference in axial wind velocity when the rotor tilt is not taken into account (left), which is the most common approach in analytical wake modelling [5], and when both the shaft tilt and tower tilt are considered in the wake computation
(right). The wake deflection due to the tower tilt is consistent with the existing literature : 0.04 to 0.1D depending on the wind speed and direction [25, 20, 26].

<figure><img src="image_6.png" /><figcaption>Figure 4: Normalised axial velocity profile ahead of WT12 (in the wake of WT11), position relative to Figure 1, for a cross-wind of 11m/s with 14% turbulence intensity and a vertical shear exponent of 0.1. Left: no tilt considered, wake center at 140m. Right : tower tilt and shaft tilt considered, leading to a total tilt of 12.3° for WT11, wake center at 172m</figcaption></figure>

### 3. Uncertainty propagation in a wake model

The wake model described in section 2 takes as input a set of variables describing the ambient wind conditions $x \in \mathbb{R}^3$ and computes the perturbed wind conditions at each WT represented by the vector $x'_l, l \in (1, .., n_{WT})$, where $n_{WT}$ is the total number of turbines in the farm:

$$g : \mathbb{R}^3 \to \mathbb{R}^{3n_{WT}} \qquad (2)$$

$$x \mapsto g(x) = (x'_1, .., x'_{n_{WT}}) \qquad (3)$$

The uncertainties associated with the ambient wind conditions are represented by a random vector $X$ following the distribution $p_0$. Note that the index 0 is a reference to the fact that these conditions are not perturbed by the wake. A parametric model has been fitted in [27] using conditional probability density functions to capture the dependence structure, with an approach similar to the one presented in [28]. The random vector $X$ is described by the following input random variables:

* Mean wind speed ($u$) is the 10-min average horizontal wind speed at hub height.

* Wind turbulence intensity ($TI$) is the 10-min wind speed turbulence intensity at hub height.

* Wind direction ($\theta$) is the 10-min average wind direction.

In the following, we assume the wind orientation variable $\theta$ to be unaffected by the wake. When perturbed by the wake of the wind farm, the WT $l$ sees a wind flow represented by the random vector $X'_l, l \in (1, .., n_{WT})$, following the distribution $p'_l$. Then, the two remaining parameters are $u_{rotor}$ and $TI_{rotor}$ to represent this modified wind flow on a vertical plane located at each WT. These two quantities of interest are averaged over the rotor while the input parameters are given at hub height. For the sake of simplicity, we will neglect this difference in what follows in order to consider the transformation $X' = g(X)$ as a simple perturbation of $X$. We will abusively denote $u$ and $TI$ both the input and output quantities. The output of the uncertainty propagation is a set of perturbed environmental distributions $p'_l, l \in (1, .., n_{WT})$. A Monte-Carlo sample $X_n = x^{(1)}, .., x^{(n)}$ of the three random input variables is generated. Then, considering the farm
<figure><img src="image_7.png" /><figcaption>Figure 5: Joint perturbation at WT 13, 19, and 25</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>Figure 6: Ambient and wake-disturbed distributions of the wind speed</figcaption></figure>

layout illustrated in Figure 1 and a constant wind shear exponent of 0.1 like in section 2, a wake simulation is run for every wind condition of the Monte Carlo design of experiments. The code output consists in a multivariate joint distribution of modified *u* and *TI* for each WT. As the Monte Carlo procedure is known to converge slowly but surely, it was chosen to perform this uncertainty propagation with a number of simulations of size *n* = 6 000 because of its simplicity and the low computational cost of the simulations.

We can plot a preview of the wake perturbations on the joint distribution for given WT in the two dimensions *u* and *TI*. Figure 5 illustrates this perturbation for three WT differently affected by the wake depending on their position in the farm (cf. Figure 1). One can notice that the WT 25 distribution (in orange) is very close to the ambient distribution (in black), as expected, this WT being located on the edge of the farm and facing the dominant wind direction. Meanwhile, the WT 13 distribution (in red) seems more affected by the wake, by getting a higher wind turbulence with lower wind speed. This analysis can be completed with the two marginals in Figure 6 and Figure 7, both describing the ambient marginal distributions (in black) and wake-disturbed distributions. In general, a small wind speed deficit is noticed as indicated by the small shifts of the probability density functions to the left on Figure 6. Also, a small added turbulence is indicated by the small shifts of the curves to the right on Figure 7. Ideally, a tool should allow us to quantify the perturbation between the ambient and perturbed distributions.

# 4. Statistical metric of wake-induced perturbations

## 4.1. *Maximum mean discrepancy: a distance between distributions*

In the literature, the maximum mean discrepancy was introduced by [29] as a statistical test to discriminate two distinct distributions. After further work on this tool, authors such as [13] showed that the MMD is a distance between two distributions embedded in a specific function
<figure><img src="image_9.png" /><figcaption>Figure 7: Ambient and wake-disturbed distributions of the turbulence intensity</figcaption></figure>

space. Therefore, this concept relies on the embedding of distributions in a convenient Hilbert space. Considering a positive-definite kernel $k: \mathbb{R}^d \times \mathbb{R}^d \to \mathbb{R}$, $d \in \mathbb{N}$ generating a unique Hilbert space $\mathcal{H}_k$ of functions equipped with inner products $\langle \cdot, \cdot \rangle_{\mathcal{H}_k}$ and norms $\| \cdot \|_{\mathcal{H}_k}$ (also called Reproducing Kernel Hilbert Space (RKHS) when the function $k(\mathbf{x}, \cdot)$ satisfies the reproducing property). Then, let us define the kernel mean embedding of the distribution $P$ in the function space $\mathcal{H}_k$:

$$ \mu_P(\mathbf{x}) := \int_{\mathbb{R}^d} k(\mathbf{x}, \mathbf{y})dP(\mathbf{y}) \approx \frac{1}{n} \sum_{i=1}^{n} k(\mathbf{x}, \mathbf{x}^{(i)}), \mathbf{x}^{(i)} \in \mathbf{X}_n. \quad (4) $$

The kernel mean embedding is approximated on sample $\mathbf{X}_n = (\mathbf{x}^{(1)}, \dots, \mathbf{x}^{(n)})$ following the distribution $P$. Figure 8 illustrates the kernel mean embedding of two distributions in the function space $\mathcal{H}_k$ defined previously. Notice that this procedure allows us to embed continuous distributions (such as $P$) as well as discrete distributions (such as $Q$).

<figure><img src="image_10.png" /><figcaption>Figure 8: Kernel mean embedding of two probability distributions $P$ and $Q$ mapped in the RKHS $\mathcal{H}_k$</figcaption></figure>

The distance between the two kernel mean embeddings $\mu_P$ and $\mu_Q$ is called the maximum mean discrepancy (MMD). This distance between two distributions $P$ and $Q$ is initially defined by the worst-case error for any function within a unit ball of a Hilbert space $\mathcal{H}_k$, generated by the kernel $k$:

$$ \text{MMD}_k(P, Q) := \sup_{\|g\|_{\mathcal{H}_k} \le 1} \left| \int_{\mathbb{R}^d} g(\mathbf{x})dP(\mathbf{x}) - \int_{\mathbb{R}^d} g(\mathbf{x})dQ(\mathbf{x}) \right| = \|\mu_P(\mathbf{x}) - \mu_Q(\mathbf{x})\|_{\mathcal{H}_k}. \quad (5) $$

The MMD fully relies on the difference of kernel mean embeddings. Moreover, according to [13], a kernel is called “characteristic kernel” when the following equivalence is true, $\text{MMD}_k(P, Q) =$
0 ⇐⇒ P = Q, making the MMD a metric on $\mathbb{R}^d$. For its good convergence behavior, the squared MMD has been used for multiple other purposes than numerical integration: statistical testing [29], sensitivity analysis [30]. When elevated to the square, it can be estimated using one n-sized representative sample of P denoted $\{\mathbf{x}^{(i)}\}_{i \in (1,...,n)}$ (and respectively one m-sized sample of Q denoted $\{\mathbf{y}^{(i)}\}_{i \in (1,...,m)}$:

$$ \widehat{\text{MMD}}_k(P, Q)^2 = \frac{1}{n^2} \sum_{i,j=1}^{n} k(\mathbf{x}^{(i)}, \mathbf{x}^{(j)}) - \frac{2}{nm} \sum_{i=1}^{n} \sum_{j=1}^{m} k(\mathbf{x}^{(i)}, \mathbf{y}^{(j)}) + \frac{1}{m^2} \sum_{i,j=1}^{m} k(\mathbf{y}^{(i)}, \mathbf{y}^{(j)}). \quad (6) $$

In the following, the idea is to compare the ambient wind distribution $p_0$ to the wake-perturbed wind conditions $p'_l$ at the WT $l$ using the previously defined squared MMD.

## 4.2. Application to the South Brittany wind farm project

Once the joint perturbed distributions of each WT are estimated by a large Monte Carlo sample (cf. section 3), the MMD with the ambient wind conditions can be computed. Figure 9 illustrates for each WT the squared MMD value computed to measure the wake-induced perturbation. Let us remind that MMD is a distance between the joint perturbed distribution at a WT computed for all wind orientations with the ambient wind distribution. Despite the wake obviously depend on the wind direction, our final goal is to define a small number of WT for RBD thus independently of the wind orientation. The lower this metric gets, the closer to the ambient wind distribution. Quite logically, the WT in the center of the farm are more affected by the wake since they are subject to the wake regardless of the wind direction.

The values of squared MMD given in the previous figure are estimated between two samples:

* the Monte Carlo sample of the free environmental distribution: $\mathbf{X}_n$,

* the wake-perturbed Monte Carlo sample at the WT $l$: $\mathbf{X}'_{n,l}$ (output of the steady-state wake numerical model).

To ensure that the Monte-Carlo estimation converged, Figure 10 plots the squared MMD between the sample $\mathbf{X}_n$ and the increasing samples $\mathbf{X}'_{n,l}$, $i \in 400,..,6000$. After a few thousands of simulations, the MMD of each WT tends to converge towards a stable value, as expected. The design of experiment with $n = 6 000$ is thus considered as sufficient.

# 5. Clustering the wake-induced perturbations

A clustering between wake-modified environmental conditions is performed, in order to gather WT that are exposed to similar wind conditions. This analysis is essential to ease the RBD at the farm scale, requiring a large number of simulations to estimate the rare failure probability of such a system (order of magnitude of $10^{-4}$). Notice that the clustering does not depend on the wind direction, since it was assumed to be unchanged by the wake.

## 5.1. Clustering methodology

At the end of previous section, we obtained for each WT a wake-modified joint wind condition. In other words a perturbed multivariate distribution of the wind speed and the turbulence intensity. The MMD is chosen as a scalar distance between two distributions, allowing to apply clustering techniques on distributions. Considering all the WT $\{1,...,n_{WT}\}$ in the farm, and their respective wake-perturbed distributions $\{p'_l\}_{l=1}^{n_{WT}}$, let us define the symmetric matrix $D$ of MMD between every pair of perturbed distributions such that $D_{i,j} = \widehat{\text{MMD}}_k^2(p'_i, p'_j)$. Note that the matrix's size is equal to the number of WT in the farm, and its diagonal is filled with zeros since the MMD is a metric. This type of data guides the clustering methods that can be used.
<figure><img src="image_11.png" /><figcaption>Figure 9: South Brittany layout and wake effects measured by the squared MMD on wind conditions. Note that the vertical direction on this plot does not represent the north direction.</figcaption></figure>

<figure><img src="image_12.png" /><figcaption>Figure 10: Convergence of the squared MMD estimation</figcaption></figure>

Among the methods compatible with this input, hierarchical and centroid based methods were explored. Hierarchical clustering consists in building groups incrementally, either by starting with as many clusters as the number of elements and then merging two clusters at each step (agglomerative approach), or by starting with one cluster containing every WT and splitting a cluster in two at each step (divisive approach). In the literature, various splitting or merging criteria are defined, leading to different methods. Among the hierarchical agglomerative family, the Ward [31] and complete-linkage [32] methods are considered. In Ward, the criterion consists of merging the two clusters for which the increase of internal inertia is minimal. The internal inertia is defined as the sum of squared distances between two WT of the merged cluster. In complete-linkage method, the criterion consists of merging the two clusters for which the minimum distance between any pair of WT of this pair of clusters is minimal.

Centroids based clustering is also a common class of methods among which the widely used k-means [33] method. In the following, the k-medoids method [34, 35] is applied to the matrix D (a version of k-means using medoids). The medoid of a given group is an existing element which minimizes the sum of the distances as follows:

$$T_r = \arg \left\{ \min_{T \in C} \sum_{i=1}^{n} d(T, T_i) \right\} \quad (7)$$
where d refers to the MMD defined in section 4, T corresponds to a turbine in a given cluster C and T<sub>r</sub> the representative turbine of that given cluster.

A drawback of those methods is that their results are highly dependent of the initial conditions. This is why, they were applied 5 000 times with random initial conditions. In the end, the solutions with the minimal inertial were kept. In order to perform the RBD on a few WT among the farm, one representative WT per cluster is choosen, defined as the medoid of the cluster. Note that the numerical experiments on the previously introduced clustering methods were all made using the Python package Scikit-learn [36]. An example of clustering obtained by the k-medoid method for four clusters is illustrated in Figure 11. The wind turbines indexes containing the indication r are the representative of their clusters. The four clusters represented by colors in Figure 11 then define groups of WTs which minimize their MMD with respect to their representative turbine. Attention should be paid that in Figure 9 the MMD is computed for each turbine against the ambient distributions while in the clustering analysis we are considering the MMD between each pair of turbines. As a result, two turbines (e.g. 5 and 21) with similar MMD with regards to ambient wind distributions are not necessarily close to one another leading them being in different clusters (1 and 4). In the meantime, two turbines (e.g. 7 and 8) that are close to one another can belong to different clusters because the clustering process is a global process with optimisation considerations over the whole wind farm.

<figure><img src="image_13.png" /><figcaption>Figure 11: Clustering solution obtained with k-medoids for N=4, the four cluster representative WTs are indicated by a ‘(r)’ in their indices.</figcaption></figure>

## 5.2. Comparison between clustering methods

After applying the different clustering methods to the distance matrix D, let us define a relative error within each cluster. Considering the wake-perturbed random variable $X'_l = (u'_l, TI'_l), l \in \{1, ..., n_{WT}\}$, with the random variable of its cluster representative $X'_r = (u'_r, TI'_r)$. Let us define the relative error $\epsilon_l$ on the expected value of the marginal $u'_l$ (respectively on $TI'_l$):

$$ \epsilon_l = \mathbb{E} \left[ \frac{|u'_l - u'_r|}{u'_r} \right] \qquad (8) $$

The clustering solutions for each method are obtained for a number of cluster N varying from to 2 to 8. Table 1 presents an average of the relative error over the wind turbines as follows: $\bar{\epsilon} = \frac{1}{n_{WT}} \sum_{l=1}^{n_{WT}} \epsilon_l$.

Probably due to the large distance between turbines in the layout studied, the mean relative errors in Table 1 are relatively small. All the clustering methods lead to a very close split
<table><caption>Table 1: Mean relative errors (in %) for the wind speed and turbulence intensity for different clustering methods.</caption><thead><tr><th rowspan="2">N</th><th colspan="2">Ward [31]</th><th colspan="2">Complete [32]</th><th colspan="2">k-medoids [34, 35]</th></tr><tr><th>u</th><th>TI</th><th>u</th><th>TI</th><th>u</th><th>TI</th></tr></thead><tbody><tr><td>2</td><td>1.53</td><td>1.01</td><td>1.53</td><td>1.01</td><td>1.43</td><td>0.96</td></tr><tr><td>3</td><td>1.31</td><td>0.88</td><td>1.31</td><td>0.88</td><td>1.15</td><td>0.78</td></tr><tr><td>4</td><td>0.95</td><td>0.62</td><td>0.95</td><td>0.62</td><td>0.89</td><td>0.56</td></tr><tr><td>5</td><td>0.84</td><td>0.53</td><td>0.84</td><td>0.54</td><td>0.70</td><td>0.43</td></tr><tr><td>6</td><td>0.73</td><td>0.46</td><td>0.76</td><td>0.50</td><td>0.60</td><td>0.32</td></tr><tr><td>7</td><td>0.62</td><td>0.37</td><td>0.64</td><td>0.41</td><td>0.52</td><td>0.27</td></tr><tr><td>8</td><td>0.57</td><td>0.35</td><td>0.57</td><td>0.35</td><td>0.46</td><td>0.23</td></tr></tbody></table>

between groups. For a given number of clusters, the clusters built by each method have most of their elements identically paired, only a few WT are in-between clusters. Then, the arbitrary choice of the number of clusters is a trade-off between reducing the mean relative error within clusters and reducing the number of clusters (and therefore the number of costly RBD studies). Looking at the evolution of the mean relative error along the increase of clusters, the highest gain in precision is reached when moving from $N = 3$ to 4. In this case, a good compromise is then offered by the k-medoids method with four clusters. With this clustering solution (cf. Figure 11), one can easily justify assessing the RBD analysis over the farm by only performing it on these four representative turbines, with their respective wake-perturbed wind conditions as inputs.

## 6. Conclusion

A steady-state engineering wake model is coupled with a hydrostatic solver to take into account the effect of the floaters position in the wake computation of a floating offshore wind farm. The main impact of the floater's position is the increased rotor tilt, which leads to a larger vertical deflection of the wake. In the South-Brittany farm, the elevation of wake center is significant and could modify the fatigue loads on the downstream WT. This low-fidelity, or engineering model is very fast (about 3 minutes on a regular computer), while higher fidelity models can simulate the wake in a wind farm more accurately, but with much higher computational cost (several days for LES Meso-NH solver, with intensive parallelisation). In this paper, we consider that the modelling error made by the engineering wake model as reasonable. Further investigations should be done on the integration of the different fidelity levels in the uncertainty propagation. In this work, the uncertainty propagation is performed for to random inputs: the wind speed and the turbulence intensity, leading to a wake perturbed distribution per WT. A metric was then defined to measure the distance between distributions in order to build clusters of WT seeing similar wind conditions. After applying the metric to the South Brittany case, a clustering approach is used to determine a limited number of WTs representing the farm for RBD. Several clustering methods are compared and provide similar results for the current case study. In this case, a solution with 4 clusters is a good compromise between low relative error with clusters and low number of clusters. Getting a low number of clusters allows us to reduce the number of representative WT on which a RBD study can be done to assess the RBD over the whole farm. The differences among a cluster and between clusters have to be studied when looking directly to the output of interest for ultimate or fatigue reliability. However the difference on load output quantities may be reduced when compared to that on wake output quantities thanks to the damping of the WT. Depending on the definition of a wind farm failure (series system:
one WT fails or parallel system: all WT fail or intermediate system), the probability of failure can be estimated from the probability of failure of the representative WT. These considerations need to be further explored to improve RBD at the farm scale.

Acknowledgments

Authors are grateful for the technical assistance from contributors to the HIPERWIND project: N. Dimitrov, C. Peyrard, E. Vanem and A. Cousin. The Authors wish to thank EDF Lab and the Saint-Venant Hydraulics Laboratory in Chatou, France, for making the ANEMOC-3 model and database available.

Funding Statement

This study was part of HIPERWIND project which has received funding from the European Union's Horizon 2020 Research and Innovation Programme under Grant Agreement No. 101006689.

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