Analyzing the mechanisms behind temporal correlation between power sources using frequency separated time scales: A Swedish case study on PV and wind

The temporal correlation between different power generation sources is important for quantifying the reduction in variability when constructing co-located hybrid power parks (HPPs) that combine multiple power sources. This study investigates the physical mechanisms behind correlation on time scales relevant for the power system using frequency separated time scales. The methodology is universally applicable to any data set consisting of at least two power sources and could be adjusted accordingly. The methodology is demonstrated and validated in a case study across Sweden for wind and PV power generation, using the meteorological reanalysis dataset CosmoREA-6. All studied time-scales (seasonal, mid-term, synoptic and diurnal) showed anti-correlated characteristics, although the magnitude of temporal correlation is highly dependent on the time-scale considered. The highest potential for useful anti-correlation is found on the seasonal cycle, followed by the diurnal cycle where existing wind turbine sites, on average, have stronger anti-correlation than the average site. The validation showed good correspondence with measurements for all time-scales. However, an underestimations of the results were found for the diurnal and seasonal cycle while this was shown to have a minor effect when analyzing the correlation on different time scales. The methodology of the case study should be generally valid for all similar climates.


Introduction
Fluctuations in photovoltaic (PV) and wind power pose challenges of different nature for the power system depending on the time scale analyzed [1].For large offshore wind farms, variations on the scale of minutes may require flexibility in the grid [2].The diurnal variability of PV may pose problems for the power system as it increases the difference in net load (Load − Renewable energy generation) from the middle of the day to the peak load in the evening, as exhibited by the so called 'duck curve' [3].Balancing the power system on the synoptic time scale, which tends to dominate the variability in wind power production, has shown to be increasingly problematic as the penetration of wind power increases [4].Seasonal variations of wind power and PV power may require large water reservoirs if hydro power is used as the primary source of balancing the seasonal differences [5].
Since the production of wind and PV power is non-dispatchable, i.e., to a high degree inadaptable to the demand, the increasing share of these power sources requires strategies to manage a potential mismatch between supply and demand [6].One approach to even out fluctuations is to find locations where wind and PV power complement each other and there install co-located hybrid power parks (HPPs) that combine the two power sources [7,8].Deployment of HPPs is also justified by increasing the capacity factor of wind power parks and PV parks.Hence, combining the power sources could provide a more effective utilization of the grid connection capacity and provide a lower cost for the park owner [7,9,10].Locations in favor of installing HPPs could be found by studying the temporal correlation between the power sources across a country or region since the collective variability, when the solar and wind resources are available in concert, is dependent on local climate and weather conditions at different temporal scales [11][12][13].
Already in 1979, Takle and Shaw [14] presented one of the first studies on temporal correlation between wind and solar based on station data in Iowa, USA.The authors found the resources to be anti-correlated, i.e., they exhibits complementary characteristics.The most pronounced anti-correlation was found on an annual basis, while small on a daily basis.Since then, temporal correlation has been studied in several countries and regions; Algeria [15], Australia [12], China [16], central and southern Europe [11], Germany [6], Portugal [17], Italy [18], Mexico [19], the UK [20] and globally [21].
The above presented studies have a varying level of complexity and model assumptions which risk straight forward comparison of the results being misleading.Firstly, some studies use a combination of wind and solar data sources that lack direct connection [6,11,15,18], e.g.satellite data for the solar resource in conjunction with reanalysis data for the wind resource without validating that the combined output accurately represents the physical coupling between wind and solar resources.Although validation might seem as an obvious first step before using a model, it is rarely done in terms of validating the temporal correlation itself.Some papers have referred to previous validation studies related to one or both resources individually, but not how well the two resources correlate with each other, e.g.Refs.[11,15,17,18,20,22].A few studies have left out validation completely [16,19].Prasad et al. [12] confirmed their results to other studies using similar datasets, and argued that it is sufficient to establish the credibility of their results.Kapica et al. [21] admit that the accuracy of the dataset was questionable, but that the important takeaway from their study is the global assessment of the temporal correlation.Secondly, when modeling the wind resource, some of the studies extrapolate wind speed close to the ground (typically 10 m) to hourly wind speed at hub height [6,15,18], failing to take into account that the diurnal and seasonal wind speeds at ground and hub height are typically anti-correlated, both on average and particularly on time scales where the atmospheric stratification plays a role for the turbulence level [23,24].Hence, there is a potential to decrease the uncertainty in studies of combined power production from renewable energy source by improving the methodology.
The aforementioned studies focus on mapping the correlation and present descriptive conclusions.The physical mechanisms behind correlation and the potential smoothing effect on different time scales relevant for the power system (e.g.diurnal, synoptic or seasonal) has thus so far been less researched.To our knowledge, it is the first time that this type of analysis is performed to study the temporal correlation for more than one resources across a country, i.e. not only for multiple resources at a single site as was done in Shi et al. [25].The mathematical methodology uses filtering, which is described in Section 3.3, and allows to isolate and study the smoothing effect at particular time scales and could also be extended to other power sources than just PV and wind power.Furthermore, meteorological reanalyses have gained interest in the field of renewable energy modeling and been improved over the last few years [26,27].However, even state-of-the-art reanalyses suffer from systematic and random model errors due to the state of knowledge in wind dynamics and limitations in numerical weather prediction on the mesoscale [28].Therefore, in order to gain confidence in our methodology and the corresponding reanalysis data set, it was first compared to measurements from four validation sites.
The methodology is demonstrated across Sweden for wind and PV power using the reanalysis dataset Consortium for Small-scale Modeling Retrospective Analysis 6 (COSMO-REA6), in which the meteorological variables are physically consistent at any given time step and spatial scale.Sweden is considered to be a good case because of the large variety in climate across the country due to the latitudinal stretch (between 55 and 69 • N) which makes it a suitable electricity market to study the impact of meteorological effects in the mid-to high latitudes.Subarctic climate is found in the northern part and the mountain region.In the south, continental climate is found in the inland and oceanic near the coasts and islands.

Scope of the present study
We identify in Section 1 that the physical mechanisms for temporal correlation between wind and PV power and potential smoothing effect on different time scales relevant for the power system should be studied in more detail.Furthermore, in previous studies, validation of the correlation of modeled power output has been left out.Thus, there is a need to validate the output from the reanalysis model (in this case COSMO-REA6) with measurements to understand the accuracy and limitation of the employed datasets.Considering this, our approach differ from previous studies on temporal correlation in two main aspects: (i) validation of COSMO-REA6 with regards to accuracy of the temporal correlation and (ii) the mathematical method for separation into different time scales to understand the physical mechanisms for temporal correlation.
Existing wind power infrastructure is a motivating factor for the deployment of HPPs and rarely included in previous studies and is therefore included when conducting part 2. Additionally, the results of this study will partially be presented as maps, which may be used to identify potential sites for HPPs to provide a more resource efficient utilization of the electricity system.

Structure of the paper
The paper is structured as follows; Section 2 presents the data used for the study.In Section 3, the theory and methodology used when modeling the power sources as well as statistical metrics used when validating and presenting the results are described.Validation of the reanalysis data and results from the correlation study are presented in Section 4. In Section 5, a discussion on the findings and suggestions for further research can be found.Finally, in Section 6, the main conclusions are put forward.

Data
This section describes the data used in this study.Section 2.1 presents the reanalysis data as well as previous literature that validate the reanalysis data.In Section 2.2, turbine sites used when studying the temporal correlation are presented.Finally, the sites used when validating the reanalysis data are presented in Section 2.3.

Modeling data
In reanalysis datasets, meteorological variables are calculated by the same physical model which means that they are physically consistent at any temporal time step and spatial scale.On top of this, the time series from reanalysis datasets are long and consistent, i.e., the quality do not change substantially over time.This study uses the reanalysis dataset COSMO-REA6 produced by the Hans-Ertel-Centre for Weather Research, which is described in Ref. [29] and freely accessible from Ref. [30].Instantaneous hourly data were used, covering January 1995-August 2019 at a horizontal resolution of 0.055 • × 0.055 • (≈6 km × 3 km) across Sweden.
The COSMO-REA6 is the only meteorological reanalysis data set available to the authors with the meteorological components needed to conduct the study, i.e. hub height wind fields for wind power calculation as well as direct and diffuse irradiance for PV power calculation.The COSMO-REA6 also has relatively high temporal and spatial resolution which is desirable to represent variations in terrain and coastline.A similar reanalysis data set is the New European Wind Atlas (NEWA).However, we found an error in the NEWA database where data labeled as diffuse radiation was in fact direct tilted irradiance, whereas data labeled direct irradiance was direct horizontal irradiance.The NEWA consortium confirmed this after the our remark.
The validity of COSMO-REA6 has been investigated in a few studies and is also extensively done in the present study in Section 4. Below follows a short literature review on previous studies that have validated COSMO-REA6.Borsche et al. [31] evaluated modeled wind speed against four wind masts in Germany, the Netherlands, the North Sea and the Baltic sea, respectively.The results showed good agreement with measurements in terms of wind speed probability distribution.However, Borsche et al. [31] showed shortcomings in the representation of the mean diurnal cycle above 40 m over land although the mean diurnal cycle is qualitatively represented near the ground.Borsche et al. [31] explains this by the parametrizations of the boundary layer and the sub-grid scale orography which were optimized to the observed 10 m wind speed.
Jourdier [27] compared COSMO-REA6 and other reanalysis datasets with wind speed measurements and historical wind power production in France.In general, COSMO-REA6 predicted wind power production with low bias compared to NEWA, ERA5 and MERRA-2.A weakness was however found in the accuracy of the diurnal cycle.Most notably COSMO-REA6 failed to capture the nocturnal increase in wind speeds at heights exceeding 80 m, leading to a negative bias during nights.
Frank et al. [32] identified a systematic bias in the global horizontal irradiance (GHI) when comparing to the Baseline Surface Radiation Network and proposed a post-processing methodology.Frank et al. [32] and Urraca et al. [33] found the GHI to be underestimated in clear-sky conditions and overestimated in cloudy conditions, which was explained by an underestimation of the cloud extinction.Urraca et al. [33] compared GHI of COSMO-REA6 and ERA5 against station data and the satellite product Surface Solar Radiation Data Set -Heliosat (SARAH) and showed that COSMO-REA6 provides relatively high bias of the GHI.It was concluded that this was probably related to the challenge of predicting cloud evolution correctly.However, Urraca et al. [33] reported that this underestimation is lowered above 45 • N.
In conclusion, COSMO-REA6 have shortages in terms of accurately modeling the wind and solar resource.Compared to tailored products focusing mainly on one resource, such as the NEWA atlas [34] it was estimated from initial comparisons with the validation data that the COSMO-REA6 reanalysis is a data set where uncertainty with regards to renewable electricity production is relatively balanced between the energy sources and therefore the most suitable at the authors disposal.

Wind turbine data
To assess the potential correlation between existing wind farms colocated with PV parks, data of all built and approved (i.e.all permits in place) wind projects in Sweden was obtained from Ref. [35].This dataset is presented as filled circles in Fig. 1, where the size represents the installed capacity.The small fraction of off-shore wind farms existing in Sweden were excluded since the study focuses on land based PV parks.Although floating PV parks are gaining attention [36], it is excluded in this study since the risk of ice formation on water bodies in the Nordics make them unsuitable.Electricity market bidding zones, referred to as SE1-4 and indicated by thick lines (and labels in Fig. 1), are shown in order to refer to certain areas for readers not familiar with the Swedish geography.

Validation data
One purpose of this study is to assess the spatial variability of the correlation between wind and PV power.To avoid representation errors in the conversion between resource and power production, the study is limited to tower measurements of wind speed and excludes power output from wind turbines.Publicly accessible observations with more than 100 m high towers and concurrent solar irradiation measurements are limited to a very small number of locations shown in Fig. 1 with cross markers.However, the sites are spread across Sweden, making them representative.
An overview of the data availability and site characteristics is given in Table 1.Although the time resolution is higher than 1 h in some cases (shortwave radiation for all sites as well as wind speed for Ryningsnäs), hourly data were used to comply with the COSMO-REA6 dataset.The wind speed measurement height that was the closest to 125 m was used, as this height was used when calculating the wind power production, see Section 3.2.On the other hand, the variability of the downwelling short wave radiation is assumed to be independent of the height.Therefore, the available shortwave-irradiance at the sites were compared to the 2 m GHI in COSMO-REA6.In between occasionally missing data points, linear interpolation was used.If more than 6 consecutive hours were The size of the circles represents the installed wind farm capacity in each locality.The overview is created using data from the County Administrative Board [35].The labeled cross-markers represents the validation sites.Note that the size of the circles do not correspond to the area the wind power park, which is much smaller.missing, the period with missing data was excluded from the calculations.

Theory and methodology
In Section 3.1, the theory and methods for the statistical analysis are presented.In Section 3.2, the methods for calculating power output from the wind and solar resources are presented.Filtering of the data into frequency time scales are explained in Section 3.3.

Statistical analysis
By combining anti-correlated power sources, the variance can be lowered compared to solely installing a single power source with the same total energy production.The sum of N power sources is calculated as: where w i is the share of the total capacity (i.e.∑ i w i = 1) and P i is the power output of time series i.The variance of P tot can be calculated as: where σ x is the standard deviation and ρ i , j is the Pearson's correlation coefficient between the time series i and j.For example, given two signals (N = 2) with unity standard deviation (σ i = σ j = 1) and equal capacity share (w i = w j = 0.5), the role of ρ i , j becomes obvious.With ρ i , j of − 1, 0 and 1, the variance of the sum of the signals becomes 0, 0.5, 1.To rephrase it, if ρ i , j < 0, which is to say that the power sources are anticorrelated, the combined variability is lowered due to cancellation of negatively correlated variations in each signal.The temporal correlation for a single power source, ρ i , j could be thought of as 1.Also, note that the correlation coefficient, ρ i , j , is not affected by the magnitude of the time series.
To study where the potential useful correlation is found, we use normalized co-spectrum at isolated frequencies.A discrete signal x t in the time domain can be decomposed into its corresponding frequency domain components X k by the fast Fourier transform (FFT).In signal processing applications, such as studying the variability of wind power, an important aspect of the FFT is to isolate and study the dominating frequencies in a noisy signal.The normalized co-spectrum is defined as: where S i , j is the cross spectral density.The reason for taking the real part (R) of S i , j is to solely study the simultaneous variations, i.e. not to consider any time lag between the two signals.The cross spectral density was evaluated using a fast Fourier routine based on hourly resolution and the frequency vector expressed as cycles per day.The spectral densities, starting from the Nyquist frequency, 12 cycles per day, as well as the frequencies, were then arithmetically averaged in 50 logarithmically spaced frequency bins.

Power production
From COSMO-REA6 wind speed data, hourly wind fields were transformed into wind power using power curves, which are given by manufacturers' in the form of power as a function of wind speed.The most important factor for the shape of the power curve is the specific rating, i.e. ratio of rated power to rotor area.Three wind speeds are of importance in the power curve, the cut-in (where the WT begins to operate), the rated (the lowest wind speed with full output) and cut-out (where the WT is shut down for protection) [47].The type of turbine was selected based on mean wind speed at the specific location according to The International Electrotechnical Commission (IEC) wind class standards using four reference wind turbines, ranging from 226 to 472 W/m 2 .The power curve gives an ideal quantitative relation between wind speed and wind power generation.Several meteorological parameters are known to have an impact on the actual power curve [48], nevertheless, in this study the ideal case is considered.This study assumes a constant hub height of 125 m.From wind project data, the average installed turbines in Sweden during the last five years had a nacelle height of roughly 117 m [35].
The PV power production was calculated using a simple PV system model [49].Included parameters were; albedo (constant at 0.3), module efficiency (constant at 17%), system efficiency (constant at 80%), solar irradiance on the tilted plane (G T ), derived from the Hay & Davies model [50], in which the following were used as inputs; direct and diffuse horizontal irradiance from COSMO-REA6, 0 • azimuth (due south) and panel tilt of 30 • , which is a commonly used tilt of PV parks in Sweden.Although temperature is known to affect PV power production, it was excluded since the surface temperature was not measured at the validation sites.Temperature is assumed to have a minor effect on the correlation in Swedish climate because of the relatively low summer temperatures.

Filtering
As was discussed in Section 1, PV and wind power fluctuations pose challenges of different nature for the power system depending on the time scale analyzed.It is important to note that hourly resolved time series contain variability on other time steps than just the hourly.Therefore, when correlation is calculated using hourly resolution, other time scales (e.g.diurnal and seasonal correlation) contribute to the calculated correlation.To better understand the variability patterns, the raw signals were separated into four frequency components: seasonal (time periods longer than 8 month), mid-term (2 weeks-8 month), synoptic (2 days-2 weeks) and diurnal (less than 2 days).
Seasonality is seen as an important aspect of both PV and wind power generation.By choosing a cut-off frequency of eight months, the components were allowed to follow the seasonal variability of the respective signals.Synoptic scale weather systems affect wind and solar variability in terms of low and high pressure systems on the scale of one to four thousand kilometers.These pressure systems result in high and low wind power production, that may last for up to ten days [51].For solar power, these pressure systems may bring more or less cloudy conditions that affect the solar power production.The cut-off frequency for the synoptic component was therefore chosen between 2 and 14 days.The diurnal variability is prominent in PV production, although it is found in wind power as well [24].The scale of eight month to fourteen days, denoted as Mid-term, followed naturally to cover all frequencies.
Finally, the correlation between wind and PV power with hourly resolution is what has mostly been used in previous correlation studies presented in Section 1 and is therefore also considered in this study to facilitate comparison.It is then referred to as correlation on hourly resolution (albeit limited by the resolution of the reanalysis).An example of frequency band separation is shown in Fig. 2.

Results
This section is divided into two parts; in Section 4.1, the correlation between the solar and wind power outputs in the COSMO-REA6 model are compared with observations.In Section 4.2, the temporal correlations between the solar and wind resource across Sweden at different time scales are presented.

Model validation
The particular skill of interest to this study is how well temporal correlation between wind and PV power is represented at different time O. Lindberg et al. scales.Correlation between wind power and PV power for seasonal, midterm, synoptic, diurnal time scales and hourly resolution are given in Fig. 3 for the four sites.The correlation coefficients were calculated after separation into the different time scales for COSMO-REA6 (dashed lines) and measurements (solid lines).The results shows that the strongest anti-correlation is found on the seasonal time scale, whereas the other time scales show weak, although consistent, anti-correlation.The largest absolute deviations between the correlation pairs are 0.12, 0.06, 0.08, 0.06 and 0.05 for the seasonal, mid-term, synoptic, diurnal and hourly average, respectively.The largest difference between measurements and COSMO-REA6 is found for the synoptic time scale indicating that the model has difficulties to exactly capture the wind speeds or exactly capture the thickness of the clouds, even tough timing might be good.However, the model predict the measured correlation with an error smaller than ±0.12 on all time scales.
To study on what cycles the potential useful correlation is found, the model was validated by studying the normalized co-spectrum as described by Equation (3.3) in Section 3.1.The result of this analysis is shown in Fig. 4. Note that we make a distinction between cycles and time scales; cycles relate to the variation on individual frequencies while time scales relate to the aggregation of several frequencies.Fig. 4a, c, e & g show the normalized co-spectrum between wind power and PV power for COSMO-REA6 (blue solid lines) and measured data (gray solid lines).The ability of the model to accurately predict the magnitude of correlation between wind and PV power on each cycle is reflected by the proximity of the two curves in Fig. 4, left panel.The normalized cospectrum is similar to Fig. 3, but divided into more frequency bands and normalized by the product of the PV and wind power standard deviations.This means that the normalized co-spectrum is related to the linear correlation, but subject to the variability on all time scales.In other words, the integral of the normalized cospectrum over all cycles is equal to the linear correlation.Due to the limited length of the measured time series, frequencies below 0.02 cycles/day are not shown.
The normalized co-spectrum (Fig. 4a, c  validation sites, while the other frequencies are shown to be more random.Since magnitude of these frequencies are large compared to the other cycles, they can be considered as useful.The diurnal cycle (1 cycle/day), which is mainly due to the obvious day-night cycle of sunlight, the modeled correlation is inconsistent with the measured correlation.For Svartberget, Norunda and Hyltemossa (Fig. 4a, c & g), the normalized co-spectrum at the diurnal cycle is underestimated.For Ryningsnäs (Fig. 4e), modeled data is opposite to the measured at the diurnal cycle.One of the reason for the deviation at the different time steps is revealed by Fig. 4b, d, f & h, which show the spectral correlation between measured and modeled wind power (dark gray solid lines) as well as the spectral correlation between measured and modeled PV power (light gray solid lines).Spectral correlation is used to study the linear correlation for each cycle, as compared to the normalized cospectrum.In this case, the variability of the different cycles are subject to the variability of the isolated cycle and the lines should be analyzed independently from each other.If the reanalysis dataset perfectly represented the measurement, the lines should reach a value of 1 at all time steps.The COSMO-REA6 performs better at lower frequencies (i.e., longer time steps) than the diurnal cycle as compared to frequencies above the diurnal cycle.For frequencies higher than the diurnal cycle, the spectral correlation is decreasing towards zero for both wind and PV power.This is expected due to the more stochastic nature of short fluctuations as well as the scales of the correlation approach the horizontal resolution of COSMO-REA6.The average spectral correlation at the different frequencies is 0.91 and 0.87 for wind and solar power, respectively.For the diurnal cycle, the spectral correlation of the wind power is 0.65 or lower for the four validation sites, confirming the problems for COSMO-REA6 to accurately predict the amplitude of the diurnal cycle winds in the studies [27,31].From visual inspection of the average diurnal variability, this study confirms that an underestimation of the amplitude of the wind power output affects the correlation on the diurnal cycle.This means that the problem of accurately predicting the diurnal cycle of wind speed is likely to carry over to other sites as well.This is likely due to parametrizations in the boundary layer, as elaborated upon in Section 2.1.For Ryningsnäs and Hyltemossa, the model inaccurately represents the diurnal cycle, which results in disparate peaks of the modeled and measured spectral correlation in Fig. 4f and h.For PV output, the diurnal cycle is modeled with close to perfect accuracy, which is reasonable since it to large extent is completely deterministic and related to the apparent motion of the sun across the sky.At 12 h and 6 h, there are also obvious peaks for the PV, which is probably due to zero irradiance at nighttime demonstrated as a superposition of several different periodic waves in the spectrum.Furthermore, this indicate that the useful correlation is found at the diurnal cycle.
Based on the results presented in Fig. 3, the model performs well for with regards to studying the correlation between wind and PV power on different time scales.However, the diurnal cycle which is driven by atmospheric stratification, is shown to contain much of the potential useful correlation and shown to be the inaccurately modeled.As will be shown later, at the annual cycle, the effect of varying atmospheric stratification again becomes important to explain the variance of wind power.For that particular cycle, a larger uncertainty should also be expected, although the measurement records used for this study were not long enough to establish that with spectral analysis.

Temporal correlation
Having established the degree to which COSMO-REA6 represents the temporal correlation of wind and PV power, in this section the temporal correlation is presented.The seasonal, mid-term, synoptic, diurnal and hourly resolved temporal correlation between wind and solar power across Sweden are shown as maps in Fig. 5.In the bottom right of each maps, the probability distributions of Sweden as a whole (dashed histograms) and turbine sites (gray as explained in Section 2.2, are shown.In general, the seasonality (Fig. 5a) shows the strongest negative correlation.This means that wind and PV power complement each other best on the yearly cycle.In the northern part of the mountains (western SE1 and northwestern SE2), some areas show weak anti-correlation or weak positive correlation.The seasonal anti-correlation is strongest along the eastern coast of SE2 as well as the coastal and central regions in SE3.Focusing on correlation coefficients for existing wind turbine O. Lindberg et al. sites, the distribution shows a higher frequency of more anti-correlated values, indicating that a turbine site is likely to be more anti-correlated than any randomly chosen site on the seasonal time scale.
For the synoptic time scale (Fig. 5c), wind power and PV power are weakly anti-correlated across the country.The correlation coefficients span from − 0.22 to 0.06 with an average correlation coefficient of − 0.10.The correlation is relatively spatially homogeneous compared to the other time scales, indicating a robust correlation between cloudiness and windiness connected to the migration of low pressure systems.
For the diurnal time scale, the correlation coefficients has a maximum value of 0.09, a minimum value of − 0.11 and an average value of − 0.04 across Sweden.This is the least negative average correlation coefficient of the time scales analyzed.The majority of the positively correlated areas are found in the mountainous area (western SE1 and northwestern SE2).
For the hourly resolution, the correlation coefficients are between − 0.19 and 0.09 with an average value of − 0.10.To separate the seasonal variability from the hourly resolution, the hourly resolved data set was split into the different months after which the correlation was calculated.The results are shown in Table 2.Although the correlation coefficients for the hourly resolution each month are not particularly strong, they are consistently negative throughout the year.The temporal correlation in the lowest 5th percentile for each month (values in parenthesis in Table 2) suggest that it is important to choose the locations where the most pronounced anti-correlated characteristics are found as these may differ between locations.Another general trend is that the average anti-correlation for hourly resolution is stronger during the summer months than in the winter months, which is mainly due to the seasonal variability in available solar irradiance.
Fig. 5 shows how the temporal correlation coefficients differs as a function of the geographical location and the time scale analyzed.To further analyze whether certain local climatic conditions affect correlations between wind power and PV power, Fig. 6 shows the seasonal, mid-term, synoptic, diurnal and hourly temporal correlation between wind and solar power across Sweden as a function of annual mean wind speed, annual mean global irradiance and altitude, respectively.For the seasonal and diurnal time scales, the anti-correlation depends on the magnitude of the annual mean wind speed which further explains why turbine sites, in general, are shown to be more anti-correlated than the average site.This trend is also seen in the correlation coefficients in the hourly resolution, highlighting that correlations found in the seasonal and diurnal time scale is impacting the total correlation.For the midterm and synoptic time scales, the trend is not clear.However, sites with a yearly amount of global irradiance exceeding 120 W/m 2 (which are situated in the southern part of Sweden) are prone to stronger anticorrelation on the seasonal and diurnal time scale as well as the hourly resolution.On top of this, among the sites with a global irradiance over 120 W/m 2 , on average roughly 2% had an annual mean wind speed below 6 m/s at the modeling height of 125 m.For sites with a yearly global irradiance below 120 W/m 2 , the correlation coefficient is more random.
As evident from Fig. 6, the characteristics of the temporal correlation between wind and PV power is different for sites with high wind speed compared to those with low wind speed.To further investigate low and high wind speed sites, 1000 randomly chosen grid points with an average yearly wind speed above 6.5 m/s as well as 1000 randomly chosen grid points with an average yearly wind speed below 5.5 m/s at (the total number of COSMO-REA6 grid points across Sweden is 11,884) were chosen and studied further.In Fig. 7, the normalized co-spectrum for these two time datasets are shown.This is similar to the analysis in Fig. 7.However, lower frequency are included since we are not restricted by the amount of validation data.In this case, the integral of the line, i.e., the area between the line and y = 0, amounts to the correlation of the hourly resolution.The figure also shows where the useful correlation is found, i.e. most prominently on the diurnal and seasonal time scale as well as to a lesser extent in the synoptic time scale.Furthermore, sites with a higher mean annual wind speed than 6.5 m/s (gray) show slightly more anti-correlation than sites with a mean annual wind speed below 5.5 m/s for both seasonal and the diurnal time scales while there is very little difference at the synoptic time scale and even less difference at the mid-term scale.In other words, virtually all of the difference in anti-correlation between high wind and low wind sites is explained by variations on the seasonal and diurnal time scales, where much of the variation is driven by changes in the atmospheric stratification.

Discussion
In Section 5.1, uncertainties in the results based on the validation are discussed followed by the implications of the results with regards to the temporal correlation in Section 5.2.Finally, in Section 5.3, a discussion related to further research directions is presented.

Validation of COSMO-REA6
The correlation between wind and PV power for different time scales showed good correspondence with measurement.Despite the fact that the model at least has a physical connection between the two, i.e., both being dynamically coupled in the model, the study found that studying the normalized co-spectrum using isolated frequencies revealed deficiencies in the diurnal cycle.Since this cycle is driven by atmospheric stratification, it is likely carried over to the seasonal cycle as well.COSMO-REA6 has a European coverage and since previous studies validating the data set for single resources also has shown deficiencies in the diurnal cycle of wind for central Europe [27,31], the validation presented here is expected to generally hold for the midlatitudes, at least with respect to the wind energy part.Previous research has also identified uncertainties with regards to the solar irradiance in southern Europe [32,33] and thus it still remains important to further validate the data set, especially if used in southern Europe.Although COSMO-REA6 showed some drawbacks in terms of temporal correlation on individual cycles, publicly available reanalysis with high horizontal and temporal resolution are limited.

Implications of the results
For the temporal correlation assessment across Sweden, seasonal correlation show the most anti-correlated characteristics with an average correlation coefficient of − 0.62.The other studied time scales showed a slight anti-correlation.Therefore, all time scales suggest the general trend that less solar power is produced during windy conditions while wind power production is lower during sunny conditions.This means that, due to the negative correlation, a more evenly distributed output could be expected when combining PV and wind power generation compared to the sources alone.Apart from the overall smoothing effect in a HPP, Lindberg et al. [7] discussed other potential benefits from anti-correlation, such as less storage needs, more reliable power supply and reduced power delivery limit (capacity of the connection point to the grid) for the aggregated production as compared to separately installed wind and PV power parks.Since the highest potential for anti-correlation was found at existing wind turbine sites, retrofitting established wind farms with PV and hence lower the connection capacity relative to the nominal capacity has the potential of providing a smoother power output as well as increasing production returns relative to the grid connection capacity.According to our results, particularly as expressed by Fig. 7 there are three main sources for the temporal anti correlation; seasonal variations, synoptic variations and diurnal variations, with the largest potential for anti correlation found in the seasonal and diurnal time scales.As was established in Section 4.1, particularly the diurnal cycle, but also the seasonal cycle were defined as the most uncertain cycles, owing to uncertainty in the representation of the variation in wind power.The reason for this might be due to difficulties in accurately representing the cross-over height as was found in Borsche et al. [31] and Jourdier [27].The cross-over height is related to the average diurnal variability of wind speed.Below the cross-over height, wind speeds are characterized by a mean daytime wind speed larger than the mean nocturnal wind speed, while the opposite is true above the cross-over height [24].Because the overall anti-correlation of wind power and PV power is sensitive to the phase and amplitude of wind power variations driven by stratification, the overall anti-correlation will thus be sensitive to the heights of wind power extraction relative to the cross-over height.
This analysis has assumed a constant hub height of 125 m which, although modern turbine reach this height, could be seen as a futuristic case as many of the installed turbines in Sweden are below this height [35].However, the 125 m height should be above the cross-over height as it has been found at around 100 m in Germany [52], 60-80 m for spring and summer in Moscow [53] and 40-80 m in the Netherlands [31].Furthermore, the results in Fig. 4 as well as results from Jourdier [27], Borsche et al. [31] indicates that the cross-over height of COSMO-REA6 is too high, suggesting that the real anti-correlation might be stronger than reported in Fig. 5d.

Directions for further research
This is the only study to the authors' knowledge that isolate and assess the temporal correlation across a country using different frequency time scales.The methodology is generally applicable for other regions.With regards to the coverage of COSMO-REA6 and the consistency between the studied validation sites, other regions within Europe could use the methodology in order deepen the knowledge of variability between co-located wind and PV power.
As stated in Section 1, varying levels of complexity in previous studies related to temporal correlation assessment risk straightforward comparisons of results to be misleading.However, there are some interesting comparisons that can be made.Miglietta et al. [11], Ren et al. [16], Monforti et al. [18] studied the average or cumulative hourly, daily and monthly resolution.In accordance with this study, the previously mentioned studies found that by removing shorter temporal variations (that is studying temporal resolutions lower than hourly resolution), the anti-correlation was amplified.In other words, their studies also indicate that the largest anti-correlation is found on time scales of one month or larger.Although these studies analyze correlation using average or cumulative daily resolution, a straightforward comparison with Fig. 5d is not possible since all time scales above the diurnal time scale is also included when assessing the daily correlation in their studies.However, since COSMO-REA6 showed shortcomings in the representation of the diurnal cycle, an important future step is to go into detail on how the diurnal variability may differ between locations.The reason is that the diurnal variability will most probably be useful for short-term operation for HPPs, such as securing production bids for capacity or ancillary services.The seasonal variability may, on the other hand, be used for providing a more constant power production on a yearly basis.Since the effect on the power system is a natural continuation of the study, future research should analyze to what extent the anti-correlation may be useful for power system applications.Since a unique feature of co-located wind and PV is that smoothing could be achieved without physically separating the parks, an interesting research direction is to analyze how this affects the accuracy of forecasts.The reason is that smoother signals are generally more straightforward to predict [54].Another promising research avenue for co-located wind and PV parks relate to analyzing the capability of co-located parks, as compared to individual power parks, to deliver power system services.The reason is that these parks are motivated as potential candidates to deliver capacity or ancillary services [7], from which profitability of individual power parks will increasingly depend on as more intermittent and non-dispatchable energy sources are installed [55].
Another interesting direction is whether or not the uncertainty in representation of the diurnal cycle carry over to meteorological prediction which would be useful in assessing whether or not there are potential synergy effects with HPPs also for meteorological prediction.Related to this, it would also be interesting to study how much of the anti-correlation that is related to the presence of clouds (and time scales) as well as how much of the anti-correlation that is related to the absence of clouds.The reason is that the diurnal profile of wind speed is amplified in clear-sky conditions since the stratification of the boundary layer is stronger.On the other hand, during cloudy conditions, the solar resource is lower which results in less pronounced diurnal profile for the wind resource.Furthermore, an interesting direction for future research is to validate the model at more heights and investigate how the temporal correlation differs between varying hub heights.Since the methodology can be extended to other power sources, it would be of interest to study the correlation for other power sources than wind and PV power.
Studies from other parts of the world have found positive correlation between PV and wind power in mountainous areas [16,18].In the case of Italy [18], positive correlation was motivated with Foehn.In China, Ren et al. [16] motivated the positive correlation on the hourly resolution by microscale winds, such as valley winds.On the averaged daily and averaged monthly resolution, the negative correlation was motivated by monsoons and fronts.However, these could be seen as rather speculative conclusions based on the methodology used, although these phenomena might be well-known in these parts of the countries.For the Swedish case, weaker anti-correlation and positive correlation is found on the diurnal and seasonal time scales.Foehn can likely be ruled out as these would have been shown on the synoptic time scale [56].As we did not observe a strong relationship with altitude itself (Fig. 6), but rather that correlation is driven by stratification, a weak anti-correlation or positive correlation is likely caused by katabatic winds or gravitational waves.To establish the cause of the weaker anti-correlation and positive correlation, further studies casting light on the diurnal and seasonal time scales in mountainous areas are recommended.

Conclusion
This study examines the temporal correlation between wind power and PV power on different time-scales, relevant for the power system, in order to understand the physical connections that result in the correlation.It was found that temporal correlation is highly dependent on the time-scale considered and the highest potential for useful anticorrelation was found on the seasonal cycle, followed by the diurnal cycle.All studied time scales (seasonal, mid-term, synoptic and diurnal) showed anti-correlated characteristics, which suggest that less PV power is produced during windy conditions while wind power production is lower during sunny conditions.Further, it was found that wind turbine sites obtain more anti-correlated characteristics for the seasonal and diurnal time scale than the average site.The reason is that these time scales are driven by stratification which seems more pronounced for windy locations.However, the data that were used in this study, COSMO-REA6, showed deficiencies in the representation of winds on cycles driven by atmospheric stratification, contributing to an uncertainty on these cycles.

Declaration of competing interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Fig. 1 .
Fig. 1.Wind farms in Sweden that are built (blue circles) and approved (green circles).Electricity market bidding zones are indicated by labels and thick lines.The size of the circles represents the installed wind farm capacity in each locality.The overview is created using data from the County Administrative Board[35].The labeled cross-markers represents the validation sites.Note that the size of the circles do not correspond to the area the wind power park, which is much smaller.

Fig. 2 .Fig. 3 .
Fig. 2. Illustration of frequency separated time scales for wind power (denoted as WP in the figure) and PV power for a randomly chosen site in Sweden from the COSMO-REA6 dataset.Note the different scales on the x-axes.

Fig. 4 .
Fig.4.The left column shows the normalized co-spectrum between wind power and PV power for COSMO-REA6 (blue solid lines) and measured data (gray solid lines).The right column shows the spectral correlation between wind power for COSMO-REA6 and measured data (dark gray solid lines) and PV power for COSMO-REA6 and measured data (light gray solid lines).In this case, the spectral correlation is calculated subject to the variability on each cycle.a) & b) corresponds to Svarberget, c) & d) corresponds to Norunda, e) & f) to Ryningsnäs and g) & h) to Hyltemossa.Note the logarithmic scale on the x-axes.

Fig. 5 .
Fig. 5. Maps of temporal correlation for each time scales analyzed across Sweden; a) Seasonal, b) Mid-term, c) Synoptic, d) Diurnal and e) Hourly resolution.The histograms in the bottom right of the maps show the corresponding distributions for Sweden (dashed histogram) as a whole and the turbine sites (gray bars).Note the different scales on the x-axes.

Fig. 6 .Fig. 7 .
Fig. 6.The figure shows the correlation as a function of annual mean wind speed, annual mean global tilted irradiance and altitude for seasonal, mid-term, synoptic, diurnal time scale as well as hourly resolution.The light gray markers indicate a wind speed below 6 m/s, while the dark blue indicate a wind speed above 6 m/s.

Table 1
Information on the validation sites.
O.Lindberg et al.

Table 2
Average temporal correlation across Sweden for hourly resolution within each month.The numbers in parenthesis shows the lowest 5th percentile of the grid points.