The proposed method is built upon previous works by Saint-Drenan (2015) and Saint-Drenan et al. (2017), where a model for the aggregated PV power produced by a fleet of PV plants installed in a region is described. The authors have showed that an accurate estimate of the German PV power generation can be obtained by using the statistical distribution of the orientation angle of PV panels as the sole plant-relevant input to the model (Fig. 1). The model is based on the simple idea that the aggregated PV power generated in a region is the sum of the normalized outputs of all plants with characteristics Ai multiplied by the proportion wi of plants having the characteristics Ai in the whole set of plants installed in the considered region. The regional PV power generation can therefore be expressed as follows: PPV(xt)=∑i=1nwifPVx,t,Gx,t,Tax,t,Ai Where PPV(x,t) is an estimate of the aggregated power produced by all PV plants located at x at time t [W Wp-1], G(x,t) is the global horizontal irradiance (GHI) received at x and t [W m-2], Ta (x,t) is the air temperature at x and t [∘C], fPV(…) is a function representing the single PV plant model used to calculate the normalized PV power [W Wp-1].

A first advantage of the chosen regional PV model is that each important configuration is considered only once and the number of configurations Ai can be optimized in order to limit the calculation costs. A second advantage is the high flexibility offered by the use of an analytical function to describe the statistical distribution of the characteristics of the plants installed in a region.

The function fPV in Eq. (1) represents a single plant model, which needs to be chosen prior to the implementation of the proposed approach. In Saint-Drenan (2015) and Saint-Drenan et al. (2017), the authors demonstrated that a simple model with a limited number of input parameters yields good results for regional application. With the chosen model, the set of characteristics Ai is only composed of the module tilt angle γ and azimuth angle α or orientation. The Ai's, which are a function of (α, γ), are hereafter referred to as reference configurations. There are two steps for the implementation of the regional PV model that are described in the following section: the estimation of the weights wi and the choice of the reference configurations. A detailed description of the chosen model which is illustrated in Fig. 1 can be found in e.g. (Saint-Drenan, 2015).

Saint-Drenan (2015) and Saint-Drenan et al. (2017) have created a dataset of peak power and module orientation angles for 35 000 PV plants located in Germany, which is used here. This amount of plants represents approximately 2 % of the number of plants installed in Germany. It is assumed that this dataset is representative of all plants in Germany. A realistic example of the relationship between wi and Ai at country level may be derived from this dataset. Figure 2 exhibits the share wi of installed capacity per module orientation evaluated from this dataset. One may note the high share of installed capacity for modules with a tilt of 20∘ southwards facing (orientation 180∘).

The use of Eq. (1) requires that the space spanned by α and γ is properly sampled in order to obtain a robust estimation of the plant shares wi corresponding to the sampled orientations Ai=(αi;γi) in that equation. The smoothness and form of the joint distribution displayed in Fig. 2 suggest that it may be possible to fit an analytical relationship, therefore reducing the number of parameters used to describe it. We propose to use the product of two Gaussian distributions: w̃iαi;γi=12πσα2exp⁡-αi-μα22σα212πσγ2exp⁡-γi-μγ22σγ2Ai=(αi;γi) In Eq. (2), the first product term corresponds to the normal distribution of α, characterised by a mean value μα and a standard deviation σα. The second product term corresponds to the normal distribution of γ, characterised by a mean value μγ and a standard deviation σγ. It appears reasonable to assume that the distribution of α is centred on a southwards orientation, so that μα can be set to 180∘. In Eq. (2), it is assumed that the distributions of the α and γ are independent – from Fig. 2 this assumption is an acceptable first-order approximation. A different notation has been used for the weights wi in Eqs. (1) and (2) since, as further explained in Sect. 5, values w̃i calculated with Eq. (2) must be normalized before being inserted into Eq. (1).

The use of the analytic form in Eq. (2) for the weights w̃i allows reducing the number of plant-relevant parameters down to three, which are the mean and standard deviation of γ and the standard deviation of α. This reduced form has been tested using the German solar plants. We found an average value of 20.6∘ for μγ and values of 19.3 and 10.8∘ for σα and σγ. The empirical histograms of the tilt and azimuth angles are compared to their fitted normal distributions in Fig. 3. The match between the histograms is not perfect but it seems an acceptable first-order approximation.

The three plant-related parameters necessary to calculate the total PV power produced in a given region from meteorological data has been determined above in the specific case of Germany. How can these three parameters be extended to other countries/regions? At this stage, one possibility may consist in using ones own expertise on the characteristics of PV plants installed in the considered regions as in Marinelli et al. (2015); Schubert (2012); another is a detailed statistical analysis of a dataset of plant information installed in the studied areas. Both ways hamper the easy use of the regional model aimed at in this work. To address this issue a parameterization of the three parameters is proposed in this section, which makes the model implementable in any region without any prior knowledge on the installed PV plants.

The statistical distribution of the plant capacity as a function of the module orientation of a region is the result of individual choices on the configurations of each single plant. It is affected by many factors of different nature such as the characteristics of the solar resource, the shading profile, architectural characteristics, different installation practices, etc. All these factors cannot be taken into consideration and we make the assumption that the most important one is the characteristics of the solar resource. We propose to take this into consideration through the use of an optimal tilt angle. The optimal tilt angle corresponds to the value of the tilt angle of a southwards oriented module yielding the largest annual output. In this work, we use the raster file of optimal tilt angles available on the PV-GIS website (http://re.jrc.ec.europa.eu/pvgis/), which is displayed in Fig. 4a as our starting point.

It can be observed in Fig. 4 that the optimal tilt angle γopt is ranging between 30 and 35∘ in Germany while the average value for the tilt angle has been found equal to 20.6∘ in the previous section. The reason for this mismatch is that a tilt angle smaller than the optimal tilt angle is commonly used to install more PV capacity per unit of surface and maximize the economic output of the plant. This practice has become more frequent with decreasing PV price and scarce available surfaces for new installations. We propose to quantify the mismatch between these two angles by a coefficient f. The average tilt angle μγ can thus be expressed as μγ=fγopt The unknown factor f can vary from one plant to another since it depends on numerous factors such as the solar resource, the plant cost per peak capacity or the land price. It may thus exhibit spatial and time variation and an accurate determination of this coefficient for all European countries may be difficult. We assume that this factor is spatially constant. Considering the average value of the tilt angles, which is equal to 20.6∘, the factor f should be chosen between 0.6 (20.6/35) and 0.7 (20.6/30). Given that the chosen dataset includes an under-representative share of large solar park, that usually have an optimal tilt angle (Saint-Drenan, 2015), we have chosen the upper bound for f (f=0.7). Similarly, we assume the standard deviations of the azimuth and tilt angles spatially constant and set them to the values found with the set of PV plants installed in Germany (19.3 and 10.8∘ for the azimuth and tilt angle).

The weights corresponding to the different orientation angles are finally estimated using Eq. (2), where the mean tilt angle is taken equal to the optimal tilt angle time the factor f=0.7 and the standard deviations of the azimuth and tilt angles are considered constant and equal to 19.3 and 10.8∘ respectively.

Some implementation details have been intentionally omitted in the previous sections for the sake of clarity and conciseness. This section provides some important details for the implementation of our method.

For the implementation of Eq. (1), the identification of a limited number of vectors Ai describing the reference module orientations is necessary. The accuracy of the model output and the computation cost will depend on the number of vectors chosen. It is thus an important step for an efficient use of our model. We used a set of 19 module orientation angles: three azimuth angles (170, 180 and 190∘) and 7 tilt angles ranging from 0 to 60∘ with a step of 10∘. These are represented by black squares in Fig. 2.

Our parameterization of the distribution of the module orientation is a function of the optimal tilt angle found on the PV-GIS website. This dataset is displayed in the left map of Fig. 4, where it can be observed that greater than average values are present in mountains (see e.g. regions of the Alps or the Pyrenees). These high values are presumably stemming from the high irradiation values present at high elevations. Since little PV plants are installed in these regions and to avoid overestimation of the tilt angle in the region neighbouring the mountains, these values have been filtered out. The resulting data are displayed in the right map of Fig. 4.

As already mentioned in Sect. 3, the expression given in Eq. (2) cannot be directly used to estimate the weights wi needed by Eq. (1). Indeed, for a finite sample of orientation angles Ai, the sum of the values w̃i evaluated with Eq. (2) is not equal to unity. To address this issue, w̃i estimated with Eq. (2) is normalized as follows to yield wi: wi=∫∫Diw̃i(α,γ)dαdγ∑i∫∫Diw̃i(α,γ)dαdγwithDi=αi-δα2;αi+δα2×γi-δγ2;γi+δγ2 The scalars δα and δγ in Eq. (6) represent respectively the resolutions of the azimuth and tilt angles, which are both equal to 10∘ in our implementation.

The model has been assessed by comparing its outputs to the PV power generated within a country. Given the approach is influenced by uncertainties in the input meteorological parameters, this comparison allows only an indirect evaluation of our model and not a quantification of the modelling accuracy. However, this approach offers a good balance between accuracy and versatility. The goal of this evaluation is thus to verify the plausibility of the model output for a particular model set up. Not only is there a lack of certainty in the input meteorological data but also there are various sources of uncertainty impacting the TSO data as well as the installed capacity used by the model, both making the conclusion of the validation difficult. To address these issues, we conduct the validation in two steps. In the first step, the validation is conducted for two countries: France and Germany, where we have long experience with both the installed capacity and the TSO data. In this first step, the impact of the uncertainty on the installed capacities and TSO estimates is under control but its spatial extension is limited. We therefore conduct a second step, where TSO data from 16 countries are considered. Given the lack of available information on the installed PV capacity in these countries, it is assumed spatially and temporally constant. The actual installed capacity being unknown, the validation is made by evaluating the correlation coefficient between TSO data and model output.

The assessment is first performed for Germany and France for the year 2014. The choice of these two countries has been strongly motivated by the comparatively high level of knowledge of their electricity supply structure and the availability of the data to conduct the validation. The PV power data was provided by the TSOs themselves with a time resolution ranging from 15 min to 1 h. A visual analysis of the time series was performed to control the data. The data was aggregated into 3 h means to conform to the temporal resolution of the meteorological data. Instants with no production by PV (night time) were excluded from the comparison.

The German case is used to validate the assumption made that the statistical quantities evaluated with 35 000 plants can be generalized to the ca. 1 500 000 plants installed in Germany at that time. France has a different level of PV development compared to Germany and is located at slightly different latitudes. This second case will test the validity of our approach to generalize the statistical quantities evaluated in Germany to another country with somewhat different meteorological conditions.

Gridded values of the normalized PV power were computed with the model using the bias-adjusted ERA-interim data proposed by the ECEM project (Jones et al., 2017) as meteorological inputs. A bias-adjusted dataset was preferred to the original ERA-Interim re-analysis dataset in order to limit the effect of error in the input meteorological data on the assessment of model performance. The bias-adjusted ERA-Interim covers the period from 1 January 1979 to 31 December 2016 and is covering Europe with a spatial resolution of 0.5∘ × 0.5∘. The domain covered by the data extends between 21.75 and 45.25∘ in longitude and between 26.75 and 72.25∘ in latitude. The two meteorological variables used for the calculation were the solar surface radiation downward (SSRD, also known as GHI) and air temperature at 2 m. As the input meteorological data has a time resolution of 3 h for SSRD and 6 h for temperature, an increase of the time resolution was needed to properly estimate the PV power generation with respect to the variation of the sun position with time. For this purpose, the temperature and clearness index (the ratio of SSRD to the irradiation at the top of atmosphere) were resampled down to a time resolution of 5 min by a linear interpolation technique. The normalized PV power was calculated with these resampled inputs and then summed up on 3 h periods, which is the original time resolution of the solar radiation data.

By using gridded maps of the installed PV capacity in each country (Fig. 5), the generated PV power was computed at each grid cell and then spatially summed to yield the production for each country. The data on the installed PV plants used for this purpose have been retrieved from the websites of the four German TSOs (PV-DE 2014) and from a data portal of the French government (PV-FR 2014). Finally, all time series have been normalized by the total installed PV capacity, which is equal to 6.17 and 36.87 GWp for France and Germany, respectively, in 2014 (PV-DE, 2014; PV-FR, 2014). Figure 6 exhibits the time series of both measured production and model outputs for France and Germany daily and 3 h resolutions. It reveals that the seasonal variations of the PV power are well assessed by the proposed model for the two countries and that the match between model output and actual values is qualitatively good. Scatter plots of the TSO data against the model outputs are displayed in Fig. 7 for France and Germany for the 3 hourly resolution and different error metrics are also displayed in Fig. 7. The data points are well centred on the identity line for Germany while an underestimation by the model can be observed for France. These observations are confirmed by the bias, which are respectively equal to -2.4×10-2 and 0.1×10-2 W Wp-1 for France and Germany. The correlation coefficient is large in both countries: 0.987 and 0.975 respectively for France and Germany. The MAE is respectively 3.8×10-2 and 2.4×10-2 W Wp-1; the RMSE is respectively 4.2×10-2 and 3.0×10-2 W Wp-1.

Some efforts were made to understand the reasons for the greater bias value observed for France. During this investigation we obtained access to the content of the bdpv.fr online portal (BDPV, 2018), which contains the main information for more than 20 000 PV plants installed in France. We used this new data source to compare the characteristics of the German and French PV plants and to verify the validity of our assumption for France.

The strongest assumption made in this work is to consider that the mean tilt angle is equal to the product of the optimal tilt angle and a constant f equal to 0.7. In order to verify this assumption, the ratio between actual and optimal module tilt angle has been analysed for the two countries. The histograms of this ratio are displayed for the two countries and for different classes of nominal capacity in Fig. 8. The assumed value for the ratio f is displayed by a dashed black line in these two plots. We can observe that the assumed ratio value matches well large German plants with an installed capacity greater than 500 kWp (no information on large PV plant is available in France). Data from both countries reveal that this value is not fitting actual tilt values of medium and small plants: an optimal ratio value of 0.4–0.6 would better match plants with an installed capacity between 50 and 500 kWp and a ratio value of 0.9–1.3 would be better for plants with an installed capacity smaller than 10 kWp. It is interesting to note that the optimal ratio changes with the size of the plants in a similar way for both countries. These observations indicate, that the variation of the share of PV plants according to their size between countries can bring about a deviation from the assumed factor of 0.7. A first possible explanation for bias observed in France may thus be that the distribution of French plants according to their size is different from the German one.

It would be interesting to exploit the trend observed in Fig. 8 in our model. However, information on the size of installed PV plants is missing in most European countries so that this is unfortunately impossible. Based on these new results, one can wonder whether the choice of a value of 0.7 for the ratio between actual and optimal tilt is still relevant? Given that larger plants have more weight for the calculation of the regional PV power generation than smaller plants, we consider that our estimate is not unfounded and we decide to keep this value.

In Fig. 8, it can also be observed that for PV plants with an installed capacity smaller than 10 kWp, the range of tilt angle values taken by French plants is larger than for German plants. To understand this difference, the tilt angle values of small plants have been displayed as a function of their geographic position (Fig. 9). In this map, a very large difference in tilt angles between North and South of France can be observed. This spatial difference is much more pronounced than the spatial variation that can be expected from the optimal tilt angle. Since such a marked spatial difference is not present in Germany, it could bed a second possible explanation to the observed bias in France.

As reported in Saint-Drenan (2015), the spatial variations of the tilt angle of small plants are resulting from regional architectural practices. It would therefore be tempting to integrate this information into our model. However, because this information is not commonly available (i.e. not even for France), it could not be accounted for in a robust way.

Though the results of this first validation can be considered as satisfactory, it is important to also demonstrate that results for Germany and France can be extrapolated to other (European) countries, also with different climates, engineering practices, etc. We therefore decided to conduct an additional validation step, in which we compared the output of our model to additional TSO data. To this end, we collected time series of solar power generation on the ENTSO-E Transparency Portal for 16 countries for the year 2015 and built 3-hourly averages to make the data comparable with the model output. The model setup is the same than in the previous validation except for the installed capacity, which is not known and thus assumed spatially and temporally constant (even in France and Germany). Indeed, the information available on the installed capacity is only updated yearly and we experiment several situations where the time series of the production were not matching with the given installed capacity (e.g. situation with production values greater than the installed capacity).

The comparison of the model output with the ENTSO-E data has been conducted for 16 countries. The scatter plot of the model output against ENTSO-E data is given in Fig. 10 for each country. As mentioned before, since the installed capacity is not known, the model output has not been scaled to the actual capacity. As a result, one should not consider the absolute error values in these plots but solely the correlation between the two time series. Accordingly, only the correlation coefficient is given in Fig. 10 and discussed in the remaining of this section. To facilitate the visualisation of the results, the correlation coefficients evaluated for the different countries are displayed as a map in Fig. 11.

With values greater than 0.97, the correlations are particularly high in Italy, France and Germany. These results confirm those obtained for France and Germany in the first validation. That the best correlation (0.982) is found for Italy is a very good surprise since no information on the PV plants installed in this country was considered in the model development. As we can see in Fig. 12, the high installed capacity in Italy (ca. 19 GWp in 2016) may account for this good performance. The correlation coefficients are high and comprised between 0.95 and 0.97 for six countries: Denmark, Belgium, Czech Republic, Slovakia, Greece and Portugal. This demonstrates that the proposed approach using the optimal tilt angle is valid at different latitudes. The low performance of the model for Spain is explained by the fact that the time series of solar generation available on the ENTSO-E website includes both photovoltaic and concentrated solar power generation. The reason for the medium performance in the remaining countries is unclear: it may stem from an intra-yearly change of the installed capacity, from lower performance of the re-analysis data in some regions or from other unidentified issues, including in the ENTSO-E generation data. It is however interesting to note that in the 16 countries, as shown in Fig. 8, the greater the installed capacity, the better the performance of our model performance. There may be several reasons to explain this observation: firstly, the relative effect of the intra-yearly new installations is lower when the installed capacity is high, and secondly, our assumption on the distribution of plants may only become valid as the number of plants exceeds a certain threshold.