The distance between the measurement and prediction location is traditionally used as an indicator of model uncertainty, and measurement campaigns are often planned to minimize the extrapolation distance. defines a “representativeness radius” as the distance from a met mast to the furthest location that can be extrapolated with tolerable uncertainty. The representativeness radius depends on the complexity of the terrain, where complex terrain is characterized by having terrain slopes greater than 0.3. Figure  illustrates two met masts located in hilly terrain. Following the recommendations of for simple and complex terrain, the measurements taken in the valley, U01, are preferable for predicting UP as the distance is small. For sites that are even more complex, MEASNET states that mast positioning should be based on a site-specific analysis.

According to , errors related to flow modelling are minimized when the predictor site (the met mast location) and the predicted site (the wind turbine location) are as “similar” as possible regarding factors like regional wind climate, roughness, orography and obstacles. refers to this as the similarity principle. The underlying assumption is that no matter how advanced a flow model, it always produces errors, and they scale with the forcing applied. Figure  illustrates the similarity principle. The wind conditions in the valley, U01, differ substantially from the conditions at the prediction site, UP, and a better predictor (according to the similarity principle) is, therefore, the hilltop met mast, U02, despite its more distant location.

Experienced wind and site engineers determine suitable locations for met masts based on the representativeness radius and their judgement. They intuitively understand that distance is not the only parameter that should be evaluated; however, since no algorithmic method for similarity exists, the industry standard is often to minimize the extrapolation distance.

To make a parameter that represents similarity, we define the directional-averaged speedup uncertainty, σS, 1σS2=∑j=1NθfjσSj2, where fj is the wind direction frequency at the met mast, Nθ is the number of wind direction sectors and σSj2 is the speedup uncertainty of the micro-scale model. As an expression for the speedup uncertainty for a particular wind direction, we use the model by : 2σSj2=λ1-exp⁡-dL12+Sj-1Sj+12, where d is the extrapolation distance, L1=1 km and λ=0.1 are empirically calibrated constants , and Sj=UPj/U0j is the speedup where U0 and UP are the measured and predicted mean wind speeds respectively. L1 and λ have not been re-calibrated in this work but are kept to their original values.

σS combines into one expression, the uncertainty associated with both extrapolation distance and the wind speed difference. Small values of σS signify that predictor and prediction conditions are similar. The inclusion of speedup in the expression ensures that any difference between predictor and prediction site included in the micro-scale flow model is evaluated (roughness, orography, obstacles, measurement height, etc.). The speedup uncertainty is, therefore, expected to be a more accurate “measure of similarity” than distance, and we define the “most similar” predictor to be the location with the smallest σS value.

For large wind farms with several met masts, a standard practice when making predictions is to either select the closest available predictor or to make a weighted average of multiple predictions using “inverse-distance weighting”. As an example, the inverse-distance-weighted mean wind speed, UP, can be determined as 3UP=∑i=1NMWiUPi∑i=1NMWi, where UPi is the predicted mean wind speed, Wi=di-2 is the predictor weight, di is the extrapolation distance and NM is the number of predictors on a particular site. The underlying reason for using inverse-distance weighting is that the standard error of a weighted mean decreases with the number of independent predictions. However, for this to be valid, the predictions are assumed independent (model errors should be random), and extrapolation distance is assumed to be the parameter that correlates the strongest with model error.

This paper aims to show that met masts should be placed following the similarity principle instead of reducing extrapolation distance. The hypothesis is verified by showing that the most similar predictor gives lower prediction errors than the closest predictor and inverse-distance weighting. In the following, we conduct a large number of predictions using the following multi-mast strategies and compare the results to measurements. 4Closestpredictor:Wi=1ifdi=min(d)0otherwise5Inv.dist.weighting:Wi=di-26Mostsimilarpredictor:Wi=1ifσSi=minσS0otherwise

The dataset used in this work has been collected specifically to validate multi-mast strategies. It is obtained through wind project developers worldwide and is considered to cover a wide spectrum of the conditions experienced in wind projects. The dataset consists of wind measurements from sites with three or more met masts. By having at least three masts, each mast location can be predicted using at least two predictors, and results can be compared against the measurements taken at the predicted mast.

The dataset consists of wind speed and wind direction measured from the top anemometer of each met mast, already screened and long-term corrected by wind project developers. The data are grouped into 36 10∘ wind direction sectors and have Weibull distributions fitted to the wind speed histogram. Wind statistics from a total of 210 met masts were provided for the study. The only additional screening that has been conducted is the removal of four sites (25 met masts) from the dataset. These sites were removed since mast-to-mast predictions of wind speed and power density led to substantial errors (>3σ) for four of the met masts. We did not investigate the reason for the significant errors but removed the sites from the investigation. Table  shows a summary of the sites and met masts used. As seen, the screened data consist of 185 met masts distributed over 40 wind turbine sites.

The data come from met masts located near potential wind energy installations, and the 185 mast locations represent varied site conditions from all over the world. Figure  illustrates the complexity of the sites using the RIX (left) and ΔRIX (right) measures where RIX is defined by the percentage fraction of the terrain along the prevailing wind direction, which is over a critical slope of 0.3 . The mean RIX value is 7.0 %, which can be considered moderately complex terrain. Compared to the closest neighbouring mast, 84 masts have |ΔRIX| values of less than 1 % and 164 masts have |ΔRIX| below 5 %. Most of the met masts are therefore placed in similar, well-exposed hills and ridges.

Figure  shows the height of the masts (right) and the distance to their closest neighbour (left). The distance is up to 10 km, but on average the distance is 3.1 km. As seen, the masts do not have the same height but vary from 20 to 120 m, and half of the sites have mast height variations. These sites often have combinations of short and tall masts, e.g. of 40/80, 60/80 or 60/100 m masts. The wind statistics have not been corrected or “sheared up” to unify the height differences. Instead, the flow model has been used directly to estimate the wind statistics at the prediction height using the statistics at the predictor height. The most similar method has an advantage compared to the other multi-mast strategies for sites with different mast heights as there will be a speedup between masts of different heights and will therefore not be valued as similar. In the current analysis, this is however of minor importance as the results, particularly in Sect. , depend mainly on a single site with 25 met masts that are all 80 m tall.

The dataset (Table ) allows for a total of 185 possible predictions with two or more predictor masts. The results section analyses how the different multi-mast strategies perform with this specific mix of multi-mast sites. Also, other combinations of multi-mast sites from the same data pool have been made to analyse how the multi-mast strategies perform on sites with a specific number of predictor masts. As an illustration of how other multi-mast combinations can be made from the same data, we can imagine that a single site with three met masts can also be viewed as three different sites with two met masts. Table  shows the many combinations of predictions possible with one, two, three, four, five or six predictor masts. As seen, it is possible to make more than 33 million different predictions with six predictor masts. The many combinations are primarily possible because of the site that has 25 met masts. The results for different numbers of predictor masts are not directly comparable since they are based on different sites. The results section, therefore, normalizes the most similar and the inverse-distance-weighting results with the “closest mast” to make comparisons of the multi-mast strategies possible.

The micro-scale flow model WAsP 12.3 has been used to make all wind climate predictions in this study. Since every mast location needs to be predicted using every possible predictor mast, there is a total of 1168 mast-to-mast predictions. A database and an automatic model script (PyWAsP) were made to conduct all the simulations, and every simulation was run with the standard WAsP 12.3 parameters. No ΔRIX or other corrections have been made to the model results, and no tuning of model parameters to improve results was performed.

The topography maps used for the flow model were provided to the authors and used directly without corrections or quality control. As all data have previously been analysed to be used for wind farm development, and scrutinized by third-party companies, the data are considered to have an industry standard quality. The current work is a statistical analysis of the multi-mast strategies. All simulations are based on identical model setups and using the digitized maps, mast locations, anemometer heights and wind measurements provided to the authors; only the method used to calculate the predicted wind statistics is different between the three multi-mast strategies.

Wind statistics are calculated at each prediction location to validate the performance of the multi-mast strategies. Specifically, the predicted mean wind speed, UP (m s-1), and mean power density, EP (W m-2), are compared to the measured values (U0 and E0). Since the measured wind data and the wind climate predictions are given in terms of the Weibull parameters A and k, the mean wind speed and mean power density are calculated following : 7Ui=∑j=1NθfjAijΓ1+1kij,8Ei=12ρ∑j=1NθfjAij3Γ1+3kij, where ρ is air density (1.225 kg m-3) and Γ is the gamma function. Having calculated the wind speed and power density for every possible single-mast prediction, the multi-mast strategies were applied to calculate the final predicted wind statistics: 9UP=∑i=1NMWiUPi∑i=1NMWi,10EP=∑i=1NMWiEPi∑i=1NMWi, where Wi is the mast weight that depends on the chosen strategy (Sect. ). It was also considered to apply a generic wind turbine power curve to allow calculation of energy yield predictions. However, as the prediction heights and wind speeds vary between the sites, using a single power curve for all predictions was not pursued.

To evaluate the multi-mast strategies, we compare the measured and predicted wind statistics at each mast position and calculate the average error of all predictions. The closest-mast results are used as a baseline. The paper does not analyse the baseline error in any detail; instead, the paper focuses on how the multi-mast strategies improve the baseline. The data and model setup used for the closest mast, inverse-distance weighting and most similar methods are identical, to make an objective comparison of the multi-mast strategies. For each multi-mast strategy, the absolute error of each mast prediction is given by 11Xi=XP-X0, where XP is either the predicted mean wind speed (UP) or power density (EP) and X0 is the measured value (U0 or E0). The mean error of a strategy is calculated by 12μ=∑i=1nXin, where n is the number of mast predictions. Finally, the standard deviation of the error is given by 13σ=∑i=1nXi-μ2n-1. The results shown in the following indicates the mean error, μ, the standard deviation of the errors, σ, and the number of predictions used to calculate the statistics, n.

The top row of Fig.  shows histograms of the absolute error of wind speed (left) and power density (right) for each of the 185 mast predictions using closest mast as a multi-mast strategy. Gaussian distributions are shown on the histograms (despite the poor fit), and the mean error and standard deviation of wind speed (μ=0.11, σ=0.63) and power density (μ=36.3, σ=149) are given in the legends. The observed mean wind speed and power density of the 185 masts are 7.36 m s-1 and 460.7 W m-2 respectively.

The results for inverse-distance weighting and most similar predictor are shown in the middle and bottom rows of Fig. . The left column shows that both multi-mast strategies significantly reduce both the mean wind speed error and the standard deviation compared to the closest mast. The standard deviation of the wind speed error is seen to decrease from 0.63 to 0.54 m s-1 for both methods. The right column of the figure shows even more substantial error reductions for power density. The standard deviation of power density decreases from 149 to 125 W m-2 for inverse-distance weighting and from 149 to 116 W m-2 for most similar mast. Therefore, by selecting the most similar location, the mean power density error is reduced by 22 % compared to the closest mast and 7 % compared to inverse-distance weighting.

To clarify the difference between strategies that rely on the representativeness radius (closest mast and inverse-distance weighting) and the similarity principle (most similar predictor), this section focuses on sites with a specific number of predictor masts. By combining the available dataset, it is possible to generate new combinations of sites with one, two, three, four, five or six predictors (see Table ).

As an example of the absolute prediction errors, Fig.  shows a histogram of the absolute error of wind speed (left) and power density (right) for predictions that use six predictors. The available dataset allows 33 million different combinations of a prediction location and six predictor masts. The histograms of the strategies have different colours, and the legends indicate the mean error and the standard deviation of each. Compared to the closest mast, the standard deviation of power density decreases from 189 to 164 W m-2 for inverse-distance weighting and from 189 to 124 W m-2 for the most similar mast.

The mean absolute error and the standard deviation for predictor combinations with one, two, three, four, five and six predictor masts are given in Tables  and ; however, the results are not directly comparable since they are based on different sites. To make a more valid comparison, we normalize all the results with the results of the closest mast. Figure  shows the normalized average reduction in the standard deviation of wind speed error (left) and power density error (right) for inverse-distance weighting and the most similar predictor compared to the closest mast. Note that the multi-mast strategies give identical results for predictions with only one predictor.

Figure  shows that both inverse-distance weighting and most similar predictors significantly reduce the average prediction error compared to the closest mast. While inverse-distance weighting reduces the error significantly with two predictors compared to one, the added improvement of using three or more predictor masts is much smaller. The most similar strategy has achieved 34 % reduction of the standard deviation of power density error with six predictor masts, and it appears that the error would keep decreasing with additional predictor masts. It should be noted that the most similar strategy only uses a single predictor mast, but having more options to choose from decreases the error. This indicates that significant improvements are gained if the most similar strategy is followed for placement of met masts, and it should also be used for single-met-mast campaigns.