We have implemented the mineral dust emission scheme of improved by the following benefits. (1) It is based on a global dataset of soil properties (size distribution, residual soil moisture). (2) It accounts for the soil dispersion state. (3) A tile approach was introduced to account for soil type heterogeneity at coarse resolutions. The scheme combines the parameterization of for the saltation flux with a parameterization of for the threshold friction velocity. When the friction velocity is above this threshold, soil erosion by wind is initiated. The resulting saltation flux is then used to calculate a dust emission flux using the parameterization of . The size distribution of the emitted mineral dust varies according to soil type and meteorological situation.

The size distributions of soil particles are crucial input parameters for the mineral dust emission scheme. In order to obtain a global coverage, HWSD data Harmonized World Soil Database; containing the global distribution of soil types are used. For each of these soil types, two limiting particle size distributions, each consisting of up to four log-normal distributions, are used see Table B1 in. The following approach to calculate dust emission fluxes applies for one single soil type. For a grid box containing different soil types, a tile approach is used and explained afterwards.

For the size distribution of soil particles ns(dp) we take the soil dispersion state into account. This can be described with the help of two limiting size distributions for weak erosion ns, m (minimally dispersed) and strong erosion ns, f (fully dispersed), in which the dispersion factor γd (between 0 and 1) determines the actual size distribution between the two limits depending on the friction velocity u* : ns(dp)=γd⋅ns, m(dp)+(1-γd)⋅ns, f(dp),γd=e-0.5⋅(u*-u*t, m)3. u*t, m is the global minimum (in a mathematical sense) of the threshold friction velocity u*t(dp). The threshold friction velocity is defined as the value of the friction velocity at an equilibrium of aerodynamic, cohesive and gravitational forces e.g.,. For values higher than the threshold friction velocity, emission of soil particles takes place. derive an expression for the threshold friction velocity from the equilibrium of moments of forces acting on the particle: u*t(dp)=fr⋅fη⋅An⋅ρsρ⋅g⋅dp+γnρ⋅dp, with An=0.0123 and γn=3⋅10-4 kgs-2. ρ is the air density, ρs=2650 kgm-3 is the density of the soil and g is the gravitational acceleration. For application in a global or regional model, we include correction factors to account for the effects of roughness elements (fr) and soil moisture (fη) on u*t e.g.,. Applying Eq. (), we can analytically derive the global minimum of the threshold friction velocity, which is needed in Eq. ().

Independent of the size distribution, no emission is possible below u*t, m. The factor to account for soil roughness elements is calculated using the expression of . This correction term is based on the percentage of plant coverage pp, a variable which is typically available in atmospheric modeling systems: fr=1-0.5⋅λ⋅1+0.5⋅90⋅λ,withλ=-0.35⋅ln⁡(1-pp).

Higher soil moisture leads to increased adhesive forces between the soil particles. Therefore the threshold friction velocity increases for higher soil moisture. To consider this behavior, the correction term by is used, which is based on gravimetric soil moisture in % η and the percentage clay content pc of the soil: fη=1+1.21⋅(η-η′)0.68,withη′=az⋅(0.0014⋅pc2+0.17⋅pc), where η′ is a minimum value of η. The factor az=5 is introduced to account for too-high soil moisture content in the model and to increase the performance of the emission scheme within ICON-ART. The derivation of pc is explained in more detail at the end of this section. The saltation flux is calculated following : Fh(dp)=Cwhite⋅ρgu*3⋅1+u*t(dp)u*⋅1-u*t2(dp)u*2, where Cwhite=0.7 is a linear scaling parameter to adapt the dust emission flux to measurements. The total saltation flux Fth is the result of an integration over all saltation particle diameters weighted by the product of the cross-sectional area and the number of particles . This weighting represents the contribution of particle surface area at a certain diameter to the total soil surface area: Fth=fre⋅∑s=14∫-∞∞Fh(dp)⋅π4dp2⋅ns(dp)∫-∞∞π4dp2⋅ns(dp)dln⁡dpdln⁡dp.

In contrast to , ns(dp) takes into account the soil dispersion state by using Eq. (). The summation is performed for the four log-normal size distributions of soil particles as described above. fre is the fraction of erodible soil calculated based on GlobCover2009 land use data . The fractions frb,i of the land use classes – bare areas, sparse vegetation, closed to open grassland, closed to open shrubland and mosaic grassland/forest-shrubland – sum up to the erodible fraction fre=∑i=15frb,i.

By impaction of the saltating particles, the saltation flux leads to the release of small particles creating a dust emission flux. use the kinetic energy of the impaction, which has to exceed the binding energies of particles in the soil to describe this process. Larger particles are less tightly bound to the soil, and hence their binding energy is smaller than that of smaller particles. This leads to the following equation that connects the saltation flux with the dust emission flux of aerosol mode l : Fv,l(dp)=π6⋅ρp⋅d3,l3⋅pl(dp)⋅βkin⋅Fh(dp)el, with βkin=163ms-2. The product βkin⋅Fh(dp) is the kinetic energy of the saltation particles and el the binding energy of particles of mode l. pl is the percentage of kinetic energy that is spent to release particles of mode l and is calculated based on the binding energies as summarized in Table 2 of . These percentages of kinetic energy are chosen such that particles in the largest mode are emitted first when the threshold friction velocity is exceeded. With increasing friction velocity, the share of smaller particles that are released increases. An integration of Eq. () over all saltation particle diameters weighted by their cross-sectional areas yields the total dust emission flux of mode l: Ftv,l=fre⋅∑s=14∫-∞∞Fv,l(dp)⋅π4⋅dp2⋅ns(dp)∫-∞∞π4⋅dp2⋅ns(dp)dln⁡dpdln⁡dp.

Similar to the saltation flux, ns(dp) takes the soil dispersion state into account (Eq. ). Equation () was derived for one specific soil type. As ICON-ART is typically used with grid spacings ranging from 100 to 102 km, a grid box may contain a mixture of different soil types. To account for this subgrid-scale heterogeneity, a tile approach is introduced.

The size distributions of soil particles from are available for soil types as defined by the US Department of Agriculture (USDA) except for silt, where silty loam is used instead. Table  provides an overview of the soil types and the percentaged clay content pc,i of soil type i. To determine the soil type in an ICON-ART grid element, information from the high-resolution (30 arc seconds) global database HWSD is aggregated to the target grid. By this, the fraction fs,i covered by soil type i with ∑i=114fs,i=1 is available to the model. There are 14 fractions as, in addition to the 13 soil types, 1 fraction accounting for water, rock and urban surfaces is considered. As stated previously, a tile approach is used to calculate the dust emission flux. For clarity, Eq. () was derived for one single soil type i. To consider the subgrid-scale heterogeneity of soil properties, the dust emission fluxes Ftv,l,i are calculated for each soil type separately and the result is then weighted with the corresponding fraction of the soil to get the final dust emission flux:

Ftv,l=∑i=113fs,i⋅Ftv,l,i.

The soil clay content used in Eq. () to calculate the residual soil moisture content in a grid element can be calculated with

pc=∑i=113fs,i⋅pc,i.

The calculation of dust emission fluxes Ftv,l,i for individual soil types within one grid element differs by the use of the individual soil particle size distributions ns,i(dp). However, it differs neither in the clay content of the soil within one grid element nor in the residual soil moisture η′. The simple reason for this is that the gravimetric soil moisture η used from the surface scheme of ICON-ART is a grid-scale variable and hence is the average value for one grid element. As Eq. () includes a difference between η and η′, grid-scale values for both variables are used.

ICON-ART includes online mineral dust–radiation interaction utilizing the Rapid Radiative Transfer Model (RRTM) as described in more detail by . To account for the influence of mineral dust on radiation, the optical properties of mineral dust are calculated offline once. For this, Mie calculations are applied, which require the complex refractive index of mineral dust as input. For the Mie calculations, a code developed by was used. This code in turn utilizes a subset developed by for the calculation of the scattering coefficients and truncation of the series. The code was adapted to allow for processing of multiple wavelengths and averaging to the RRTM wavebands in a post-processing step. A new polynomial parameterization was introduced to account for the change in median diameter during transport. The local radiative transfer parameters (extinction coefficient, single scattering albedo and asymmetry parameter) are then calculated online within ICON-ART through multiplication of the mass-specific mineral dust optical properties derived from Mie calculations with the current mineral dust mass concentration from ICON-ART. The actual radiative transfer parameters are then used by the RRTM radiation scheme, thereby accounting for the local mineral dust effect on radiation. Thus, changed radiative fluxes from the RRTM scheme feedback on the meteorological conditions, which themselves can influence the mineral dust processes again.

The values of the refractive index of mineral dust used for the Mie calculations are of great importance for the radiative properties extinction coefficient, single scattering albedo and asymmetry parameter. give a summary of the various influences. For example, peaks in the real part show as maxima of the extinction coefficient, whereas the single scattering albedo and thereby absorption is determined by the imaginary part of the refractive index.

In order to assess the uncertainties related to the refractive indices we have conducted a sensitivity study. The data sources used for the refractive indices are shown in Table  and the values for the real part and the imaginary part of the refractive index are presented in Fig. . The refractive indices used in ICON-ART (BUSE) are the same as those used by for COSMO-ART.

For the longwave part of the spectrum down to 4 µm we use values as published by . For wavelengths smaller than 4 µm the shape of the curve is still replicated, however, with a fit through smaller values for the imaginary part which are obtained from . This part of the spectrum is especially important as solar radiation intensity is highest at these wavelengths. The smaller values correspond to weaker absorptive and enhanced scattering properties of mineral dust in ICON-ART for this part of the spectrum. These smaller values are in better agreement with , who by inversion of AERONET (AErosol RObotic NETwork) retrievals determined values similar to . This is further supported by several studies: report inconsistencies between popular dust models and remote measurements due to uncertainties in the imaginary part of the refractive index. show a better representation of observed mineral dust radiative forcing in models using these refractive indices obtained from AERONET, confirming that mineral dust is less absorptive than previously thought. point out that in situ measurements reporting higher absorption values possibly measured a mixture of dust and absorbing aerosol. obtained imaginary part refractive index values similar to those used in ICON-ART.

The data presented in Fig.  exhibit a large scatter of the refractive indices. This has to be considered alongside other sources of uncertainty, such as the size distribution of the dust particles. To account for these uncertainties, we conducted Mie calculations for the refractive indices differing most from the ones we are using (Fig. a), namely that of and , as well as for varying median diameters of the log-normal distribution of the dust particles (Fig. b). The results show that changes in the size distribution lead to a stronger signal than changes in the refractive index. This is in agreement with findings of .

A recent study by provides the first regionally detailed values of refractive indices for the longwave part of the spectrum. Although for the direct effect of mineral dust on the PV-power forecast the shortwave part of the spectrum is of greater importance, a usage of this dataset can lead to changes in the mineral dust radiative effect on meteorological conditions and to a further improvement of the forecast.

In the operational version of ICON used at DWD, a bulk scheme is applied to treat the cloud microphysical processes. For this study, we are using the two-moment microphysics scheme of . This scheme solves prognostic budget equations for number and mass concentrations of six hydrometeor classes (cloud, rain, ice, snow, graupel, hail). For the size distribution of hydrometeors, generalized gamma distributions with constant shape parameters are used. It considers the microphysical processes of autoconversion, accretion, self-collection and breakup in the warm phase. For cold clouds, diffusional growth, freezing, aggregation, self-collection, riming and melting are taken into account.

For the nucleation of ice particles, a competition between heterogeneous and homogeneous freezing occurs. Homogeneous freezing describes the formation of an ice particle without the involvement of a solid ice nucleus (IN). It takes place at temperatures below 235 K at high supersaturations with respect to ice on the order of 40 to 80 %. As this process does not depend on aerosol characteristics and there are always sufficient liquid droplets available e.g.,, a large number of ice particles form nearly instantly as soon as these ambient conditions are met. In accordance with these assumptions, we set the number of liquid droplets available for homogeneous freezing to 1000 cm-3. This leads to a strong increase in water vapor depletion and therefore a fast decline of supersaturation, which in turn leads to small ice particles. Heterogeneous freezing occurs at surfaces of IN that grant favorable conditions for freezing. Depending on IN characteristics, heterogeneous freezing can even occur at temperatures close to 0 ∘C at ice saturation for a review see. Due to comparatively low concentrations of IN at heights where freezing occurs, the nucleation rate of pure heterogeneous freezing is typically 1 to 2 orders of magnitude lower than that of pure homogeneous freezing. Hence, the depletion of supersaturation takes longer and larger ice particles are formed. For ICON-ART, the empirical parameterization of is used to describe heterogeneous formation of ice particles. In an ascending air parcel, heterogeneous freezing occurs earlier (i.e., at higher temperatures and lower supersaturation) than homogeneous freezing. Hence, a sufficient number of IN can suppress homogeneous freezing due to depletion of water vapor. To account for these competing mechanisms, ICON-ART uses the parameterization of . The resulting discrepancy in number concentration and size distribution of ice particles between the two freezing mechanisms leads to differences in the radiative properties of the clouds. Mineral dust is one of the most ubiquitous types of aerosol and acts as IN at temperatures as high as -10 ∘C. Measurements show that other IN can usually be neglected for modeling studies .

The coupling of microphysics and the parameterization to account for competing effects of the freezing mechanisms was performed in a similar way to and with two exceptions. The SD of the assumed subgrid-scale Gaussian distribution of the vertical velocity was reduced by a factor of 0.3 to σw=0.3TKE, where TKE is the (prognostic) turbulent kinetic energy. The value of 0.3 was derived by tuning based on ML-CIRRUS measurements . Additionally, a budget variable for mineral dust acting as IN was introduced to prevent double counting. We followed the approach of by using a characteristic relaxation timescale of 4 h in which only part of the mineral dust is available for further heterogeneous freezing.

As mineral dust serves as ice nuclei in ICON-ART it modifies the physical properties of the simulated clouds. Consequently, the optical properties of the ice clouds are modified when mineral dust is present. The optical properties of the ice crystals are calculated according to based on the cloud ice effective radius.

Since in the operational setup a one-moment scheme for the microphysical processes is used, the effective radii are calculated as a function of the ice mass concentration only. Instead, for this study, the ice particle size distribution obtained from the two-moment microphysics scheme of is used. For the calculation of the ice particle effective radii, we apply the formula of .

To calculate PV power we apply the open-source PV modeling environment PV_LIB for python . It converts direct and diffuse radiation, temperature and wind speed into normalized PV power. Here the normalization is done with respect to the nominal capacity. Therefore, a specific PV module and PV inverter combination, as well as the module's tilt and orientation, need to be specified. Furthermore, the surface albedo and station height are necessary input parameters.

All evaluations concerning PV power presented in the following assume a south-oriented PV module with a nominal power of 220 W and a size of 1.7 m2. The chosen configuration represents a typical system for applications on residential or industrial rooftops in Germany. In more detail, a “Canadian Solar CS5P-220M” PV module and the micro-inverter “ABB: MICRO-0.25-I-OUTD-US-208” are selected. The software PV_LIB retrieves the corresponding module and inverter properties automatically from an online database provided by NREL (National Renewable Energy Laboratory of the US Department of Energy).

Mineral dust emitted from the Sahara during 1 and 2 April is transported towards central Europe along the forward flank of a trough. After being transported across France, it reaches Germany on the 4 April 2014. This can be seen in Fig. , which shows the spatial distribution of mineral dust optical depth at 500 nm (in the following abbreviated with AOD) at different dates. Already during the night of 4 April 2014, the southern part of Germany is covered by a mineral dust plume, which leads to an AOD between 0.25 and 1. Over France, higher values between 1 and 1.5 are simulated. During the day, the mineral dust is transported to the north so that in the evening all of Germany is affected by the mineral dust, with the highest AOD values of about 1 in the northwest. In the remaining parts the AOD values are between 0.25 and 0.5.

A qualitative comparison to satellite, ceilometer and lidar observations shows that the spatial distribution and temporal evolution of mineral dust as simulated over Europe is in good agreement with the available measurements. Unfortunately, these observations of mineral dust are hampered by the presence of clouds. The areas with high mineral dust loads coincide also with cloudy conditions and only a few observation time steps within the period of interest are available for a quantitative comparisons. In Fig. , the mineral dust aerosol optical depth as forecasted at 20 km grid spacing for case TT is shown for 4 April 2014. On top of that, filled circles provide the corresponding AERONET measurements. The observations are averaged within a time interval of 1 h before target time and represent level 2 coarse-mode AOD at 500 nm derived with Direct Sun Algorithm version 2 and Spectral Deconvolution Algorithm version 4.1; for a description see. The arrival of the dust cloud in eastern Germany is observed by the station Lindenberg. There is only a small spatial discrepancy with the forecasted location of the dust cloud. Note that the region with rapid increase of mineral dust concentration also visualizes the weak frontal zone spanning from the North Sea over eastern Germany to the southeast of Europe (see Sect. ).

The surface incoming shortwave irradiance (SIS, or global radiation) is the key parameter for adequate PV-power forecasts. In order to evaluate the numerical simulations we use in situ and remote sensing observations. Surface measurements of global radiation are available from SYNOP stations hourly and, with a temporal resolution of 1 min, from the pyranometer network of the DWD. Additionally, the surface shortwave radiation as retrieved from the SEVIRI (Spinning Enhanced Visible and InfraRed Imager) sensor on board the geostationary METEOSAT (Meteorological satellite) second-generation (MSG) satellite number 10 is available for comparison with model results. In particular, hourly SARAH-2 data Surface Solar Radiation Data records – Heliosat; are used, which are provided by the EUMETSAT (European Organisation for the Exploitation of Meteorological Satellites) Satellite Application Facility on Climate Monitoring (CM SAF). It should be noted that the retrieval algorithm for the SARAH-2 data employs a modified MACC (Monitoring Atmospheric Composition and Climate) aerosol climatology , which deviates from the actual concentrations especially during mineral dust episodes. Therefore, the satellite-derived SIS can be expected to overestimate the real SIS in the considered time period.

Figure  shows the horizontal distribution of SIS for the simulations TT and FF and for the satellite product SARAH-2 on 4 April 2014, 12:30 UTC. In addition to the simulation results, the SIS measured at SYNOP and pyranometer stations are depicted in circles. Pyranometer stations are indicated by bold circles. There are two main synoptic features that were not correctly captured by both numerical simulations. On the one hand, the cloud band along the Alps is missing (see Fig. ). On the other hand, the activity of the frontal system in the northern part of Germany and the related clouds are represented in a different way (see Fig. ). The cloud cover and corresponding precipitation is overestimated in eastern Germany, whereas the rainfall in northern Germany in the afternoon is underestimated (not shown). Such fine structures are challenging for day-ahead numerical weather predictions. Inherently, deterministic NWP forecasts are afflicted with errors. These arise from inaccurate initial conditions as well as from deficiencies in the NWP model, whereas small errors in the finer structure, such as the position of individual clouds, tend to grow more rapidly . Ensemble forecasts could provide an estimate of the reliability of the forecast and of individual synoptic patterns such as the discussed frontal system. However, this is not within the scope of this study, and we concentrate on the differences between the simulations TT and FF.

In the northern part of Germany both simulations show very low SIS values, which were also observed by the satellite and by the ground-based stations. In the western part, where the AOD reaches the highest values, simulation TT gives noticeably lower SIS than in case of simulation FF, which is in better agreement with the observations. These improvements can be attributed to the consideration of the interactions between mineral dust and meteorology.

The improvements in TT are confirmed by Fig. a), which shows a histogram of the differences (simulation – ground observation) for 4 April 2014 using hourly data at SYNOP stations (see Fig. ). In case of underestimation (negative ΔSIS), the errors for TT are larger than for FF. The opposite happens in the range of overestimation (positive ΔSIS), where the error is remarkably lower for TT. Furthermore, the difference between TT and FF is larger in the overestimation sector than in the underestimation sector. This becomes even more clear in the joint histogram (Fig. b). When the difference is negative in the FF case, the green regression line is almost identical to the one-to-one line indicating no systematic differences between TT and FF in the underestimation sector. In the overestimation sector, however, the overestimation is strongly reduced by the TT simulation as indicated by the regression line. This finding is confirmed by Table . The mean error and the SD is substantially reduced in case of TT. While the 5th percentile slightly increases in case of TT, the 95th percentile indicates that the overestimation of SIS is drastically reduced. The same holds for minimum and maximum values. In summary, this means that in the TT case, overestimations are substantially reduced compared to the FF case whereas the results remain similar for underestimations.

Beside the technical characteristics of the solar panels, PV-power output depends mainly on SIS, temperature and panel geometry such as orientation with respect to the sun. From that, it becomes clear that reliable day-ahead forecasts of PV power depend on accurate weather forecasts. The following example demonstrates that the PV-power forecasts for Germany failed tremendously for 4 April 2014.

Figure  shows the day-ahead PV-power forecast for 4 April 2014. The illustrated day-ahead PV-power forecast is the so-called meta-forecast of the German TSOs. It is a multi-model and multi-method product, which combines many different NWP and power forecast models as well as many different post-processing methods. The day-ahead PV-power forecast for 4 April 2014 overestimated the actual power production for Germany by up to 5.3 GW. The forecast error is also divided into the control areas of the four TSOs. In the areas of TenneT and Amprion, which cover west, central and southeast Germany see Fig. 1 of, the largest forecast errors occurred on 4 April 2014. These are not only the regions where large contributions of PV capacity are installed see Fig. 1b of but also the forecast of incoming solar radiation was challenged by the presence of clouds and aerosols as already discussed in Sect. . These discrepancies between energy demand and day-ahead forecast of supply need to be compensated, for example on the intra-day market of the European power exchange EPEX SPOT, where electricity is traded within the Austrian, French, German or Swiss transmission systems. Corresponding market data are openly published on the EEX Transparency Platform . Wrong forecasts may cause economic costs on the order of tens of millions of euros per day.

The transport of Saharan dust and the interactions of mineral dust particles with the atmosphere are not explicitly considered within conventional NWP forecasts. Most likely, this has contributed to the large PV-power forecast errors on 4 April 2014 (Fig. ). We will quantify the effect of the mineral dust outbreak on PV-power forecast for our own model results.

PV_LIB (see Sect. ) is used to transfer the observed and simulated meteorological variables into normalized PV power. Computed PV power based on the observed radiation by the 26 pyranometer stations throughout Germany (see Appendix ) is taken as reference. To convert meteorological variables from the NWP simulations into PV power, the closest grid point to each pyranometer station is considered.

Figure  shows the observed and simulated SIS as well as normalized PV power for the stations Mannheim, Meiningen, Trier and Weihenstephan. A reduction in positive forecast error (overestimation) can be observed for the stations Mannheim, Trier and Weihenstephan. For stations with considerable cloud cover, for example Meiningen, smaller differences between the TT and FF simulations are observed. To quantify the improvement of PV-power simulations of scenario TT, Table  summarizes error quantities for all 26 pyranometer stations and for different lead times. For 4 April 2014, the root mean square error (RMSE) for example is reduced by about 17 % from 0.124 to 0.103. Bold values in Table  indicate better results and confirm that the TT simulation showed better performance with respect to the observations.

As mentioned before, a substantial number of clouds were present on 4 April 2014. From that, the question arises of which portion of the changes we found in PV power are due to the direct aerosol effect and which are due to the indirect effect. In our case we are using a factorial method (FM) to separate the direct radiative effects from the indirect radiative effects on PV-power generation. For this purpose, an unreplicated 2k factorial design with k=2 is used . A short description of this FM is given in Appendix . This method was also used for other similar problems in atmospheric science e.g.,.

Figures  and show the results of the FM calculation for four selected SYNOP stations (Mannheim, Meiningen, Trier and Weihenstephan-Duernast). The geographical positions of the stations are given in Table . The figures show the temporal evolution of the difference in normalized PV power between TT and the measurements (green line, right ordinate) and the corresponding difference between FF and the measurements (red line, right ordinate). In contours, the percentage contribution of direct radiative (beige), indirect radiative (blue) and synergistic interaction (orange) effects to the change between TT and FF based on the FM calculation is given (left ordinate).

The more westerly stations (Mannheim, Trier) show a systematic overestimation of PV-power generation in the FF case. At both stations, the PV-power forecast is improved significantly in the TT simulation. For Mannheim, the contributions of the different effects is alternating, whereas in Trier the direct radiative effect dominates the result, nearly always accounting for more than 50 %. In Meiningen, the contributions of the different effects vary strongly, accompanied by several intersections of the FF and TT curves. However, the green TT curve looks like a damped version of the red FF curve. For overestimations of the PV power in the FF case, the TT result is lower and for underestimations the TT result is higher, leading to a better agreement with measurements and therefore an improvement in the forecast. One could argue that the overestimations are damped mostly by direct radiative effects (e.g., at 10:00 UTC and 14:00 UTC) whereas the underestimations are damped mostly by indirect effects (e.g., at 08:00 UTC and 13:00 UTC). This becomes more obvious at Weihenstephan. Between 06:00 UTC and 11:00 UTC, the overestimation of PV power is nearly exclusively caused by direct radiative effects. Starting from 12:00 UTC, the subsequent underestimation is nearly completely related to indirect radiative and synergistic interaction effects. From 14:00 UTC on, FF shows an overestimation, whereas in the TT case the forecast is improved mostly by direct radiative effects.

The results for all SYNOP stations with high-quality radiation measurements in Germany are shown in Fig. . For the classification of the results at these stations, five different characteristic measures are calculated. A mathematical description of these characteristic measures is given in Appendix . The integrated difference ID is a measure for the magnitude of the difference between TT and FF without any information about the sign of the difference. The mean improvement ratio IR‾ is a measure for whether the TT result is better or worse than the FF result, where IR‾=0 characterizes a perfect FF result, IR‾=1 a perfect TT result and IR‾=0.5 an indifferent result. Hence, IR‾>0.5 shows an improvement of the results due to the impact of mineral dust on cloud formation and radiation. CR‾, CC‾ and CI‾ state the mean percentage contribution of direct radiative, indirect radiative and synergistic interaction effects on the difference between FF and TT. The synergistic interaction represents the nonlinear feedbacks, which are acting between the two factors direct radiative effect and indirect radiative effect, when both effects are active at the same time. The contributions of the different effects are derived by the FM formulae B1–B7 given in Appendix B.

Of the 26 stations in total, 17 show an improvement of the forecast in the TT case compared to the FF case (i.e., IR‾>0.5). However, this includes also stations with small differences between the results of TT and FF where the significance of the measure IR‾ is low as it does not contain any information about the magnitude of the change. Focussing on stations with a high difference (here defined as ID>1), 14 out of 16 stations show an improvement. For very high differences (here defined as ID>2), nearly every station (8 out of 9) shows improvements in the TT forecast. The only exception is Nuernberg, where there is a strong underestimation of the PV power in the early morning which is not compensated by the improved forecast afterwards (not shown). In total, the indirect effect contributes for 45.41 % to this underestimation in Nuernberg. Averaging over all stations, 63.86 % of the differences are caused by the direct radiative effect, whereas 20.22 and 15.92 % are caused by the indirect and synergistic interaction effects, respectively. For stations with very high differences, this shifts to higher contributions of the direct effect (68.08, 16.21 and 15.71 %).

Taking a look at outlying stations, Hohenpeissenberg, Saarbruecken, Weihenstephan and Hamburg show comparatively high contributions of the direct effect of more than 80 %. They have in common that, in the FF case, the PV power was nearly always overestimated. The direct effect in the TT case leads to a decrease in the PV-power forecast, which in turn leads to an improvement. At Luedenscheid and Nuernberg, indirect effects contribute more than direct effects to the changes in PV-power forecast. These are the only two stations that show a worsening of the PV-power forecast out of the 16 stations with high differences. For Luedenscheid, strong contributions of indirect and synergistic interaction effects in the afternoon lead to an overestimation of the PV power in the afternoon while the FF forecast already shows a good result (not shown). The forecast for Nuernberg in the FF case already shows a systematic underestimation of the PV power. A mixture of direct and indirect radiative effects in the morning leads to a strong further underestimation, dominating the IR‾ score. The improvements mostly due to indirect and synergistic interaction effects afterwards are smaller (not shown).