Multidecadal offshore wind power variability can be mitigated through optimized European allocation
Charlotte Neubacher
Dirk Witthaut
Jan Wohland
Wind power is a vital ingredient for energy system transformation in line with the Paris Agreement. Limited land availability for onshore wind parks and higher wind speeds over sea make offshore wind energy increasingly attractive. While wind variability on different timescales poses challenges for planning and system integration, little focus has been given to multidecadal variability. Our research therefore focuses on the characteristics of wind power on timescales exceeding ten years. Based on detrended wind data from the coupled centennial reanalysis CERA20C, we calculate European longterm offshore wind power potential and analyze its variability focusing on three locations with distinct climatic conditions: the German North Sea, the Greek Mediterranean and the Portuguese Atlantic coast. We find strong indications for two significant multidecadal modes that are identified consistently using two independent spectral analysis methods and in the 20year running mean time series. In winter, the longterm evolution of wind power and the North Atlantic Oscillation (NAO) are directly linked in Germany and Portugal. While German North Sea wind power is positively correlated with the NAO (r=0.82), Portuguese Atlantic coast generation is anticorrelated with the NAO ($r=\mathrm{0.91}$). We evaluate the corresponding potential for spatial balancing in Europe and report substantial benefits from European cooperation. In particular, optimized allocations off the Portuguese Atlantic coast and in the German North Sea allow to reduce multidecadal generation variance by a factor of 3–10 compared with countrylevel approaches.
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A fundamental transformation of our energy system towards renewable energy sources is inevitable (Rogelj et al., 2015). Wind power is an essential element of this transition: It is CO_{2} neutral during operation, costs are competitive (IRENA, 2019), and resource availability exceeds demand substantially (IEA, 2019, p. 50). Offshore wind power outperforms onshore wind power in some relevant aspects. First, the wind blows more steadily resulting in a higher number of full load hours (IEA, 2019). Second, land availability and public acceptance is less limiting if wind turbines are placed offshore far away from residential areas. Until 2020 7.5 GW offshore wind power capacity had been installed in Germany and approximately 25 TWh of electricity were generated in 2019 (WindGuard, 2019). Moreover, the German government has increased its offshore wind power capacity expansion target by 5 to 20 GW in 2030 (BMWi, 2020). The longterm strategy of the European Union includes 240–450 GW of offshore wind power capacity in 2050 (IEA, 2019), such that a better understanding of wind power generation variability is of great interest.
The main technical challenge of wind power integration is its high variability on many different timescales, from minutes (Milan et al., 2013; Anvari et al., 2016; Haehne et al., 2018) over days, weeks and seasons (Heide et al., 2010; Staffell and Pfenninger, 2018; Bloomfield et al., 2018) to years (Collins et al., 2018), decades and multiple decades (Wohland et al., 2019a). Appropriate forecasts are essential to cope with shortterm variability (Foley et al., 2012), in order to control backup power plants (Faulwasser et al., 2018), to minimize curtailment of wind turbines and redispatch (Wohland et al., 2018) and to avoid large deviations of the power grid frequency (Gorjão et al., 2020). While short time scales are well explored, less literature is available for low frequency variability. Earlier studies establish mechanisms through which low frequency climate variability can induce low frequency variability in wind power generation. For instance, the troposphere and stratosphere respond to the low frequency component of the North Atlantic Oscillation (NAO) as shown in Omrani et al. (2016) and NAO variability has been linked with wind speeds and wind power generation in Great Britain (Brayshaw et al., 2011; Ely et al., 2013). Further results on natural low frequency variability of near surface wind speed over Europe have been given in Bett et al. (2017, 2013).
Wohland et al. (2019a) provide a detailed analysis of such low frequency modes and their impacts on wind power potential in Germany. They use the multi taper method (MTM) and report significant multidecadal variability with a period 25 to 50 years. A strong influence of the NAO has been shown for generation during the winter months. These effects have a nonnegligible influence (approximately ±5 %) on the overall lifetime generation of wind turbines and consequently impact economic performance and investment decisions.
The present study investigates multidecadal variability in wind power potential, complementing and extending the available literature in three ways. First, it focuses on offshore wind power while Wohland et al. (2019a) investigated mainly onshore. Offshore wind conditions are different from onshore, most importantly wind speeds are higher. It would thus be conceivable that multidecadal variability does not matter offshore if wind speed changes dominantly occur in the upper part of the wind speed distribution where the turbine is in the rated regime. Second, it uses different statistical approaches which allow to verify whether earlier results are robust to methodological changes. Third, it looks at a larger domain and thereby allows to check whether intercountry transmission helps to mitigate renewable generation variability also on multidecadal timescales.
Our general approach is to isolate periodic signals from time series of the twentieth century offshore wind power potential, using two independent spectral analysis methods and an adapted noise hypothesis. We compare three different locations to verify whether multidecadal variability is a local phenomenon and to reveal the impact of largescale climate dynamics. Spectral analysis can not answer all questions as periodic signals can have offsets or may be fully outofphase. We therefore complement our analysis by calculating longtime running means to gain visual insight and to improve the interpretation of the spectral results. We provide a flowchart (see Fig. 1) to follow the data processing more easily.
2.1 Power potential
We compute wind power potential from 1900 to 2010 over Europe based on the wind components (u, v) from the coupled reanalysis CERA20C (ECMWF) at a height of 10 m. CERA20C is an ensemble reanalysis consisting of 10 members with a spatial resolution of 1.125 ${}^{\circ}\times \mathrm{1.125}$ ^{∘} and a temporal resolution of 3 h (Laloyaux et al., 2018). We use all ensemble members even though ensemble spread is limited.
As in Tobin et al. (2016), we assume a wind turbine hub height of 80 m and calculate hub heightwind speeds from 10 m wind components as
Wind power potential c_{P} is subsequently computed from the wind speed using a standard power curve (cf. Tobin et al., 2016) as
assuming a cutin velocity of s_{in}=3.5 m s^{−1}, a rated velocity of s_{r}=12 m s^{−1} and a cutout velocity of s_{out}=25 m s^{−1}. To isolate the effect of changes in resource availability, we leave the allocation of wind parks constant over the 20th century. In other words, we calculate the wind power potential during the twentieths century that could have been generated from a fixed wind park setup.
CERA20C features strong upward trends in wind speeds, in particular over the oceans (Wohland et al., 2019b). The trends are linked to the assimilation of marine wind observations and are likely spurious. We therefore subtract the linear trend from the wind power potential time series as in Wohland et al. (2019a). It was shown before that multidecadal wind generation variability of CERA20C, ERA20C and 20CR is very similar after subtracting the linear trend at least in Germany (Wohland et al., 2019a), thus justifying the use of a single reanalyses in this particular case. All three centennial reanalyses also show the same relationship with multidecadal winter NAO variability, suggesting that the underlying physical processes are equally well captured in the detrended data sets. Therefore, this study is based on CERA20C only.
2.2 Definition of the regions
In the German North Sea we use realworld installed capacities from the Open Power System Database, totalling 4.5 GW (OPSD, 2018). Real world data is lacking for the Greek Mediterranean, hence we follow (Karanikolas et al., 2011) who provide plausible estimates of suitable wind park locations and sizes. Wind parks are distributed mainly over seven grid boxes in the Aegean and capacity totals at 8.3 GW. Short of real world data and good scenarios in the Portuguese Atlantic Coast, we utilize each grid box in the area 10.25–11.375^{∘} W and 33.375–41.25^{∘} N with turbines of 100 MW. As verified later on, the precise siting of the turbines has negligible impact on multidecadal generation characteristics (see Fig. 4), justifying the different approaches taken in different regions. In Sect. 3.2, we assume uniform capacities of 100 MW in each grid box as we seek to investigate generation variability independent of installed capacity.
2.3 Spectral analysis
To gain insight into the statistical properties of the time series, we use two methods to analyze their spectrum, namely the periodogram and singular spectrum analysis (SSA). Additionally we define a noise model to quantify the significance of spectral peaks.
2.3.1 Power spectral density
A periodogram shows the power spectral density (PSD) of a time series which is the squared Fourier transformed (Vaughan et al., 2011)
where the power spectral density P_{xx}(ν) denotes the variance which occurs at a frequency ν for a time series with the values $\mathit{\{}{X}_{n}:\mathrm{0}\le n\le N\mathit{\}}$, length N and temporal resolution Δt.
As usual in real world time series, we expect the wind potential time series to be a superposition of several (periodic) signals and an underlying stochastic process. When investigating spectral properties, this means that the spectrum consists of several peaks plus background noise. The background together with effects of finitelength time series and nonsinusoidal signals complicates the identification of real signals. A test for statistical significance is therefore needed and we introduce a noise model in the following to which we apply a χ^{2}test to quantify the significance of single peaks.
2.3.2 Noise model
White or red noise is often assumed for the background signal in atmospheric time series analysis. The autoregressive process of order 1 (AR(1) process) is the simplest statistical model for a red noise time series (Mann and Lees, 1996). We therefore choose an AR(1) reference process ${Y}_{t}={\mathrm{\Phi}}_{\mathrm{1}}{Y}_{t\mathrm{1}}+{\mathit{\u03f5}}_{t}$ to model red noise contribution of the original time series X_{t}. The coefficients Φ_{1} and ϵ_{t} are obtained by the YuleWalker equations (Kirchgässner et al., 2012, p. 49–51) and fully determine the process (see Appendix A1 for details).
2.3.3 Singular spectrum analysis
Singular spectrum analysis (SSA) decomposes a signal in its eigenmodes and is particularly well suited for relatively short and noise time series as the wind potential time series we assess. Shortness is to be understood relative to the investigated time periods of multiple decades. Furthermore SSA enables detection of nonsinusoidal signals in contrast to Fourier transformed based analysis since the generating function is not restricted to sinusoidal functions. The decomposition is achieved by solving the eigenproblem of the signal's correlation matrix. Our approach follows Ghil et al. (2002) where the application of SSA is demonstrated for the Southern Oscillation Index. For consistency, we also normalize the time series by its standard deviation. To focus on longterm variability, we mute higher frequency variability, including the seasonal cycle, by resampling from 3 h to annual values. The window size M is set to 50 years due to the limited sample size of N=110 in line with Ghil et al. (2002) suggestion to avoid the hard limit of $N/M<\mathrm{1}$. This reduces the outcomes of SSA to modes with periods smaller than M (Vautard et al., 1992). We choose the singular value decomposition (SVD) estimate (Broomhead and King, 1986) to calculate the correlation matrix C_{X} (see Appendix A2 for detailed revision).
Diagonalizing C_{X} yields the eigenvalues $\mathit{\{}{\mathit{\lambda}}_{k}:\mathrm{1}\le k\le M\mathit{\}}$ which are ranked in decreasing order ${\mathit{\lambda}}_{\mathrm{1}}\ge {\mathit{\lambda}}_{\mathrm{2}}\ge \mathrm{\dots}\ge {\mathit{\lambda}}_{N}\ge \mathrm{0}$. Each eigenvalue λ_{k} of C_{X} corresponds to the variance of the time series in the direction of the corresponding eigenvector ρ_{k}, also known as empirical orthogonal functions or EOFs.
Ghil et al. (2002) also give a noise filtering method for SSA reported as Monte Carlo SSA (MCSSA): Basically we apply SSA to 1000 synthetic time series {Y_{t}} that were generated by the noise model, in our case it is the AR(1) process (see Appendix A3). For each synthetic time series, we calculate the correlation matrix C_{Y} using SVD and project it onto the eigenvectors ρ_{k} of the real data. Thus each diagonal entry of the projected matrices is a surrogate of the corresponding eigenvalue λ_{k} of the real data. Eigenvalues within the inner 90 % of the surrogates are rejected as noise, eigenvalues out of this range are reported as significant signals.
2.4 20 year running mean
Our spectral analysis is complemented by analysing the long term evolution of wind power potential. Therefore we take the 20year running mean of the time series. By averaging over 20 years, highfrequency variability is eliminated and the longterm evolution can be assessed. This provides a good proxy for lifetime energy generation since 20 years roughly correspond to wind turbine life times.
We define a 20year forward running mean as
where G(t) denotes the detrended time series and G_{20}(t) is the 20year running mean covering the period of 1900 to 1990. Extension beyond 1990 (i.e., mean from 1990 to 2009) is impossible due to data availability. The seasonal potential (${G}_{\mathrm{20}}^{\text{season}}\left(t\right)$) is calculated analougously by using the seasonal time series instead.
2.5 North Atlantic Oscillation
Following Wohland et al. (2019a) we compare winter North Atlantic Oscillation (NAO) with the long term evolution. This yields further insight into the relation of local wind power potential to the general circulation of the atmosphere. The NAO dominates climate variability in the North Atlantic sector, particular in winter, affecting weather as well as climate allover Europe (Marshall et al., 2001). We refer here to it as the first principle component of sealevel pressure over the area 20–80^{∘} N and 90^{∘} W – 40^{∘} E as detailed in Omrani et al. (2016). Our NAO index is computed from sealevel pressure data from the Hadley Center (Rayner et al., 2003) over the winter months December, January and February.
3.1 Spectral analysis reveals multidecadal variability
We show the PSD for three separated regions of Europe, namely (a) the Portuguese Atlantic Coast, (b) the Greek Mediterranean and (c) the German North Sea (see Fig. 2). All three reveal significant peaks on multidecadal timescales. The strongest peaks occur in Portugal where the periods ${P}_{\mathrm{1}}^{\text{POR}}\approx \mathrm{100}$ year and ${P}_{\mathrm{2}}^{\text{POR}}\approx \mathrm{20}$ year are statistically significant at the 99 % level. A third peak is visible around P≈8 years but will not be studied here as we focus on multidecadal peaks. While only one 99 % significant peak exists for the Greek Mediterranean (${P}_{\mathrm{1}}^{\text{GRE}}\approx \mathrm{50}\phantom{\rule{0.25em}{0ex}}\mathrm{years}$), there are two for the German North Sea (${P}_{\mathrm{1}}^{\text{GER}}\approx \mathrm{45}\phantom{\rule{0.25em}{0ex}}\mathrm{years}$ and ${P}_{\mathrm{2}}^{\text{GER}}\approx \mathrm{10}\phantom{\rule{0.25em}{0ex}}\mathrm{years}$). In other words, multidecadal variability is a large scale phenomenon that impacts different parts of Europe differently.
We investigate the evolution in all regions in more detail which provides further evidence for the existence of multidecadal peaks. Focusing on the seasonal evolution in Portugal, the peaks of seasonal PSD (see Fig. 3a–d) support the periods ${P}_{\mathrm{1}}^{\text{POR}}$ and ${P}_{\mathrm{2}}^{\text{POR}}$. The PSD of summer (b) and winter (d) both show peaks at ${P}_{\mathrm{1}}^{\text{POR}}$ and ${P}_{\mathrm{2}}^{\text{POR}}$. In spring (a), we observe a mode with period P^{POR}≈50 years which might be a harmonic of ${P}_{\mathrm{1}}^{\text{POR}}$. In winter another significant peak occurs with a period ${P}_{\mathrm{3}}^{\text{POR}}\approx \mathrm{11}$ years which is not mirrored in the annual values. The PSD of autumn (c) shows no peaks that are significant at the 99 % level but there is one which exceeds the 95 % significance threshold P≈30 years.
In addition, we report the eigenspectrum of the SSA for Portuguese Atlantic Coast. In Fig. 3e the eigenvalues (EV) are ordered by their rank and in (f) by the frequency of the belonging EOF. The structure of the eigenvalues in (e) show three groups formed by gaps and different slopes. The first six form a group that carries a large part of the variance (≈33 %) despite the small group size and is therefore of higher interest than the second group (7th–14th EV ≈26 % of total variance) or the last group (15th–34th EV ≈33 % of total variance). Lower eigenvalues (rank > 34) are indistinguishable from noise and consequently rejected by the noise hypothesis, which is represented by the surrogates and a χ^{2}test, providing another reason to focus on high eigenvalues. A closer look to the frequency ordered EVs (f) shows that the first and second EV correspond to the frequency ν_{1}=0.031–0.022 yr^{−1} (P≈30–45 years) and the fifth and sixth to the frequency ${\mathit{\nu}}_{\mathrm{2}}=\mathrm{0.052}\phantom{\rule{0.125em}{0ex}}{\mathrm{yr}}^{\mathrm{1}}$ (P≈20 years). While the latter matches perfectly with ${P}_{\mathrm{2}}^{\text{POR}}$, the former hardly fits ${P}_{\mathrm{1}}^{\text{POR}}$. This lacking fit is a consequence of the resolvable frequencies in SSA because a window size of M=50 years restricts the EOFs to frequencies $\mathit{\nu}\ge \mathrm{0.02}\phantom{\rule{0.125em}{0ex}}{\mathrm{yr}}^{\mathrm{1}}$. The first and second EV consequently correspond to the lowest resolvable frequency, further validating variability on very long time scales. Moreover, P≈30–45 years agrees well with the seasonal PSD results in spring and winter (see Fig. 3a) and autumn (see Fig. 3c). The third and fourth EV correspond to a relatively high frequency $\mathit{\nu}=\mathrm{0.333}\phantom{\rule{0.125em}{0ex}}{\mathrm{yr}}^{\mathrm{1}}$ (P≈3 years) and therefore do not contribute to multidecadal variability.
SSA of German North Sea and Greek Mediterranean confirm the reported periods of PSD as well (see Supplement Figs. S1 and S2). We have refrained from a detailed analysis of these sites in favour of a more detailed discussion of their interconnections, because we focus on balancing potential.
3.2 Significant multidecadal peaks everywhere across Europe
Whether or not multidecadal variability affects large areas determines its systemwide relevance. A PSD investigation reveals the existence of at least one significant multidecadal peak in every grid box in Europe (see Fig. 4). We show in (a) a map of the maximal amplitude in the multidecadal range for each grid box, additionally we give in (b) the variance of the full time series and in (c) the ratio of both as normalized amplitude. The normalized amplitude quantifies the relative importance of multidecadal variability and short term variability. In (a) the amplitudes located offshore are particularly high in contrast to those onshore. This clear demarcation of offshore and onshore areas weakens in the normalized amplitudes (c) because high amplitudes coincide with high variances in some areas (e.g., the North Sea). Both, amplitude and normalized amplitude, host a maximum in the Portuguese Atlantic coast (around 250 MWh^{2}). The amplitudes in the German North Sea are lower (around 50 MWh^{2}), and the high variance there is leading to even lower normalized amplitudes. In the Mediterranean isolated higher amplitudes (around 100 MWh^{2}) are shown as around the Greek islands in the Aegis. Due to moderate variance in the Mediterranean, the normalized amplitudes reveal isolated areas of medium height there, too. In order to ensure comparability between the locations, we have assumed the same capacity (100 MW) for each grid box in this investigation. This distinguishes the amplitudes here from those shown in Fig. 2 for Greek Mediterranean and German North Sea, where we use capacities of installed or planed power plants (see Sect. 2.2).
3.3 Timing of multidecadal variability differs vastly throughout Europe
The 20year mean confirms that offshore wind power potential in Portuguese Atlantic Coast, Greek Mediterranean and German North Sea show a similar period in multidecadal variability. Nevertheless, their timing differs vastly (see Fig. 5 for the normalized running mean of winter and Fig. 6 for the annual values). All three panels in Fig. 5 show multidecadal variability with an amplitude of ±6 %–10 % reaching maxima and minima at least once over the last century. The peaks and dips, however, do not occur simultaneously: While Portugal shows 8 % above average potential in 1960, Germany is 7 % below average at the same time (a, c). In fact, the longterm evolution in Portugal is inverse to the evolution in Germany. This similarity strongly suggest a common dynamical origin by a larger climate pattern. This common forcing could be the NAO as high negative/positive correlations of 20year mean NAO and Portuguese/German wind power illustrate ($\text{r}=\mathrm{0.91}$ and r=0.82, respectively). The link between NAO and wind potential is less pronounced in Greece (see Fig. 5b), likely because local and regional effects play a more important role in Greece compared to largescale atmospheric variability over the North Atlantic region.
3.4 Different multidecadal evolution allows for efficient balancing
The pronounced timing differences between wind power potential in the individual countries and their relation to the NAO suggests capability for spatial balancing. To study the positive effects from combining potential in different countries through inter country transmission, we define the total potential (G_{tot}) of wind turbines in two countries as
where $\mathit{\lambda}\in [\mathrm{0},\mathrm{1}]$ is a mixing parameter that describes the share of wind parks installed in country 1 and G_{1,2}(t) is the 20year running mean potential in the countries 1 and 2, respectively. A simple minimization of the variance of G_{tot} yields an optimum mixing λ_{opt}. Figure 6 shows total potential G_{tot} using λ_{opt} and the potential of the individual countries G_{1,2}. The total potential of the combinations in (a) and (b) has distinctly smaller variance than the potential of single locations (by a factor of 3–10). In contrast, combining Greece and Portugal (c) leaves the variance essentially unchanged. These different effects on variance can be conceptually understood: Assuming comparable values for the variances of both locations $\text{Var}\left({G}_{\mathrm{1}}\right(t\left)\right)\approx \text{Var}\left({G}_{\mathrm{2}}\right(t\left)\right)\approx \text{Var}\left({G}_{\mathrm{1},\mathrm{2}}\right(t\left)\right)$ and the optimal values of λ being of the order of $\mathrm{1}/\mathrm{2}$ (${\mathit{\lambda}}_{\text{opt}}\approx \mathrm{1}/\mathrm{2}$), the total variance is reduced to:
The values of Var(G_{1,2}(t)) and Var(G_{tot}(t,λ_{opt})) are given in Table 1 respectively for the combinations shown in Fig. 6.
The introduced metric in Eq. (6) means that the effectiveness of the variability mitigation depends on the sign of the covariance. If the combined 20year running mean variance is higher than half of the individual variances, the covariance is positive and reinforces the joint variance. This is true for combining Portuguese and Greek potential (see Table 1), where joint variance is higher than half of Portuguese or Greek variance. However, if the joint variance is smaller than half of the 20year running mean variance, which is correct for balancing between for Greece and Germany as well as for Portugal and Germany (see Table 1), the covariance is negative thus balancing is effective. The joint variance of Portugal and Germany is a magnitude lower than the variance in Germany. The joint variance of Greece and Germany is significantly lower than half of Var(G_{Ger}) as well. However, the benefit through balancing of Germany and Portugal is substantially higher in comparison.
Using a stateofthe art centennial coupled oceanatmosphere reanalysis, CERA20C, and correcting for likely spurious trends due to marine wind speed assimilation, we report significant multidecadal offshore wind potential variability in Portugal, Greece and Germany. Our results are robust to the chosen statistical approach as Power Spectral Density, Singular Spectrum Analysis and visual inspection of longterm time series reveal consistent results. Portuguese and German multidecadal wind power evolution are strongly coupled with the NAO in winter but with different signs. This emphasises mitigation potential through intercounty transmission.
In particular, we report two significant multidecadal modes at the Portuguese Atlantic Coast indicated by significant peaks at periods of ${P}_{\mathrm{1}}^{\text{POR}}=\mathrm{50}$–100 years and ${P}_{\mathrm{2}}^{\text{POR}}=\mathrm{15}$–20. They occur consistently in both spectral methods. We report similar modes in the Greek Mediterranean (${P}_{\mathrm{1}}^{\text{GRE}}=\mathrm{50}\phantom{\rule{0.25em}{0ex}}\mathrm{years}$) and the German North Sea (with ${P}_{\mathrm{1}}^{\text{GER}}=\mathrm{45}$ and ${P}_{\mathrm{2}}^{\text{GER}}=\mathrm{10}\phantom{\rule{0.25em}{0ex}}\mathrm{years}$). Moreover, we demonstrate the existence of significant multidecadal modes all over Europe, which are connected to slowly varying climate patterns, such as the NAO.
The longterm development of offshore wind potential shows strong differences, especially between North and South Europe. While the German offshore potential correlates strongly with NAO (r=0.82), the Portuguese and Greek anti correlate ($r=\mathrm{0.91}$ and $r=\mathrm{0.41}$, respectively). Furthermore, Portuguese multidecadal wind power evolution is inverse to German evolution: a positive NAO leads to above average potential in Portugal and below average potential in Germany (each by around 7 %). These differences allow mitigation of multidecadal variability through spatial balancing. In particular, we uncover that combining Portuguese and German offshore potential reduces multidecadal fluctuations around the mean value from ±5 % to only about ±1 %. Combining Germany and Greece also results in a reduction of variability but only to around ±2 %. In contrast, combining locations in Portugal and Greece yields no reduction. Our research therefore suggest that multidecadal wind power variability can be tackled with optimized wind park allocation and intercountry transmission, methods that were proved effective at synoptic scales (Grams et al., 2017; Rodriguez et al., 2014). At the same time, local solutions to balance generation variability, such as storage (Tröndle et al., 2019), will most likely fail to deliver on multidecadal timescales. Intercountry transmission is therefore a promising approach for future fully renewable power systems because it allows stable supply despite multidecadal resource fluctuations.
A1 Autoregressive process
In general an autoregressive processes (AR(p)) of order p could approximate an original time series X_{t} by a stochastic process Y_{t} defined by
where Φ_{i} are fixed coefficients which characterize the deterministic process and ϵ_{t} is a sample drawn from a Gaussian distribution with zero mean and variance σ^{2} (Percival and Walden, 1993). Note that an AR(p) process is special case of an AR(p+1) process, where ${\mathrm{\Phi}}_{p+\mathrm{1}}=\mathrm{0}$. In our spectral analysis an AR(1) process estimates the underlying red noise. Thus estimating Φ_{1} an σ^{2} fully determine the process. The corresponding power spectral density S_{Y}(ν) of Y_{t} is the Fourier transformed lagl auto covariance γ_{l} given as,
The parameters σ^{2} and Φ_{1} are obtained by the YuleWalker equations (Kirchgässner et al., 2012, p. 49–51). For p=1 they simplify to
where γ_{0} and γ_{1} are the auto covariances of the time series with lag zero and one.
A2 SVD approach
Following Broomhead and King (1986), the SVD estimates C_{X} from the trajectory matrix D. The rows of D are the ${N}^{\prime}=NM+\mathrm{1}$ lagged Mdimensional vectors $\mathit{\{}{\stackrel{\mathrm{\u0303}}{\mathit{X}}}_{t}=({X}_{t},{X}_{t+\mathrm{1}}$, …, ${X}_{t+M\mathrm{1}}):t=\mathrm{1}$, …, $NM+\mathrm{1}\mathit{\}}$ of length M given by sliding a window of size M over the time series. In particular D is given by
and C_{X} is defined as
where the superscript T denotes the transpose of a matrix.
A3 MCSSA
As suggested in Ghil (1997), we construct a time series using the stochastic process of the chosen noise model (here AR(1)process with the parameters given by the YuleWalker equation ${Y}_{t}={\mathrm{\Phi}}_{\mathrm{1}}{Y}_{t\mathrm{1}}+{\mathit{\u03f5}}_{t}$) and project this to the eigenstates of the signal with the SVD approach. This is done by first calculating the correlation matrix C_{Y} of the surrogate time series Y_{t} as shown in Eq. (A2). And second by multiply with the eigenvectors {ρ_{k}} of the original process X_{t} to the right and left of C_{Y} :
Each element of Δ is a surrogate Δ_{i} corresponding to an eigenvalue λ_{i}. This projection is repeated 1000 times generating 1000 surrogates per eigenvalue derived from the same noise hypothesis.
The code is written in Python and is available upon request to Charlotte Neubacher. CERA20C reanalysis data is provided free of charge by ECMWF through a web portal https://www.ecmwf.int/en/forecasts/datasets/reanalysisdatasets/cera20c (ECMWF, 2019). Wind park locations of Germany are provided by OPSD through a web portal https://openpowersystemdata.org/ (OPSD, 2018). We have taken the locations for Greece from Karanikolas et al. (2011). And for Portugal we utilize each grid box in the area 10.25–11.375^{∘} W and $\mathrm{33.375}{\mathrm{41.25}}^{\circ}\phantom{\rule{0.125em}{0ex}}$N with turbines of 100 MW.
The supplement related to this article is available online at: https://doi.org/10.5194/adgeo542052021supplement.
CN analyzed the data, produced all figures and wrote the manuscript. DW and JW initialized the project, supported each step with feedback and suggestions and improved the manuscript. The methodology was developed in close collaboration of all authors.
The authors declare that they have no conflict of interest.
This article is part of the special issue “European Geosciences Union General Assembly 2020, EGU Division Energy, Resources & Environment (ERE)”. It is a result of the EGU General Assembly 2020, 4–8 May 2020.
We would like to thank the conveners of the EGU 2020 Energy Meteorology session for enabling valuable feedback. Jan Wohland is funded through an ETH Postdoctoral Fellowship and acknowledges support from the ETH foundation and the Uniscientia foundation. Dirk Witthaut acknowledges support by the Helmholtz Association via the joint initiative “`Energy System 2050 – A Contribution of the Research Field Energy” and the grant no. VHNG1025. The authors want to thank ECMWF for making CERA20C publicly available.
This research has been supported by ETH foundation, Uniscientia foundation and the Helmholtz association (via the joint initiative “Energy System 2050 – A Contribution of the Research Field Energy” and the grant no. VHNG1025).
The article processing charges for this openaccess
publication were covered by a Research
Centre of the Helmholtz Association.
This paper was edited by Maren Brehme and reviewed by Andrea Hahmann and Laurens Stoop.
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