Correlations of power output fluctuations in an offshore wind farm using high-resolution SCADA data

The correlation of power output fluctuations of wind turbines in free field are investigated, taking into account the challenge of varying correlation states due to variable flow and wind turbine conditions within the wind farm. Based on eight months of 1 Hz SCADA data, measured at an offshore wind farm with 80 wind turbines, the influence of different parameters on the correlation of power output fluctuations is analysed. It is found that the correlation of power output fluctuations of wind turbines depends on the location of the wind turbines within the wind farm as well as the inflow conditions (free-stream or 5 wake). Wind direction investigations show that the correlation is highest for streamwise aligned pairs and decreases towards spanwise pairs. Most importantly, the highly variable measurement data in a free-field wind farm has considerable influence on the identification of different correlation states. To account for that, the clustering algorithm k-means is used to group wind turbine pairs with similar correlations. The main outcome is that next to the location of a wind turbine pair in the wind farm the standard deviation in their power output and their power differences are suitable parameters to describe the correlation of 10 power output fluctuations.


Introduction
Wind energy continues to be a growing source of energy. In 2019, 15.4 GW of new wind power capacity was installed in Europe, 24 % thereof offshore (WindEurope, 2020a). Considering offshore wind power in 2019, the capacity in Europe has increased by 3.627 GW, and a total of 7 wind farms were connected to the grid, and the average size of wind farms increased 15 to 621 MW (WindEurope, 2020b).
With the continuously increasing share of wind energy in the grid, the challenge of handling this highly fluctuating energy source becomes more important as discussed in Ren et al. (2017). To convert wind energy into electrical energy, wind turbines are installed, generally in groups (wind farms) at onshore and offshore sites. Fluctuations in their power output are the result of environmental influences such as changes in wind speed or wind direction, influences from neighbouring wind turbines, but 20 also their own state of operation. These power output fluctuations create challenges regarding the grid stability and therefore are an important field of investigation, (cf. Sorensen et al., 2007;Bossuyt et al., 2017b).
In order to achieve the maximum power output for the respective site, wind turbines within a wind farm are placed as efficiently as possible. The spacing of wind turbines is determined by the terrain of the site and the influence of wind turbines onto each other (their wake). Wakes cause energy losses through reduced wind speeds and at the same time greater power output 25 fluctuations and loads through increased turbulence (Crespo and Hernàndez, 1996;Vermeer et al., 2003).
Wake and wind farm flow effects on different spatial and temporal scales are reviewed by Porté-Agel et al. (2020). Many studies do not take power output fluctuations of wind turbines into account which have a high impact on the power output of a wind farm and the electrical grid. Thus, for further improvement of wind turbine control strategies like active power control (Vali et al., 2019) and grid stability by minute-scale prediction of offshore wind farm power (Valldecabres et al., 2020), the 30 occurrence of wind turbine power output fluctuations and their correlation within a wind farm are of great interest. Andersen et al. (2017) investigated the influence of large coherent structures on the power output of wind turbines in large wind farms. They were found to cause a high correlations in the power output of streamwise aligned wind turbines. Research on wind speed correlations and power output correlations has shown that wind turbines within a wind farm influence each other's power output fluctuations. Bossuyt et al. (2017a) showed that for a wind farm of 100 porous disc models in a wind tunnel, significant 35 correlations of the power output can be found for a streamwise aligned set up of the discs. Next to an increased turbulence intensity throughout the wind farm, the correlation of the power output reduced with increasing distance of the discs to each other. In an LES study by Lukassen et al. (2018) velocity space-time correlations within an infinitely large wind farm were analysed and modelled analytically. The velocity fluctuations which are directly related to power output fluctuations showed pronounced space-time correlations. Furthermore, the variance of the wind velocity turned out to be an important parameter in 40 the modelling set up. Stevens and Meneveau (2014) investigated spectra of wind turbine power output fluctuations in LES of finitely sized and infinitely large wind farms. The spectra were found to be depended on non-trivial correlations of streamwise placed wind turbines. The correlation of two wind turbines was significantly influenced by the wind direction, i.e. lowest correlation for spanwise placed wind turbines and highest correlation for streamwise aligned wind turbines. Dai et al. (2017) analysed 1 Hz wind farm SCADA data with respect to the influence of wind speed fluctuations and wind direction fluctuations 45 on wind turbine power output fluctuations of single wind turbines. They showed a direct relation of wind speed fluctuations and power output fluctuations in the partial load regime whereas wind direction fluctuations have only little effect. By using 10 minute averaged wind farm SCADA data, Braun et al. (2020) derived a stochastic model for the power time series of wind turbines which was based on the temporal autocorrelation of single wind turbines.
In our work, we analyse 1 Hz wind farm SCADA data to describe space-time correlations of power output fluctuations of wind 50 turbine pairs. In contrast to the wind tunnel measurements by Bossuyt et al. (2017a) and the LES analysis by Lukassen et al. (2018) mentioned above, the data set processed here includes unstable inflow conditions, dynamically operating wind turbines as well as changing flow conditions within the wind farm. Furthermore, there may be potential measurement inaccuracies.
The result is a highly complex data set. In this paper we investigate the influence factors which determine different correlation states. For that we group the correlations based on wind turbine statistics.

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Starting with the description of the evaluated data set in Sect. 2, the processing of the data is explained in Sect. 2.1 and 2.2.
The space-time correlation of power output fluctuations per wind turbine pair for time intervals of 600 s is introduced. Using a filtered data set with less varying flow conditions, the correlations are analysed for different wind directions in Sect. 3.1. The correlation for wind directions with streamwise aligned wind turbines is evaluated in more detail. In Sect. 3.2, the location dependency of the power output fluctuation correlation is determined by the comparison of wind turbine pairs located in 2 Reference wind farm and data processing The analysis performed in this work is based on measurements from the offshore wind farm Global Tech I (GT I). It is located in the North Sea more than 100 km off the coast of Northern Germany. Its total capacity of 400 MW is provided by 80 wind turbines spread over an area of about 41 km 2 . The wind turbines of type Adwen AD 5-116 have a rated power of 5 MW, a rated wind speed of 12.5 ms −1 , a hub height of 92 m and a rotor diameter (D) of 116 m. They are installed in a grid like, 70 non-axisymmetric pattern with a triangular shape towards south (see Fig. 1).
The analysed data set was measured in a period of about eight month, from January 1st, 2019 until September 9th, 2019 and consists of 1 Hz wind turbine SCADA data. The processed signals include the generated power P , the azimuth angle of the wind turbine (i.e. the nacelle direction) θ, the nacelle based wind direction ϕ (measured relative to θ), the pitch angle β of each blades, and the nacelle based wind speed U . It has to be noted that the wind speed U is not directly measured but recalculated 75 from the measured power and the control settings of the wind turbine. Due to that, U is an approximated and idealised value which does not include wind speed independent power reduction, e.g. by misalignment of the wind turbine. However, it can   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  36  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62  63  64  65  66  67  68  69  70  71  72   73  74  75   76  77  78 79 80 Figure 1. Layout of GT I. Each wind turbine is labelled with its corresponding number. The spacing of the wind turbines is inhomogeneous.
The wind directions 90°and 270°(marked in the figure) will be analysed in detail in the subsequent sections. The red square depicts the set of wind turbines which will later be used during the location-dependency analysis in Sect. 3.2 due to their symmetric arrangement. In the clustering analysis in Sect. 4 the whole wind farm will be used. The blue ellipses exemplarily show the definition of streamwise wind turbine pairs. The definition of all pairs is listed in Tab. A1. still be used for assessing the effect of the wind speed on the wind turbine. The azimuth angle θ of the wind turbine refers to the direction it is facing in its preset reference system. This system does not necessarily exactly match to the global geographical one due to the measurement inaccuracies of the azimuth angle and a potentially inaccurate north orientation of the reference 80 system of each wind turbine (cf. Bromm et al.). The nacelle based wind direction ϕ is estimated based on the measurements of two 2D sonic anemometers installed behind the rotor of each wind turbine. These measurements have to be treated with care as the measured flow behind the rotor is disturbed by the rotation of the rotor and the nacelle itself. Thus, it is only an estimation of the wind direction facing the wind turbine. The combined measurements of θ i and ϕ i define the wind direction Φ i facing the i-th wind turbine.

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For assessing an average wind direction for the wind farm, we average over Φ i of all wind turbines. Due to the size of the considered wind farm, the wind direction is not expected to be consistent throughout the whole wind farm. Single wind turbines could still be facing different wind directions compared to the average wind direction of the wind farm (cf. Schneemann et al., 2020;Sanchez Gomez and Lundquist, 2020). The average wind direction over all available wind turbines is defined as Φ av .

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Wind tunnel experiments and LES simulations as described in Sect. 1, pose controllable conditions for evaluating correlations.
Such conditions can not be met in a free-field wind farm. Next to temporally and spatially varying wind conditions, the layout of the wind farm leads to unequal conditions to wind turbines due to their positions, e.g. changing wind direction throughout the wind farm, especially for large wind farms. Further, each wind turbine is acting independently of the other wind turbines including yawing, pitching, start up or shut down. Next to this, single wind turbines can be set to operate in a down-rated state 95 or be shut down due to maintenance or other reasons. All these factors multiply to an order of unpredictable variability within a wind farm which causes highly dynamical flow conditions. To cope with these issues, the data set is filtered for reasonable parameters, which limits the variability in the data set. For each 600 s interval, the parameters defined in the following have to be met, cf. Tab. 1. Depending on the operational state of a wind turbine, the power output fluctuation characteristics and the influence on other wind turbines changes. Therefore, the data set is filtered for different operation states creating a cleaned data 100 set with comparable operation state conditions for all wind turbines.
In general, a wind turbine operating in partial load is not pitching and the velocity in its wake is always below rated wind speed. A wind turbine operating in full load aims at keeping a constant rotor speed and power by pitching its blades, where the wind speed in its wake can be larger than rated wind speed. To avoid the effects of pitching and the different wake behaviour on the correlation of wind turbines, the data set is limited to partial load. To further avoid effects from the transition from idle 105 Table 1. Filters applied to the raw data of each wind turbine within the wind farm.

Signal Power Pitch Yaw
Settings 0.5 MW ≤ P ≤ 4.5 MW β < −1.3 • no yawing mode into operation or the transition from partial load to full load, only the data of wind turbines generating power in the range of 0.5 MW and 4.5 MW is considered.
The previously defined, limited power range still includes derated wind turbines. Wind turbines being derated means that the maximum power of the wind turbines is limited to a certain value lower than their rated power. Due to that wind turbines might start pitching already in the previously defined load range. To fully exclude pitching wind turbines, the data is filtered for any 110 pitching activity, only allowing pitch values smaller than -1.3°.
Furthermore, yawing wind turbines are excluded from the analysis as well. The adjustment of wind turbines to the wind direction is managed by each wind turbine individually. Thus, wind turbines could be facing slightly different wind directions Φ i and start yawing at different times. The yawing activity of a wind turbine transfers to its wake, i.e. changes its deflection (cf. Bromm et al.). This would affect the correlation of two wind turbines. To exclude yawing wind turbines, no change of θ is 115 allowed in the regarded 600 s time interval.
To further filter the data for wind directions, the average wind direction Φ av of all wind turbines is calculated for each time step of the regarded 600 s time interval. The average wind direction Φ av has to fit the wind direction of interest within a tolerance of ±10 • for all time steps in the regarded 600 s. Note that the borders of the interval are including the lower limit and excluding the upper limit. Since this filter only applies to the average wind direction Φ av , individual wind turbines might have a slightly 120 deviating relative wind direction for this specific time interval due to a false wind direction measurement, a yawing process which has taken place asynchronously to the majority of the other wind turbines, or a wind direction deviation due to local changes over the area of the wind farm.
As a summary, the overall filtering procedure is as follows. Each time interval of 600 consecutive seconds where the two wind turbines of a wind turbine pair (as defined in Fig. 1) both fulfil all the above described filtering parameters, i.e. power range, 125 pitch, yawing and wind direction, is used in the correlation analysis. This means that for different time intervals a different set of wind turbine pairs is considered and that furthermore, wind turbine pairs can be considered for multiple time intervals.

Correlation of power output fluctuations
Power output fluctuations of individual wind turbines are defined as deviations of the instantaneous power from the average power of the regarded wind turbine i within a certain time interval ∆t. We analyse time intervals of ∆t 600 = 600 s: where P i (t) ∆t600 is the average of the measured power P i (t) over an interval ∆t 600 including all 600 values for t in the discretised interval [t j , t j + 599 s]. P i,tj (t) is the power output fluctuation within the interval ∆t 600 (the index t j is omitted in the following). Depending on the data availability, the next interval of 600 consecutive seconds could go from [t j + 1 s, t j + 1 s + 599 s], and thus, partly overlap the previous one.

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The selection of the interval size of 600 s is based on the layout of the wind farm and the considered power ranges with the corresponding wind speeds. For example, considering the spacing of up to 9 D for westerly winds, the cut-in wind speed of 4 ms −1 and rated wind speed of 12.5 ms −1 , a particle moving with the undisturbed wind would take from about 84 s up to farm into account, a considered interval length of 600 s captures potential correlations of interest. This means, each time step 140 followed by 599 consecutive time steps forms an interval, individually for each wind turbine. For all available intervals of all wind turbines, the power output fluctuations are then calculated based on Eq. 1.
To analyse the influence of wind turbines onto each other, the space-time correlation is calculated. This is done using the Pearson correlation coefficient (Pearson, 1896) where ... ∆t300 is the average over an interval ∆t 300 = 300 s including all 300 values for t in the discretised interval is the power output fluctuation of the upstream wind turbine A following Eq. 1 at a time t, P B (t + τ ) is the power output fluctuation of the downstream wind turbine B at a time t + τ with a time lag τ .
The Pearson correlation coefficient is a value between -1 and 1, where 1 depicts the maximum possible linear correlation, -1 is 150 the maximal linear anti-correlation and a value of 0 depicts no linear correlation. The correlation coefficient is evaluated for a fixed period of 300 s from P A (t) to P A (t + 300 s) and likewise P B (t + τ ) to P B (t + 300 s + τ ). This allows a maximum time lag of τ = 300 s for each considered 600 s interval.
Dependent on the wind speed, wind structures responsible for power output fluctuations measured at an upstream wind turbine A take some time to travel the distance to the neighbouring downstream wind turbine B. To compare correlations at different 155 wind speeds and different wind turbine distances, the time lag τ is normalised for each time interval starting at t j individually where τ norm is the normalized time lag, U B (t) ∆t300 is the average wind speed measured at a certain (downstream) wind turbine B for a time interval ∆t 300 = 300 s for t in the discretised interval [t j , t j +299 s] and x AB is the distance between wind turbine A and wind turbine B.

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The Pearson correlation coefficient in dependence of the normalised lag τ norm is given as follows If the average travelling velocities of the wind structures matches the average wind speed measured at wind turbine B, τ norm = 1. However, in situations where wind turbine B is in the wake of wind turbine A, τ norm is expected to be larger than 1. The wind speed in the wake is reduced and recovers slowly, so that the wind speed measured at wind turbine B, i.e.,

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U B is already partly recovered and hence larger than the average wind speed between wind turbine A and B.
For averaging correlation curves for different wind speeds, the correlation curves are discretised using a histogram with a reference time lag of where τ is the time lag (0 s to 300 s), U max is an artificially introduced velocity which has to be larger than the maximum 170 possible wind speed (since the rated wind speed is 12.5 ms −1 , here U max = 13 ms −1 ) and x AB is the distance between wind turbine A and wind turbine B. Afterwards the average value of r(τ norm ) for each bin in the histogram is calculated.
3 Wind-direction dependency and location dependency    the peak magnitude. The correlation is not as pronounced as for the streamwise case above (i.e. around 90°and 270°) which confirms simulation results by Stevens and Meneveau (2014). We will not investigate the spanwise correlations in further detail 205 here. Figure 4 shows the average power output fluctuation correlation around 90°and 270°as cuts through Fig. 3 in detail. The absolute peaks are at 90°and 270°. For wind directions where the wind turbines in a pair are less streamwise aligned the peak decreases and the correlation curve flattens. In contrast to 90°, the correlations for 270°are more defined and show slightly larger peak values. This maybe due to the non-axisymmetric wind farm layout (cf. Fig. 1).

Location-dependent space-time correlation 210
The location dependency of the averaged power output fluctuation correlations is investigated for wind directions 90°and 270°.
As mentioned before, for these two wind directions the wind turbines are streamwise aligned. The most northern wind turbines 1 to 8, and wind turbines 58 to 80 in the lower triangle of the wind farm, do not follow the symmetric pattern of the square consisting of wind turbines 9 to 57. The following results are limited to this symmetric square marked in Fig. 1.  . It has to be marked that the upstream wind turbine A is standing in free 230 stream while the downstream wind turbine B is affected by the wake of the upstream wind turbine. Thus, the two wind turbines Table 2. Averaged wind turbine statistics computed for wind direction intervals around 90°and 270°with A as upstream wind turbine and B as downstream wind turbine. P 2 A ∆t 600 is the standard deviation of the power output fluctuations of wind turbine A over 600 s intervals ∆t600 (analogue for wind turbine B for the same 600 s intervals, respectively). PA ∆t 600 and PB ∆t 600 are the average power of wind turbines A and B over the same 600 s intervals. ... all denotes the average of the statistics over all available time intervals of the wind turbine pairs. Note that Φc depicts the centre of 20°wind direction intervals, here from 80°to 100°and from 260°to 280°.  have very different inflow conditions. For wind turbine pairs further downstream both wind turbines are standing in the wake of upstream wind turbines. Here, a clear correlation is found. For the second to last row, the peaks become more defined as their width decreases.

Φc
As described by Bossuyt et al. (2017a), the turbulence intensity increases with the flow towards the back of the wind farm. intensity in the wind tunnel results mentioned above. The normalised power difference is largest for the first row which is caused by the previously described deviating inflow conditions of the upstream and downstream wind turbines. This was also found in the experiment by Bossuyt et al. (2017a) where the first row generates the maximum power, the second and following rows show a significant reduction. Table 3. Averaged wind turbine statistics per wind farm row computed for wind direction intervals around 90°and 270°with A as upstream wind turbine and B as downstream wind turbine. P 2 A ∆t 600 is the standard deviation of the power output fluctuations of wind turbine A over a 600 s interval ∆t600 (analogue for wind turbine B for the same 600 s intervals, respectively). PA ∆t 600 and PB ∆t 600 are the average power outputs of wind turbines A and B over the same 600 s intervals. ... row denotes the average of the statistics over all available time intervals of the wind turbine pairs in a row. Note that 90°and 270°again refer to 20°wind direction intervals from 80°to 100°and from 260°to 280°. The results of section 3.2 reveal that the standard deviation of the power output fluctuations as well as the power difference of the wind turbines change depending on the location of the wind turbine (pairs) within the wind farm. As explained in section 2.1, conditions in a wind farm are never ideal due to the variety of influence factors such as wind direction and wind speed fluctuations or influences of surrounding wind turbines. Turbines within the wind farm which are turned off or derated might create free-stream like inflows for downstream wind turbine pairs. Such irregularities might influence the standard deviations 250 and power differences calculated for the wind turbine pairs. Even though such wind turbines are filtered out for the analysis, they still influence the surrounding wind turbines in an unpredictable way. To identify these locally abnormal conditions and the resulting deviations in the power output fluctuations and their correlations, the k-means clustering algorithm is used to sort the correlations based on the previously defined statistics of the power output fluctuations, namely the standard deviation and the normalised power difference. In the following, we investigate these influences for the directions 90°and 270°. Furthermore, 255 the triangular shape of the lower part of the wind farm (wind turbines 58 to 80) as well as the most northern wind turbines 1 to 8 are incorporated now (cf. Fig. 1)  based on Lloyd (1982), using random sample points as initial centroids, the squared euclidean distance as distance metric, with a maximum of 300 iterations and five clusters. The number of the clusters was empirically chosen as the data was grouped into a reasonable set of groups and a greater number of cluster did not lead to further clusters of importance for the present analysis (see appendix B). To avoid the generation of local centroids the clustering is repeated ten times and the run with the 265 clusters with the lowest sum of point-to-centroid distances within the clusters is chosen. Different orderings of the intervals have been tested, namely random sorting, data sorted for increasing standard deviation of the downstream wind turbine B, and a chronological sorting according to the available time intervals. The results have been found to be equal including the first decimal place of the centroids for all cases, thus, the random sorting is used in the further analysis. Table 4 lists the centroids (centres of the clusters). The standard deviations of both wind turbines A and B are significantly de-270 creasing while the normalised power difference of A and B is significantly increasing from cluster 1 to 5. To further investigate these findings, we analyse the correlation curves corresponding to the clusters. Figures 8 and 9 show the average correlations for both wind directions for each of the five clusters (upper plots) and the percentage frequency of each pair within each of the five clusters (lower plots). As expected from Fig. 7, the average correlations for 270°are higher than for 90°. Cluster 1 includes nearly 6% of the data and has the highest correlation. This is a significant increase compared to the average correlation 275 shown in Fig. 7. From cluster 2 to 4 the correlation is decreasing while the amount of data per cluster increases. For cluster 5 no correlation is found. Looking at the occurrence of wind turbine pairs within each cluster, a clear trend is visible. While cluster 1 with the highest correlation is dominated by wind turbine pairs where the upstream wind turbine is located towards the back of the wind farm, cluster 5 with no correlation is dominated by wind turbine pairs with its upstream wind turbine located in the first row of the wind farm. From Cluster 2 to 4 the dominating wind turbine pairs shift from the back rows towards the 280 front rows whereas the percentage frequency become mores balanced throughout the wind farm (i.e. more light green coloured turbines).
Comparison of the results of Fig. 8 and 9 and Tab. 4 clearly depicts that the greater the standard deviations and the smaller the normalised power difference the higher the correlation of the wind turbine pairs. The slight row dependence which was already indicated in Tab. 3 can be confirmed here. This is illustrated by a colour coding of frequency of occurrence of wind turbine 285 Table 4. Cluster centroids for wind direction intervals around 90°and 270°with A as upstream wind turbine and B as downstream wind turbine. P 2 A ∆t 600 is the standard deviation of the power outputs fluctuations of wind turbine A over 600 s intervals ∆t600 (analogue for wind turbine B for the same 600 s intervals, respectively). PA ∆t 600 and PB ∆t 600 are the average power output of wind turbines A and B over the same 600 s intervals. ... cluster denotes the average of the statistics over all available time intervals of the wind turbine pairs within a cluster. Note that 90°and 270°again refer to 20°wind direction intervals from 80°to 100°and from 260°to 280°.  The lower plot shows the percentage frequency of each wind turbine pair within the respective cluster given as colour. As the wind turbines are analysed in pairs of two, the last row of wind turbines is unlabelled as these wind turbines do not have a downstream partner.
pairs in each cluster in the lower subplot of Fig. 8 (respectively Fig. 9). The sum of all frequencies of all wind turbines within one cluster add up to 100% meaning a yellow coloured wind turbine pair makes up about 3% of the respective cluster and a green marked wind turbine pair makes up about 1.5% of the respective cluster. For example, the correlation peak for cluster 1 Note that the values do not add up exactly to 100% due to rounding. The lower plot shows the percentage frequency of each wind turbine pair within the respective cluster given as colour. As the wind turbines are analysed in pairs of two, the last row of wind turbines is unlabelled as these wind turbines do not have a downstream partner.
of more than 0.3 for 90°(respectively 0.4 for 270°) partly includes pairs with the upstream turbine in the last row but also some turbine pairs in the rows before. This is considerably larger than the correlation curve of row 6 of Fig. 6. As an outlook, further analysis on the space-time correlations within an offshore wind farm could help in the control of wind turbines, e.g. for power output fluctuation management or active wake control. Also, knowledge about the correlation of wind turbine pairs allows short-term power output fluctuation forecast within the wind farm as well as interactive wind turbine 315 control.
The presented findings can be enhanced in the future by additional Lidar or Radar measurements to access independent wind direction and wind speed measurements. Also the analysis of correlations might be extended to include the correlation of wind turbine pairs with multiple inter-turbine distances and the correlation of non-aligned wind turbine pairs. The clustering of correlation states can be further investigated by increasing the number of clusters to k > 5 as the results for k = 6 indicated 320 that the statistics of the upstream and downstream wind turbine of a pair has different influence on its correlation. Also, it is worth considering alternative clustering methods like k-medoids (Kaufman and Rousseeuw, 2008) or Density-Based Spatial Clustering of Applications with Noise (DBSCAN) (Ester et al., 1996). Furthermore, measurements on the boundary layer conditions help to assess the influence of wind turbine wakes on the space-time correlations of power output fluctuations with the additional knowledge on the atmospheric stability.
turbine 2 is the upstream wind turbine and turbine 1 is the downstream wind turbine. Appendix B: Effect of the numbers of clusters As mentioned in Sect. 4, the number of clusters chosen for the present analysis was k = 5. This decision was made based on the results for k = 6, as an increasing number of clusters revealed further features outside of the scope of this work. Figure   B1 and Fig. B2 present the clustering results for k = 6. For wind direction 90°, six clearly separable correlation curves are 335 found. Cluster 2 shows an abnormal characteristic compared to cluster 1 and 3 as here, mostly wind turbines in the second and third row are dominating the cluster. The result for wind direction 270°are 4 clearly separable correlation curves and two very similar ones. Cluster 3 displays an abnormal characteristics compared to cluster 2 and 4 as here, again wind turbines from the second row are dominating the cluster. Looking at the statistics of the correlation curves listed in Tab. B1 it further can be found that the standard deviation of the upstream and downstream wind turbines influences the correlation curves differently.

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For wind direction 90°, cluster 2 and 3 have a different order in the standard deviation of the power output fluctuation of wind turbine A and B. While the average standard deviation of the downstream wind turbine B is decreasing from cluster 2 to cluster 3, the average standard deviation of the upstream wind turbine A is higher for cluster 3 than for cluster 2. The same is found for wind direction 270°. The average standard deviation of the downstream wind turbine B is decreasing from cluster 3 to cluster Figure B1. Clustering for wind direction interval around 90°with randomly sorted parameters and k = 6. Φc depicts the centre of the wind direction interval. Table B1. Averaged wind turbine statistics computed for wind direction intervals around 90°and 270°and k = 6, with A as upstream wind turbine and B as downstream wind turbine. P 2 A ∆t 600 is the standard deviation of the power output fluctuations of wind turbine A over a 600 s interval ∆t600 (analogue for wind turbine B for the same 600 s intervals, respectively). PA ∆t 600 and PB ∆t 600 are the average power of wind turbines A and B over the same 600 s intervals. ... cluster denotes the average of the statistics over all available time intervals of the wind turbine pairs. Note that here 90°and 270°again refer to 20°wind direction intervals from 80°to 100°and from 260°to 280°. 4, the average standard deviation of the upstream wind turbine A is higher for cluster 4 than for cluster 3.

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This implies that a further separation of the statistics with k > 5 might result in a distinction of the effect of the upstream and downstream wind turbine of the correlation a pair. This is here not further investigated as this effect is not included in the scope of the work presented here.