Assessment of Spatiotemporal Wind Complementarity

Abstract

This study investigates whether combining singular value decomposition with wavelet analysis can provide new insights into the spatiotemporal complementarity between wind turbine sites, surpassing previous findings. Earlier studies predominantly relied on various forms of correlation analysis to quantify complementarity. While correlation analysis offers a way to compute global metrics summarizing the relationship between entire time series, it inherently overlooks localized and time-specific patterns. The proposed approach overcomes these limitations by enabling the identification of spatially explicit and temporally resolved complementarity patterns across a large number of wind turbine sites in the study area. Because complementarity information is derived from orthogonal components obtained through singular value decomposition of a wind power density matrix, there is no need to adjust for phase shifts between sites. Moreover, the complementary contributions of these components to overall wind power density are expressed in watts per square meter, directly reflecting the magnitude of the analyzed data. This facilitates a site-specific, complementarity-optimized strategy for further wind energy expansion.

1. Introduction

Wind power is subject to fluctuations across various spatial and temporal scales. This spatiotemporal variability is caused by differences in large-scale airflow regimes, which are superimposed by regional and local circulations. As a result, the wind speed (WS) variation near the ground ranges from less than a second to several decades [1,2,3].

The influences of airflow regimes are reflected in complex time series of wind speed, which are composed of multiple components [4]. Among the components that shape the spatiotemporal variability of wind speed are the daily cycle [5], components associated with weather changes [6,7,8], seasonal components reflecting intra-annual variability [5,9], interannual variability [2,10,11], and long-term changes quantified as trend components [12,13,14,15], as well as components that cannot be attributed to any determinism [16].

The proportions and characteristics of the different time series components are not the same across the globe [17]. They vary regionally. While wind speed time series in the tropics are dominated by a daily cycle, outside the tropics, wind speed time series are typically characterized by a pronounced annual cycle. For example, in Europe, wind speed time series are largely shaped by an annual cycle that peaks in winter, primarily due to the intensified westerlies prevailing in the Northern Hemisphere during that season [11]. As summer approaches, mean wind speed decreases significantly.

While large-scale atmospheric flow regimes determine the overarching spatiotemporal characteristics of the European wind resource, the continent’s geographic extent means it is simultaneously influenced by multiple weather patterns, which are among the main drivers of spatiotemporal variability in wind resources [18,19]. These regimes not only exert a strong influence on the day-to-day variability of wind speed across Europe but also modulate the intra-annual variation of wind power availability [11].

In the context of wind energy utilization, complementarity refers to the interplay of wind conditions at different sites or times, where fluctuations in wind availability balance each other out [20,21]. This means that when wind power is low at one location and little electricity is generated, complementary sites may have high wind availability and generate more power—and vice versa.

The types of complementarity among renewable energies are referred to as spatial complementarity, which arises from the fact that different regions and sites have varying wind resources. Another is temporal complementarity, which results from sub-hourly, hourly, daily, multi-day, or seasonal differences in the availability of wind resources. When spatial and temporal complementarity are considered together, this is referred to as a spatiotemporal analysis of complementarity [22].

It has been proposed to use wind energy in a complementary way because the spatiotemporal variability of wind power generation can lead to supply shortage [23,24], electricity surplus [25], and poses a barrier to maintaining a consistently high share of wind energy in the electricity mix [26,27,28,29,30].

Among the approaches reviewed to reduce wind resource variability through complementary use [31], cross-border and interregional utilization of wind energy has been proposed to smooth regional and local fluctuations. The potential to mitigate multi-day variability of wind resources in Europe—driven by changing weather patterns—has been demonstrated by either installing wind turbines in regions with contrasting weather regimes or distributing them across Europe’s peripheral areas [32].

Various methods and metrics were used to determine the complementarity between renewable energy sources like wind, solar, hydro, and biomass. In most cases, the complementarity between wind and solar energy was quantified by different types of correlation coefficients. Among the applied coefficients are the Pearson correlation coefficient [5,9,11,33,34,35,36,37,38,39,40,41,42,43,44,45,46], the Spearman correlation coefficient [47,48,49,50,51,52], and the Kendall correlation coefficient [52,53,54,55,56,57,58,59,60,61,62,63,64,65,66].

The Pearson correlation coefficient measures linear co-movement, regardless of scale. The Spearman and Kendall correlation coefficients measure monotonic co-movement, considering the rank order of values. None of these coefficients directly capture the magnitude of differences and the dependence structure between different variables [67].

Cantor et al. [68] summarized issues that arise from the use of correlation coefficients in the quantification of complementarity. These include, among others, that negative correlation coefficient values do not always imply complementarity, that the magnitude of the correlated quantities matters, and that correlation coefficients are limited to the comparison of two time series.

Interpreting the correlation results reported for wind complementarity is often challenging. In many cases, the reported values of the correlation coefficients fluctuate between −0.5 and 0.0 on time scales ranging from minutes to days [35,59], indicating weak to no correlation. Lower correlation coefficient values up to −0.9 were reported for monthly wind–solar complementarity across Europe [69]. In any case, correlation does not allow for a temporally localized interpretation of the results.

Other commonly applied approaches to determining complementarity include variation-based indices [45,53,56,68,70,71], copula-based correlation methods [11,52,65,72,73], and approaches that additionally apply normalization to enhance the comparability of time series [35,63,65,66,71].

These methods have been proposed to optimize various aspects of wind energy utilization. This includes the selection and interconnection of complementary wind power plant sites [74], the planning of hybrid power systems, such as wind–solar [75], wind–solar–hydro [35], and wind–hydro–biomass combinations [76], the stabilization of wind power feed-in [77], the reduction of grid congestion [78,79], and curtailment [80]. The stabilization of wind power feed-in through reduction of spatiotemporal wind power variability leads to a more continuous electricity supply, which improves grid operation and management [81].

Building on previous approaches—which are limited in their ability to quantify wind resource complementarity in large wind turbine fleets due to the complex, nonlinear superposition of various wind time series components—this study investigates whether combining singular value decomposition (SVD) and spectral methods [82], such as Fourier [83] and wavelet analysis [4], can yield new insights into wind resource complementarity in large turbine fleets by identifying dominant wind patterns and their variability.

The tested hypothesis is that by reducing the dimensionality in the wind power density (WPD) time series, essential structures are summarized in a small number of components, making global and latent patterns in the data visible that cannot be revealed through pairwise comparisons between wind turbine sites. The component matrices are intended to enable the visualization of complex interrelationships that remain hidden when using large similarity matrices from pairwise comparisons. The aim of the study is to find wind turbine sites that provide complementary contributions to wind energy generation at times when other wind power plants deliver little or no output.

Germany is chosen as the study area for this novel approach because, compared to many other countries, it has a large off- and onshore wind turbine fleet and long-term, hourly resolved WS data from the Wind Speed Complementarity Model (WiCoMo) are available at a very high spatial resolution [44].

2. Materials and Methods

2.1. Study Area and Wind Speed Data

The study area Germany covers about 357,000 km², and its wind turbine fleet includes both offshore and onshore wind turbines. The analysis incorporates 29,807 actual wind turbine sites, of which 1,664 sites are offshore—mostly grouped in wind farms.

Since operational data from wind turbines are considered trade secrets and are therefore not available for a comprehensive complementarity analysis, WiCoMo output was used for this study. WiCoMo is a three-dimensional statistical-empirical model that provides hourly WS data on a 25 m × 25 m grid for Germany [44].

The analysis period covers ten years, from 1 January 2015 to 31 December 2024. Since the analysis also included the calculation of mean annual cycles, wind speed values from 29 February 2016, 2020, and 2024 were removed. As a result, a matrix with dimensions of 87,600 (hourly values) × 29,807 (wind turbine sites) wind speed values was available for analysis.

In addition to the WiCoMo validation described in a previous study [44], which was based on data from 508 measurement stations of the German Meteorological Service, an operator of wind turbines (Das Grüne Emissionshaus, Freiburg im Breisgau, Germany) provided hourly wind speed data from 83 turbines grouped at 12 sites. These data were used to strengthen the significance of the following results. The validation was carried out for the period 2017–2021 and yielded results for the mean absolute error of MAE = 0.50 (mean value) ± 0.33 (standard deviation) m/s and a Pearson correlation coefficient of R = 0.91 ± 0.02.

The distribution of wind turbines within the study area is shown in Figure 1a. Wind turbine density is highest in the northwest and west, decreasing towards the east and south. The mean wind power density (WPD_mean), calculated from WiCoMo output for the study period, is presented in Figure 1b. It shows that the highest WPD_mean values at 100 m above ground, reaching up to 575 W/m², are achieved by offshore wind turbines in the North Sea and Baltic Sea. Further south, WPD_mean values are generally lower, down to 40 W/m², due to increased surface roughness and greater distance from the coast. Only at higher-elevation sites, mostly in low mountain ranges, are similarly high WPD_mean values observed as those found near the coast onshore.

To investigate wind spatiotemporal complementarity within the study area, the following main steps were carried out (Figure 2): (1) Calculation of WPD from hourly wind speed data at actual wind turbine sites; (2) application of SVD to the WPD matrix (dimensions 87,600 × 29,807); (3) computation of the spatial (Sp) and temporal (Te) information derived from the SVD components; (4) derivation of spatial and temporal complementarity from Sp and Te; (5) quantification of spatiotemporal wind complementarity at different scales; (6) execution of wavelet analysis to identify dominant time scales in the SVD components.

2.2. Analysis of Wind Complementarity

Wind power density was calculated from hourly wind speed at 100 m altitude (WS_100m) at the actual wind turbine sites. To minimize the influence of wind turbine characteristics on the analysis of complementarity, a uniform hub height of 100 m was used for all analyzed sites:

$$WPD={\displaystyle \frac{1}{2}}·\rho ·{WS}_{100m}^3$$

(1)

where $\rho$ is the air density kept constant at a value of 1.225 kg/m³ [11]. WPD_mean in the study area was calculated from all WPD time series.

For all hours with the wind turbine location-specific WS_100m > 25 m/s, WS_100m is set to 0 m/s to account for wind turbine shutdown due to excessive wind loading.

The WPD calculation was then restricted to 3 m/s ≤ WS_100m ≤ 12 m/s, representing the typical cut-in and rated speed range and considered turbine saturation effects. This restriction resulted in maximum hourly WPD values of 1058.4 W/m².

The complementarity between the analyzed wind turbine sites was assessed based on SVD [84] of the WPD matrix consisting of 87,600 hourly WPD values and 29,807 location-specific WPD time series:

$$WPD=U·\mathrm{S}·V^T=\sum _{j=2}^{10}{s}_j·u_j·v_j^T$$

(2)

where S is a diagonal matrix with singular values s. The columns of U represent the left singular vectors (temporal modes) of the SVD components (u); the columns of V contain the right singular vectors, i.e., the spatial component-specific modes (v). $V^T$ is the Hermitian transpose of V.

The WPD matrix was decomposed into j = 2, …, 10 orthonormal (orthogonal and normalized to a length of 1) and independent components to gain an overview of the distribution of explained variance (EV) across the components. The sorted component-specific EV_k (k = 1, …, 10) was calculated as

$${EV}_k={\displaystyle \frac{s_k^2}{\sum s_k^2}}·100.$$

(3)

To enable meaningful interpretation of the orthonormal SVD components, the singular vectors were multiplied by the corresponding singular values. As a result, they are no longer normalized but remain orthogonal and independent from one another.

Based on the ${EV}_k$ values (k = 2, …, 10), the most suitable number of SVD components for analyzing wind complementarity was determined. The key criteria were the EV distribution across the components and the spatial patterns revealed by the components within the study area.

The spatial patterns Sp associated with the SVD components were analyzed as

$${Sp}_k=s_k·v_k.$$

(4)

The spatial complementarity (${SpC}_k$) of ${Sp}_k\left(i\right)$ for k > 1 with ${Sp}_1\left(i\right)$ (${SpC}_k\left(i\right)$) was assessed as

$${SpC}_k\left(i\right)=\left\{\begin{array}{cc}{Sp}_k\left(i\right),& ifsign\left({Sp}_k\left(i\right)\ne {Sp}_1\left(i\right)\right)\\ 0& \end{array}\right.,$$

(5)

where i = 1, …, 29,807 and represents the analyzed wind turbine sites.

The temporal structure Te of the SVD components was determined as

$${Te}_k=s_k·u_k.$$

(6)

Regardless of the number of components, ${Te}_1$ is non-negative due to the non-negative nature of WPD. It contains a consistently positive WPD pattern that largely aligns with known physical processes. Its contribution to total WPD can be meaningfully expressed in absolute units (i.e., W/m²).

All other ${Te}_k$, with k > 1, contain both positive $\left({Te}_{k,pos}\right)$ and negative $\left({Te}_{k,neg}\right)$ values. These values do not represent independent energy sources but indicate deviations from the dominant pattern. Specifically, they describe

-

amplifications of ${Te}_1$ in certain time periods, and wind power density exceeds what is explained by ${Te}_1$ alone (positive contributions);

-

attenuations of ${Te}_1$ at other times, and wind power density is reduced compared to ${Te}_1$ (negative contributions).

These characteristics are physically meaningful, as near-surface wind fields inherently exhibit complex spatiotemporal patterns that can lead to simultaneous amplifications and attenuations, e.g., due to regional and local pressure systems or terrain.

While the amplification of ${Te}_1$ = ${Te}_{1,pos}$ by ${Te}_{k,pos}$ (k > 1) is interpreted as complementary boost of ${Te}_{1,pos}$, the regional and local attenuations of ${Te}_{1,pos}$ induced by ${Te}_{2,neg}$ to ${Te}_{4,neg}$ lead to a reduction of ${Te}_{1,pos}$.

The SVD can thus provide valuable insights into the structure of the WPD data in the study area. While ${Te}_{1,pos}$ can be directly interpreted as a contribution to physical energy production, the other components can be understood as modulators of the dominant pattern. Their importance lies in making the complexity [85] and complementarity of the wind power patterns visible.

The sum of ${Te}_{k,pos}$ was equated to the magnitude of the original data to quantify the contributions of the wind turbine sites in W/m² to the total sum of WPD in the study area:

$$\sum _{k=1}^K{Te}_{k,pos}≔WPD$$

(7)

where K is the total number of SVD components.

The quantification of the SVD component-specific contributions ($C_k\left(i,t\right)$, t is the time) to WPD follows the approach

$$C_k\left(i,t\right)={\displaystyle \frac{{Te}_{k,pos}\left(i,t\right)}{\sum _{k=1}^K{Te}_{k,pos}\left(i,t\right)}}·WPD\left(i,t\right)$$

(8)

The procedure used to calculate WPD from the WiCoMo data and to decompose the resulting WPD matrix was repeated for the measured 83 wind speed time series available from the 12 wind turbine sites used for additional validation. The SVD components from these measured time series were compared to the SVD components obtained from the decomposition of the WiCoMo matrix.

The continuous wavelet transform [86] was used to obtain information about the time-frequency content of ${Te}_k$:

$$W_k\left(a,b\right)=\underset{-\infty }{\overset{\infty }{\int }}{Te}_k\left(t\right){·\psi }_{a,b}^{*}\left(t\right)·dt$$

(9)

where W are the SVD component-specific wavelet coefficients available as complex numbers, a is the scale factor, which defines the frequency of the wavelet function, b is the translation factor indicating the position of the wavelet on the time (t) axis, $\psi$ is the analytic Morlet wavelet function, and the * indicates the complex conjugate.

The analytic Morlet wavelet was used because it is based on a complex exponential oscillation that is modulated with a Gaussian function. It allows optimal time–frequency localization, has very good frequency resolution, is analytical, i.e., phase and amplitude can be extracted cleanly, and is easy to scale and interpret [86].

The wavelet coefficients were calculated for a = 1, …, 448 scales, producing information on the time–frequency behavior of the SVD components from 2 h to 3.7 years. This scale range includes the highest frequency (0.5 Hz, cycles/hour) that can be resolved from hourly data. The analysis is limited to 3.7 years because the wavelet function is not only dilated but also translated along the WPD time series. Wavelets near the beginning and end of the time series are affected by edge effects, depending on the scale. The longer the scale, the greater the impact of the edges.

The impact of the edges on $W_k\left(a,b\right)$ was quantified using the cone of influence [86]. This approach was applied to exclude all wavelet coefficients potentially affected by edge effects, ensuring that uncertainty in the resulting time–frequency representation of the WPD data is minimized.

The real part of $W_k$ $\left({Re}_k\left(a\right)\right)$ represents the scale-specific temporal course of the wavelet coefficients in the study period.

Wavelet variance spectra (${VS}_k$) were then calculated from $W_k$ [87]:

$${VS}_k\left(a\right)={\displaystyle \frac{1}{N}}{\sum }_b{\left|W_k\left(a,b\right)\right|}^2,$$

(10)

where N = 448 is the number of scales. As ${VS}_k$ differ greatly in magnitude, they were normalized to [0, 1] for better comparability.

All calculations and analyses were performed with Matlab 2024b software (The MathWorks Inc., Natick, Massachusetts, United States).

3. Results and Discussion

3.1. Singular Value Decomposition of Wind Power Density Matrix

Using only two components (k = 2) did not sufficiently capture the spatial WPD variability. The first component accounted for more than 94% of EV, making it overly dominant. Starting from a decomposition of the WPD matrix into five or more components (k = 5, …, 10), the spatial WPD patterns became highly fragmented and were shaped by small-scale, local characteristics, which were associated with ${EV}_k$ values of less than 1.0%.

Since the decomposition into four components (k = 4) accounted for the wind energy sites located in the low mountain ranges in the central and southern parts of the study area, this decomposition was preferred over the three-component version (k = 3) for further analysis. The decomposition into three components failed to identify productive sites known in the low mountain ranges common in the study area, as the three components primarily represented sites in the northern regions.

Using four SVD components, ${EV}_1$ accounts for 89.1% of the explained variance, dominating the spatiotemporal variability of WPD at the analyzed wind turbine sites. The shares of explained variance for the subsequent components are much lower: 5.5% for the second component, 3.6% for the third, and 1.7% for the fourth.

The procedure described for selecting the number of SVD components was chosen because standard methods for determining the optimal number of components, such as the Kaiser criterion [88] or the Scree test [89], did not adequately reproduce the spatiotemporal characteristics of the wind field in the study area. This is particularly the case when one component dominates EV.

The selection of the number of components is therefore based on both the EV distribution among the components and expert knowledge of the near-surface wind field in the study area. The appropriate number of components is dataset-specific and must be re-evaluated for each analysis, depending on the characteristics of the near-surface wind field.

3.2. Temporal Structure of Wind Power Density

The first SVD component, which explains the largest share of variance in the WPD matrix, is dominated by the annual cycle of WS_100m, which largely determines the wind energy availability across Central Europe [8,11,32]. The annual cycle is well represented by the real part (${Re}_1$) of the coefficients at scale a = 388 resulting from the wavelet transform of ${Te}_1$ (Figure 3a). The units on both axes are arbitrary to allow comparability between the different quantities. All values of ${Te}_1$ and ${Re}_1$ are positive.

This non-negative property of ${Te}_1$ does not apply to ${Te}_2$ (Figure 3b) through ${Te}_4$ (Figure 3d), which fluctuate around the zero line and thus also take on negative values (green lines). The positive values (blue lines) of ${Te}_2$ to ${Te}_4$ are interpreted as amplifying contributions to ${Te}_1$, since ${Te}_2$ to ${Te}_4$ are, by definition, orthogonal—and thus independent—from ${Te}_1$. This amplification is considered a complementary contribution. The negative values of ${Te}_2$ to ${Te}_4$ attenuate the expression of ${Te}_1$ and cannot contribute to complementarity.

In Figure 3d, in addition to ${Te}_4$, the real part of ${Te}_4$ (${Re}_4$) combined with the wavelet scale a = 148 is also shown. This wavelet scale corresponds to a time scale of 48 h, which reflects a typical frequency of weather changes in Europe [6,7,8,18]. As a result, it exhibits significantly higher-frequency fluctuations compared to the wavelet scale used to approximate the annual cycle of ${Te}_1$.

The four normalized wavelet variance spectra calculated from ${Te}_1$ to ${Te}_4$ are shown in Figure 4. They reveal that ${Te}_1$ exhibits two distinct ranges with particularly high shares of variance: one corresponding to scales between half a week and several weeks, similar to the wavelet spectra presented in previous studies [89,90], and another associated with the annual cycle of wind speed [82,89].

In contrast, ${Te}_2$ shows significantly lower wavelet variance shares at the annual cycle scales compared to ${Te}_1$. Its highest variance contributions are found at scales ranging from several days to a few weeks.

Only very small shares of variance at the annual cycle scales are observed for ${Te}_3$ and ${Te}_4$. For these two components, the dominant scales lie between half a day and one week, indicating that their temporal variability is largely driven by weather changes.

To illustrate the central tendencies of ${Te}_{1,pos}$ to ${Te}_{4,pos}$ throughout the year, Figure 5 shows their mean monthly daily cycles and their relative contributions to the sum of ${Te}_{1,pos}$ through ${Te}_{4,pos}$ ($\sum {Te}_{k,pos}$).

The mean monthly daily cycles of ${Te}_{1,pos}$ show maximum values during the nighttime hours in all months (Figure 5a). During the daytime, its share decreases on average by between 2.8% in December and 8.6% in April. The shares of ${Te}_{1,pos}$ are by far the highest throughout the year compared to ${Te}_{2,pos}$ to ${Te}_{4,pos}$. Over the study period, the average ${Te}_{1,pos}$ share is 73.0 (mean) ± 2.5% (standard deviation).

The pattern of reduced WPD during the day results from a marked wind speed decrease. Daytime wind speeds during the summer months have been reported in earlier studies for the flat, northeastern part of the study area, showing a similar pattern [91,92]. This diurnal cycle is related to the development of the atmospheric boundary layer over flat land surfaces, transitioning from turbulent mixed conditions during the day to stable, weakly turbulent or even laminar flows at night. Particularly during the morning transition period, strong WPD gradients can occur.

${Te}_{2,pos}$ shows significantly lower values (Figure 5b). Its average share over the study period is 5.6%. Here as well, nighttime values are higher than daytime values in all months. The mean daily difference ranges from 1.8% in September to 3.1% in August. It is higher in the spring (March, April, May) and summer months (June, July, August) than in the fall (September, October, November) and winter months (December, January, February).

The daytime decreases in ${Te}_{1,pos}$ and ${Te}_{2,pos}$ are compensated by higher values of ${Te}_{3,pos}$ (Figure 5c) and ${Te}_{4,pos}$ (Figure 5d). These two components show their highest values during the day. The difference in the mean monthly daily values of ${Te}_{3,pos}$ ranges from 1.6% (December) to 5.8% (August), showing a pronounced annual cycle. The average share of ${Te}_{3,pos}$ over the study period is 9.5 ± 1.5%. A similar pattern is found for ${Te}_{4,pos}$. Its mean daily differences between day and night range from 2.2% (January) to 6.0% (April). With an average share of 11.9 ± 2.2%, ${Te}_{4,pos}$ contributes more than ${Te}_{3,pos}$.

The patterns of ${Te}_{3,pos}$ and ${Te}_{4,pos}$, featuring a daytime WPD maximum, resemble the average diurnal cycle of wind speed during summer as presented in a previous study based on measurements from the southern part of the study area [92].

In addition to the mean monthly daily cycles, mean annual cycles are evident in ${Te}_{1,pos}$ to ${Te}_{4,pos}$. While ${Te}_{1,pos}$ and ${Te}_{2,pos}$ show the highest values in the winter months, when WPD also reaches its peak, ${Te}_{3,pos}$ and ${Te}_{4,pos}$ show the highest values in the summer months, indicating that their relative importance increases during the summer—when WPD is at a minimum across Europe—and decreases during the winter months, when the westerly flow in the northern hemisphere over Europe is strongest [93].

The higher ${Te}_{3,pos}$ and ${Te}_{4,pos}$ contributions in the summer months can be attributed to regional and local circulations whose properties differ from the large-scale circulation described by ${Te}_{1,pos}$. Since the $WPD$ data originate from WiCoMo, which was parameterized based on wind speed measurements conducted by the German Meteorological Service, among other sources, and these do not systematically capture mountain–valley wind systems, it is not possible to assert that the high contributions of ${Te}_{3,pos}$ and ${Te}_{4,pos}$ result from specific tertiary circulations.

The corresponding values of the first component from the SVD decomposition of the 83 wind speed time series measured at wind turbines show a similar pattern to ${Te}_{1,pos}$ and ${Te}_{3,pos}$. In addition to the annual cycle, the average diurnal cycles of ${Te}_{1,pos}$ and ${Te}_{3,pos}$ are well represented for most months. Although the agreement between the evolution of ${Te}_{1,pos}$, ${Te}_{3,pos}$, and the SVD component from the validation dataset is not always exact—likely due to differences between the datasets—the observed similarity is remarkably high, especially considering the limited number of validation time series available.

Agreement with ${Te}_{2,pos}$ is absent because there are no validation sites located within the ${Te}_{2,pos}$ region, which implies that this region cannot be represented. Agreement with ${Te}_{4,pos}$ is noticeable with respect to the annual cycle. However, the average monthly diurnal cycles differ, which can be attributed to the low number of validation sites within the ${Te}_{4,pos}$ area, particularly in the Baltic Sea region. A larger number of available validation sites distributed throughout the study area would be desirable for future analyses of this type to increase the range of validation possibilities and make the validation statements more robust.

The mean annual shares of ${Te}_{k,pos}$ are not always the same (Figure 6). They fluctuate from year to year over the study period, which is a consequence of the interannual WPD variability in Europe [2,94,95]. The largest fluctuations are observed for ${Te}_{1,pos}$, which also accounts for the greatest WPD share. The largest year-to-year change of ${Te}_{1,pos}$ occurs between 2015 and 2016, amounting to 13.5%. Over the study period from 2015 to 2021, the range of annual ${Te}_{1,pos}$ values is approximately 15.2%.

The ranges of year-to-year fluctuations are significantly smaller for ${Te}_{2,pos}$ (4.4%), ${Te}_{3,pos}$ (2.0%), and ${Te}_{4,pos}$ (0.9%). This indicates that ${Te}_{1,pos}$ primarily reflects the interannual WPD variability and determines the general level of wind energy availability in the study area.

To illustrate the significance of ${Te}_{4,pos}$ for WPD_mean by example, Figure 7 shows the course of WPD_mean and ${Te}_{4,pos}$/${\sum Te}_{k,pos}$ for the year 2023, which was the year with the highest WPD_mean values in 2015–2024.

During the winter months, when the large-scale westerly flow is most pronounced and ${Te}_{1,pos}$ dominates the structure of WPD_mean, the share of ${Te}_{4,pos}$ is lower in most hours compared to the summer months, when the westerly flow weakens. At such times, ${Te}_{4,pos}$ can contribute up to 60% of the available WPD_mean, as illustrated by the inset for the months of May and June.

From this representation, it can be concluded that ${Te}_{1,pos}$ determines the general annual level of WPD_mean, but ${Te}_{2,pos}$ to ${Te}_{4,pos}$ make substantial contributions to WPD_mean at certain times. From the perspective of complementarity, wind turbine sites that contribute substantially to WPD during such periods are certainly an important complement to those where ${Te}_{1,pos}$ contributes little at the same time.

3.3. Spatial Pattern of Wind Power Density

The spatial patterns associated with the four SVD components are shown in Figure 8. As the prevailing westerlies determine the spatial distribution of wind speed over Europe [32], dominant ${Sp}_1$ exhibits the highest WPD values at offshore sites and at near-coastal onshore sites in the northern parts of the study area.

Toward the east and south, WPD values decrease markedly (Figure 8a). A clearly discernible north–south gradient in WPD is revealed by ${Sp}_2$, with the northern sites showing the lowest WPD values. The highest WPD values occur in the southwest and central parts of the study area (Figure 8b). The western sites dominate in ${Sp}_3$, where WPD decreases from west to east (Figure 8c). The WPD pattern of ${Sp}_4$ suggests that the elevated areas of the Black Forest—the largest and highest low mountain range in the study area, located in the southwest—as well as the offshore sites in the Baltic Sea, exhibit the highest WPD values in this component (Figure 8d).

The patterns of ${Sp}_2$ to ${Sp}_4$ are interpreted as either amplifying or attenuating ${Sp}_1$. For example, in ${Sp}_2$, the southwestern wind turbine sites contribute most strongly to the amplification of ${Sp}_1$. In ${Sp}_4$, the highly elevated sites in the southwestern low mountain ranges and the offshore sites in the northeast of the study area contribute most to this amplification.

The wind turbine sites that behave complementarily to ${Sp}_1$ are shown in Figure 9. All wind turbine sites that do not contribute to the amplification of ${Sp}_1$ are not displayed. For ${SpC}_2$, this includes all offshore and onshore turbines located in the northernmost part of the study area (Figure 9a). These turbines do not contribute to ${Sp}_1$ because ${Sp}_2$ is negative for them, and ${Sp}_1$ dominates. Toward the south, the complementary contribution of wind turbines to ${Sp}_1$ increases. The highest contributions—those exceeding the 90th percentile (P₉₀)—are provided by turbines located in the southwestern part of the study area.

A completely different pattern emerges for the complementary contributions from ${Sp}_3$. Wind turbines located in the western part of the study area provide the highest complementary contributions to ${Sp}_1$ (Figure 9b). Turbines located in the eastern part make no complementary contribution to ${Sp}_1$.

For the complementary contributions from ${Sp}_4$, a pattern emerges that is oriented toward the south and east (Figure 9c). In this component, wind turbines located in the south and offshore in the northeast contribute most strongly to the amplification of ${Sp}_1$. All onshore turbines in the northwest provide no complementary contributions, as ${Sp}_1$ also dominates at their sites.

The reduction of site-specific WPD time series to a few SVD components enables rapid, large-scale assessment for further planning of wind energy expansion. It immediately becomes apparent which parts of the study area are suitable for the development of complementary sites. This implies that sites contributing to wind energy generation at the same time are also identified. At such sites—for example, in the northern part near the North Sea coast—additional wind turbines significantly contribute to wind energy yield. However, their contribution is of a mostly supplementary nature, contributing to ${Sp}_1$.

The assumption that all wind turbine sites where ${Sp}_k\left(i\right)$ with a negative sign does not provide a complementary contribution to the amplification of ${Sp}_1\left(i\right)$ opens a wide range of possibilities for analyzing the spatiotemporal contributions of the wind turbine fleet within the study area.

For example, Figure 10a shows all turbine sites where ${Te}_{2,pos}$ to ${Te}_{4,pos}$ contribute only very little (combined contributions less than 20%) to the amplification of ${Te}_1$.

As another example, Figure 10b displays all wind turbine sites where no amplification of ${Te}_1$ occurs through ${Te}_2$ to ${Te}_4$, as all of them are negative at those sites.

3.4. Spatiotemporal Pattern of Wind Power Density

The wind turbine sites at which $C_2$ to $C_4$ exceed the respective 90th percentile during the study period are shown in Figure 11. The displayed values, normalized by WiCoMo values (Equation (7)), are given in W/m². $C_2$-sites contributing most strongly are located from the center of the study area down to its southern border (Figure 11a). The $C_3$ sites with the highest complementary contributions are found in the western part of the study area (Figure 11b). The $C_4$ sites with the highest contributions are situated in the south at elevated low mountain range sites and in the far northeast in the Baltic Sea (Figure 11c). The wind turbine sites in the Baltic Sea show the highest complementary contributions across all components.

4. Conclusions

The presented results show that the combination of wavelet analysis and singular value decomposition can provide valuable and expansion-relevant information on the spatiotemporal variability of wind resources in a large area with numerous wind turbines. In contrast to correlation-based approaches, which only allow for global statistical statements on complementarity over entire time periods mostly for pairs of sites, the proposed approach enables site-specific and temporally localized assessments of complementarity for many individual operating wind turbines and candidate turbine sites.

The singular value decomposition provides valuable insights into the structure of the wind power density data. While the first component can be directly interpreted as a contribution to physical energy production, the subsequent components should be understood as modulators of the dominant pattern. Their significance lies in revealing the complexity and complementarity of the wind power patterns. They are modulation patterns: they indicate how much more or less wind power is present relative to the main structure.

The spatial specification of complementary contributions makes it possible to identify sites where, at first glance, lower average wind speed values do not suggest a productive wind resource. Through the decomposition into components, which highlight their site-specific importance, it becomes possible to better assess the value of such sites for wind power supply, as the approach enables high-resolution temporal and spatial analyses that allow for conclusions beyond global statistical site metrics.

The area-wide overview of site characteristics and wind potentials can be used for the optimizing strategic planning of grid infrastructure, storage systems, and combined generation structures—such as hybrid wind–solar plants—since the approach is easily transferable to solar energy site assessment as well. In a resource-centered approach, it helps to find zones in which the hybrid use of renewable energies should be prioritized to protect climate as efficiently as possible [85].