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Power train models are required to simulate hence predict energy consumption of vehicles. Efficiencies for different components in power train are required. Common procedures use digitalised shell models (or maps) to model the efficiency of Internal Combustion Engines (ICE) and manual gearboxes (MG). Errors are connected with these models and affect the accuracy of the calculation. The accuracy depends on the configuration of the simulation, the digitalisation of the data and the data used. This paper evaluates these sources of error. The understanding of the source of error can improve the results of the modelling by more than eight percent.
In contrast to the German power supply, the energy supply in many West African countries is very unstable. Frequent power outages are not uncommon. Especially for critical infrastructures, such as hospitals, a stable power supply is vital. To compensate for the power outages, diesel generators are often used. In the future, these systems will increasingly be supplemented by PV systems and storage, so that the generator will have to be used less or not at all when needed. For the design and operation of such systems, it is necessary to better understand the atmospheric variability of PV power generation. For example, there are large variations between rainy and dry seasons, between days with high and low dust levels - caused by sandstorms (harmattan) or urban air pollution.
West Africa has great potential for the use of solar energy systems, as it has both a high solar radiation rate and a lack of energy production. West Africa is a very aerosol-rich region, whose effects on photovoltaic (PV) use are due to both atmospheric conditions and existing solar technology. This study reports the variability of aerosol optical properties in the city of Koforidua, Ghana over the period 2016 to 2020, and their impact on the radiation intensity and efficiency of a PV cell. The study used AERONET ground (Giles et al., 2019) and satellite data produced by CAMS (Gschwind, et al., 2019), which both provide aerosol optical depth (AOD) and metrological parameters used for radiative transfer calculations with libRadtran (Emde, et al., 2016). A spectrally resolved PV model (Herman-Czezuch et al., 2022) is then used to calculate the PV yield of two PV technologies: polycrystalline and amorphous silicon. It is observed that for both data sets, the aerosol is mainly composed of dust and organic matter, with a very increased AOD load during the harmattan period (December-February), also due to the fires observed during this period.
This study investigates the initial stage of the thermo-mechanical crystallization behavior for uni- and biaxially stretched polyethylene. The models are based on a mesoscale molecular dynamics approach. We take constraints that occur in real-life polymer processing into account, especially with respect to the blowing stage of the extrusion blow-molding process. For this purpose, we deform our systems using a wide range of stretching levels before they are quenched. We discuss the effects of the stretching procedures on the micro-mechanical state of the systems, characterized by entanglement behavior and nematic ordering of chain segments. For the cooling stage, we use two different approaches which allow for free or hindered shrinkage, respectively. During cooling, crystallization kinetics are monitored: We precisely evaluate how the interplay of chain length, temperature, local entanglements and orientation of chain segments influence crystallization behavior. Our models reveal that the main stretching direction dominates microscopic states of the different systems. We are able to show that crystallization mainly depends on the (dis-)entanglement behavior. Nematic ordering plays a secondary role.
Interne Audits können mehr
(2024)
Dieser Beitrag zeigt, wie das Deutsche Zentrum für Luft- und Raumfahrt e. V. (DLR) Zufriedenheitsanalysen aus zwei Sichtweisen durchführt: Aus Sicht der Auditoren und aus Sicht der Managementbeauftragten der auditierten Institute und Einrichtungen. Die Ergebnisse fließen in die jährliche Auditprogrammplanung ein. Damit wird der Nutzen von internen Audits gesteigert.
Im Interview mit Florian Kluge und Daniel Lohmann geht es um Interventionen als Teil einer Phase Null. Die beiden Architekturprofessoren der Alanus-Hochschule in Alfter und der TH Köln sehen die Intervention als Schnittstelle zwischen Architektur und Gesellschaft.
The cube in cube approach was used by Paul and Ishai-Cohen to model and derive formulas for filler content dependent Young's moduli of particle filled composites assuming perfect filler matrix adhesion. Their formulas were chosen because of their simplicity, and recalculated using an elementary volume approach which transforms spherical inclusions to cubic inclusions. The EV approach led to expression of the composites moduli that allows introducing an adhesion factor kadh ranging from 0 and 1 to take into account reduced filler matrix adhesion. This adhesion factor scales the edge length of the cubic inclusions, thus reducing the stress transfer area between matrix and filler. Fitting the experimental data with the modified Paul model provides reasonable kadh for PA66, PBT, PP, PE-LD and BR which are in line with their surface energies. Further analysis showed that stiffening only occurs if kadh exceeds [Formula: see text] and depends on the ratio of matrix modulus and filler modulus. The modified model allows for a quick calculation of any particle filled composites for known matrix modulus EM, filler modulus EF, filler volume content vF and adhesion factor kadh. Thus, finite element analysis (FEA) simulations of any particle filled polymer parts as well as materials selection are significantly eased. FEA of cubic and hexagonal EV arrangements show that stress distributions within the EV exhibit more shear stresses if one deviates from the cubic arrangement. At high filler contents the assumption that the property of the EV is representative for the whole composite, holds only for filler volume contents up to 15 or 20% (corresponding to 30 to 40 weight %). Thus, for vast majority of commercially available particulate composites, the modified model can be applied. Furthermore, this indicates that the cube in cube approach reaches two limits: (i) the occurrence of increasing shear stresses at filler contents above 20% due to deviations of EV arrangements or spatial filler distribution from cubic arrangements (singular), and (ii) increasing interaction between particles with the formation of particle network within the matrix violating the EV assumption of their homogeneous dispersion.
The cube in cube approach was used by Paul and Ishai-Cohen to model and derive formulas for filler content dependent Young´s moduli of particle filled composites assuming perfect filler matrix adhesion. Their formulas were chosen because of their simplicity, recalculated using an elementary volume approach which transforms spherical inclusions to cubic inclusions. The EV approach led to expression for the composites moduli that allow for introducing an adhesion factor kadh ranging from 0 and 1 to take into account none perfect reduced filler matrix adhesion. This adhesion factor scales the edge length of the cubic inclusions, thus, reducing the stress transfer area between matrix and filler. Fitting the experimental data with the modified Paul model provides reasonable kadh for PA66, PBT, PP, PE-LD and BR which are in line with their surface energies. Further analysis showed that stiffening only occurs if kadh exceeds <span class="math-tex">\( { \ \sqrt{E^M/E^F} \ }\) and depends on the ratio of matrix modulus and filler modulus. The modified model allows for a quick calculation of any particle filled composites for known matrix modulus EM, filler modulus EF, filler volume content vF and adhesion factor kadh. Thus, finite element analysis (FEA) simulations of any particle filled polymer parts as well as materials selection are significantly eased. FEA of cubic and hexagonal EV arrangements show that stress distributions within the EV exhibit more shear stresses if one deviates from the cubic arrangement. At high filler contents the assumption that the property of the EV is representative for the whole composite, holds only for filler volume contents up to 15 or 20 % (corresponding to 30 to 40 weight %). Thus, for vast majority of commercially available particulate composites, the modified model can be applied. Furthermore, this indicates that the cube in cube approach reaches two limits: i) the occurrence of increasing shear stresses at filler contents above 20 % due to deviations of EV arrangements or spatial filler distribution from cubic arrangements (singular), and ii) increasing interaction between particles with the formation of particle network within the matrix violating the EV assumption of their homogeneous dispersion.
Solar energy is one option to serve the rising global energy demand with low environmental impact.1 Building an energy system with a considerable share of solar power requires long-term investment and a careful investigation of potential sites. Therefore, understanding the impacts from varying regionally and locally determined meteorological conditions on solar energy production will influence energy yield projections. Clouds are moving on a short term timescale and have a high influence on the available solar radiation, as they absorb, reflect and scatter parts of the incoming light.2 However, the impact of cloudiness on photovoltaic power yields (PV) and cloud induced deviations from average yields might vary depending on the technology, location and time scale under consideration.
In this study, we investigate the thermo-mechanical relaxation and crystallization behavior of polyethylene using mesoscale molecular dynamics simulations. Our models specifically mimic constraints that occur in real-life polymer processing: After strong uniaxial stretching of the melt, we quench and release the polymer chains at different loading conditions. These conditions allow for free or hindered shrinkage, respectively. We present the shrinkage and swelling behavior as well as the crystallization kinetics over up to 600 ns simulation time. We are able to precisely evaluate how the interplay of chain length, temperature, local entanglements and orientation of chain segments influences crystallization and relaxation behavior. From our models, we determine the temperature dependent crystallization rate of polyethylene, including crystallization onset temperature.
The mechanical properties of plastic components, especially if they are made of semi-crystalline polymers, are considerably influenced by the process conditions. The degree of crystallization influences thermal and mechanical properties. Even more important is the orientation of molecules due to stretching of the polymer melt. Anisotropic material properties are the result of such orientations. Up to now all these effects are not considered within the simulation models of blow molded parts.
Photovoltaic (PV) power data are a valuable but as yet under-utilised resource that could be used to characterise global irradiance with unprecedented spatio-temporal resolution. The resulting knowledge of atmospheric conditions can then be fed back into weather models and will ultimately serve to improve forecasts of PV power itself. This provides a data-driven alternative to statistical methods that use post-processing to overcome inconsistencies between ground-based irradiance measurements and the corresponding predictions of regional weather models (see for instance Frank et al., 2018). This work reports first results from an algorithm developed to infer global horizontal irradiance as well as atmospheric optical properties such as aerosol or cloud optical depth from PV power measurements.
The electricity grid of the future will be built on renewable energy sources, which are highly variable and dependent on atmospheric conditions. In power grids with an increasingly high penetration of solar photovoltaics (PV), an accurate knowledge of the incoming solar irradiance is indispensable for grid operation and planning, and reliable irradiance forecasts are thus invaluable for energy system operators. In order to better characterise shortwave solar radiation in time and space, data from PV systems themselves can be used, since the measured power provides information about both irradiance and the optical properties of the atmosphere, in particular the cloud optical depth (COD). Indeed, in the European context with highly variable cloud cover, the cloud fraction and COD are important parameters in determining the irradiance, whereas aerosol effects are only of secondary importance.
Solar photovoltaic power output is modulated by atmospheric aerosols and clouds and thus contains valuable information on the optical properties of the atmosphere. As a ground-based data source with high spatiotemporal resolution it has great potential to complement other ground-based solar irradiance measurements as well as those of weather models and satellites, thus leading to an improved characterisation of global horizontal irradiance. In this work several algorithms are presented that can retrieve global tilted and horizontal irradiance and atmospheric optical properties from solar photovoltaic data and/or pyranometer measurements. The method is tested on data from two measurement campaigns that took place in the Allgäu region in Germany in autumn 2018 and summer 2019, and the results are compared with local pyranometer measurements as well as satellite and weather model data. Using power data measured at 1 Hz and averaged to 1 min resolution along with a non-linear photovoltaic module temperature model, global horizontal irradiance is extracted with a mean bias error compared to concurrent pyranometer measurements of 5.79 W m−2 (7.35 W m−2) under clear (cloudy) skies, averaged over the two campaigns, whereas for the retrieval using coarser 15 min power data with a linear temperature model the mean bias error is 5.88 and 41.87 W m−2 under clear and cloudy skies, respectively.
During completely overcast periods the cloud optical depth is extracted from photovoltaic power using a lookup table method based on a 1D radiative transfer simulation, and the results are compared to both satellite retrievals and data from the Consortium for Small-scale Modelling (COSMO) weather model. Potential applications of this approach for extracting cloud optical properties are discussed, as well as certain limitations, such as the representation of 3D radiative effects that occur under broken-cloud conditions. In principle this method could provide an unprecedented amount of ground-based data on both irradiance and optical properties of the atmosphere, as long as the required photovoltaic power data are available and properly pre-screened to remove unwanted artefacts in the signal. Possible solutions to this problem are discussed in the context of future work.
Solar photovoltaic power output is modulated by atmospheric aerosols and clouds and thus contains valuable information on the optical properties of the atmosphere. As a ground-based data source with high spatiotemporal resolution it has great potential to complement other ground-based solar irradiance measurements as well as those of weather models and satellites, thus leading to an improved characterisation of global horizontal irradiance. In this work several algorithms are presented that can retrieve global tilted and horizontal irradiance and atmospheric optical properties from solar photovoltaic data and/or pyranometer measurements. Specifically, the aerosol (cloud) optical depth is inferred during clear sky (completely overcast) conditions. The method is tested on data from two measurement campaigns that took place in Allgäu, Germany in autumn 2018 and summer 2019, and the results are compared with local pyranometer measurements as well as satellite and weather model data. Using power data measured at 1 Hz and averaged to 1 minute resolution, the hourly global horizontal irradiance is extracted with a mean bias error compared to concurrent pyranometer measurements of 11.45 W m−2, averaged over the two campaigns, whereas for the retrieval using coarser 15 minute power data the mean bias error is 16.39 W m−2.
During completely overcast periods the cloud optical depth is extracted from photovoltaic power using a lookup table method based on a one-dimensional radiative transfer simulation, and the results are compared to both satellite retrievals as well as data from the COSMO weather model. Potential applications of this approach for extracting cloud optical properties are discussed, as well as certain limitations, such as the representation of 3D radiative effects that occur under broken cloud conditions. In principle this method could provide an unprecedented amount of ground-based data on both irradiance and optical properties of the atmosphere, as long as the required photovoltaic power data are available and are properly pre-screened to remove unwanted artefacts in the signal. Possible solutions to this problem are discussed in the context of future work.
ISHIKAWA Diagramm
(2021)
In diesem Video wird der Aufbau und Nutzen eines Fischgräten-Diagramms zur Visualisierung von Ursachen oder Einflüssen auf eine Wirkung (Problem) oder ein Ziel (Lösung) gezeigt.
Die Vorteile liegen in der stammbaumähnlichen Darstellung aller möglichen Ursachen oder Einflüsse als Übersichtssammlung und in der fallweisen Eingrenzungsmöglichkeit je nach Relevanzeinschätzung.
Kinetic Inductance Detectors with Integrated Antennas for Ground and Space-Based Sub-mm Astronomy
(2009)
Very large arrays of Microwave Kinetic Inductance Detectors (MKIDs) have the potential to revolutionize ground and space based astronomy. They can offer in excess of 10.000 pixels with large dynamic range and very high sensitivity in combination with very efficient frequency division multiplexing at GHz frequencies. In this paper we present the development of a 400 pixel MKID demonstration array, including optical coupling, sensitivity measurements, beam pattern measurements and readout. The design presented can be scaled to any frequency between 80 GHz and >5 THz because there is no need for superconducting structures that become lossy at frequencies above the gap frequency of the materials used. The latter would limit the frequency coverage to below 1 THz for relatively high gap materials such as NbTiN. An individual pixels of the array consist of a distributed Aluminium CPW MKID with an integrated twin slot antenna at its end. The antenna is placed in the in the second focus of an elliptical high purity Si lens. The lens-antenna coupling design allows room for the MKID resonator outside of the focal point of the lens. The best dark noise equivalent power of these devices is measured to be NEP = 7×10-19 W/[square root]Hz and the optical coupling efficiency is around 30%, in which no antireflection coating was used on the Si lens. For the readout we use a commercial arbitrary waveform generator and a 1.5 GHz FFTS. We show that using this concept it is possible to read out in excess of 400 pixels with 1 board and 1 pair of coaxial cables.
Korrelationsdiagramm
(2021)
Wo Laborexperimente zu aufwendig, zu teuer, zu langsam oder zu gefährlich oder Stoffeigenschaften gar nicht erst experimentell zugänglich sind, können Computersimulationen von Atomen und Molekülen diese ersetzen oder ergänzen. Sie ermöglichen dadurch Reduktion von Kosten, Entwicklungszeit und Materialeinsatz. Die für diese Simulationen benötigten Molekülmodelle beinhalten zahlreiche Parameter, die der Simulant einstellen oder auswählen muss. Eine passende Parametrierung ist nur bei entsprechenden Kenntnissen über die Auswirkungen der Parameter auf die zu berechnenden Größen und Eigenschaften möglich. Eine Gruppe von Standardparametern in molekularen Simulationen sind die Partialladungen der einzelnen Atome innerhalb eines Moleküls. Die räumliche Ladungsverteilung innerhalb des Moleküls wird durch Punktladungen auf den Atomzentren angenähert. Für diese Annäherung existieren diverse Ansätze für verschiedene Molekülklassen und Anwendungen. In diesem Teilprojekt des Promotionsvorhabens wurde systematisch der Einfluss der Wahl des Partialladungssatzes auf potentielle Energien und ausgewählte makroskopische Eigenschaften aus Molekulardynamik-Simulationen evaluiert. Es konnte gezeigt werden, dass insbesondere bei stark polaren Molekülen die Auswahl des geeigneten Partialladungssatzes entscheidenden Einfluss auf die Simulationsergebnisse hat und daher nicht naiv, sondern nur ganz gezielt getroffen werden darf.
Kreatives Schreiben
(2023)
The lattice Boltzmann method (LBM) stands apart from conventional macroscopic approaches due to its low numerical dissipation and reduced computational cost, attributed to a simple streaming and local collision step. While this property makes the method particularly attractive for applications such as direct noise computation, it also renders the method highly susceptible to instabilities. A vast body of literature exists on stability-enhancing techniques, which can be categorized into selective filtering, regularized LBM, and multi-relaxation time (MRT) models. Although each technique bolsters stability by adding numerical dissipation, they act on different modes. Consequently, there is not a universal scheme optimally suited for a wide range of different flows. The reason for this lies in the static nature of these methods; they cannot adapt to local or global flow features. Still, adaptive filtering using a shear sensor constitutes an exception to this. For this reason, we developed a novel collision operator that uses space- and time-variant collision rates associated with the bulk viscosity. These rates are optimized by a physically informed neural net. In this study, the training data consists of a time series of different instances of a 2D barotropic vortex solution, obtained from a high-order Navier–Stokes solver that embodies desirable numerical features. For this specific text case our results demonstrate that the relaxation times adapt to the local flow and show a dependence on the velocity field. Furthermore, the novel collision operator demonstrates a better stability-to-precision ratio and outperforms conventional techniques that use an empirical constant for the bulk viscosity.
In dieser Arbeit werden neuartige methodische Erweiterungen der Lattice-Boltzmann-Methode (LBM) entwickelt, die effizientere Simulationen inkompressibler Wirbelströmungen ermöglichen. Diese Erweiterungen beheben zwei Hauptprobleme der Standard-LBM: ihre Instabilität in unteraufgelösten turbulenten Simulationen und ihre Beschränkung auf reguläre Rechengitter. Dazu wird zunächst eine Pseudo-Entropische Stabilisierung (PES) entwickelt. Diese kombiniert Ansätze der Multiple-Relaxation-Time (MRT)-Modelle und der Entropischen LBM zu einem expliziten, lokalen und flexiblen Stabilisierungsoperator. Diese Modifikation des Kollisionsschritts erlaubt selbst auf stark unteraufgelösten Gittern stabile und qualitativ korrekte Simulationen. Zur Erweiterung der LBM auf irreguläre Rechengitter wird zunächst eine moderne Discontinuous-Galerkin-LBM untersucht und um stabilere Zeitintegratoren ergänzt. Diese Studie demonstriert die drastischen Schwächen existierender LBMAnsätze auf irregulären Gittern. Basierend auf den gewonnenen Erkenntnissen gelingt die Formulierung einer neuartigen Semi-Lagrangeschen LBM (SLLBM). Diese ermöglicht in einzigartigerWeise sowohl die Verwendung irregulärer Gitter und großer Zeitschritte als auch eine hohe räumliche Konvergenzordnung. Anhand von Beispielsimulationen wird demonstriert, wieso dieser Ansatz anderen aktuellen Off-Lattice-Boltzmann-Methoden (OLBMs) in Effizienz und Genauigkeit überlegen ist. Weitere neuartige Aspekte dieser Arbeit sind die Entwicklung eines modularen Off-Lattice-Boltzmann-Codes und die Erweiterung der LBM um implizite Mehrschrittverfahren, mit denen eine Erhöhung der zeitlichen Konvergenzordnung gelingt.
In dieser Arbeit wird eine kompressible Semi-Lagrangesche Lattice-Boltzmann-Methode neu entwickelt und erprobt. Die Lattice-Boltzmann-Methode ist ein Verfahren zur numerischen Strömungssimulation, das auf einer Modellierung von Partikeldichten und deren Interaktion untereinander basiert. In ihrer Ursprungsform ist die Methode jedoch auf schwach kompressible Strömungen mit niedriger Machzahl beschränkt. Wesentliche Nachteile der bisherigen Versuche zur Erweiterung auf supersonische Strömungen sind entweder mangelhafte Stabilität der Verfahren, unpraktikabel große Geschwindigkeitssätze oder die Beschränktheit auf kleine Zeitschrittweiten. Als Alternative zu bisherigen Ansätzen wird in dieser Arbeit ein Semi-Lagrangescher Strömungsschritt eingesetzt. Semi-Lagrangesche Verfahren entkoppeln mittels Interpolation die Orts-, Zeit- und Geschwindigkeitsdiskretisierung der ursprünglichen Lattice-Boltzmann-Methode. Nach der Einleitung wird im zweiten und dritten Kapitel dieser Arbeit zunächst auf die Grundlagen und Prinzipien der Lattice-Boltzmann-Methode eingegangen sowie bisherige Ansätze zur Simulation kompressibler Strömungen aufgeführt. Im Anschluss wird die kompressible Semi-Lagrangesche Lattice-Boltzmann-Methode entwickelt und beschrieben. Die Erweiterung erfolgt im Wesentlichen durch die Verknüpfung der Methode mit geeigneten Gleichgewichtsfunktionen und Geschwindigkeitssätzen. Im vierten Kapitel der Arbeit werden neue Kubatur-basierte Geschwindigkeitssätze entwickelt und getestet, darunter ein D3Q45-Geschwindigkeitssatz zur Berechnung kompressibler Strömungen, der den Rechenaufwand gegenüber konventionellen Geschwindigkeitsdiskretisierungen erheblich verringert. Im fünften Kapitel der Arbeit werden zur Validierung Simulationen von eindimensionalen Stoßrohren, zweidimensionalen Riemann-Problemen und Stoß-Wirbel-Interaktionen durchgeführt. Im Anschluss zeigen Simulationen von dreidimensionalen, kompressiblen Taylor-Green-Wirbeln sowie von wandgebundenen Testfällen die Vorteile der Methode für kompressible Strömungssimulationen. Zu diesem Zweck werden die Überschallströmung um ein zweidimensionales NACA-0012-Profil und um eine dreidimensionale Kugel sowie eine supersonische Kanalströmung untersucht. Dem Simulationsteil folgt eine umfangreiche Diskussion der Semi-Lagrangeschen Lattice-Boltzmann-Methode im Vergleich zu anderen Methoden. Die Vorteile der Methode, wie vergleichsweise große Zeitschrittweiten, körperangepasste Netze und die Stabilität der Methode, werden hier herausgearbeitet.
The lattice Boltzmann method (LBM) is an efficient simulation technique for computational fluid mechanics and beyond. It is based on a simple stream-and-collide algorithm on Cartesian grids, which is easily compatible with modern machine learning architectures. While it is becoming increasingly clear that deep learning can provide a decisive stimulus for classical simulation techniques, recent studies have not addressed possible connections between machine learning and LBM. Here, we introduce Lettuce, a PyTorch-based LBM code with a threefold aim. Lettuce enables GPU accelerated calculations with minimal source code, facilitates rapid prototyping of LBM models, and enables integrating LBM simulations with PyTorch's deep learning and automatic differentiation facility. As a proof of concept for combining machine learning with the LBM, a neural collision model is developed, trained on a doubly periodic shear layer and then transferred to a different flow, a decaying turbulence. We also exemplify the added benefit of PyTorch's automatic differentiation framework in flow control and optimization. To this end, the spectrum of a forced isotropic turbulence is maintained without further constraining the velocity field.
Long-term variability of solar irradiance and its implications for photovoltaic power in West Africa
(2020)
This paper addresses long-term changes in solar irradiance for West Africa (3° N to 20° N and 20° W to 16° E) and its implications for photovoltaic power systems. Here we use satellite irradiance (Surface Solar Radiation Data Set-Heliosat, Edition 2.1, SARAH-2.1) to derive photovoltaic yields. Based on 35 years of data (1983–2017) the temporal and regional variability as well as long-term trends of global and direct horizontal irradiance are analyzed. Furthermore, at four locations a detailed time series analysis is undertaken. The dry and the wet season are considered separately.
Long-term variability of solar irradiance and its implications for photovoltaic power in West Africa
(2020)
West Africa is one of the least developed regions in the world regarding the energy availability and energy security. Located close to the equator West Africa receives high amounts of global horizontal irradiance (GHI). Thus, solar power and especially photovoltaic (PV) systems seem to be a promising solution to provide electricity with low environmental impact. To plan and to dimension a PV power system climatological data for global horizontal irradiance (GHI) and its variability need to be taken into account. However, ground based measurements of irradiances are not available continuously and cover only a few discrete locations.