Calculation of monthly average daily insolation on tilted, variously oriented surfaces using analytically weighted factors

A new, analytic method for calculating monthly average daily insolation on surfaces with various tilts and orientations is proposed. Beside the usual geometrical expressions concerning diurnal sun path, this method also takes into account changes in atmospheric transmissivity. Results are compared with those obtained by the widely accepted Klein's method also based on daily radiation data, as well as with more detailed and more accurate hourly calculations. Our results for two climatically very different locations in Yugoslavia stand in close agreement with hourly calculations for surfaces of arbitrary orientation and tilt, whereas Klein's method leads to significant error for large azimuth angles.     

Calculation of monthly average insolation on tilted surfaces

Several simplified design procedures for solar energy systems require monthly average meteorological data. Monthly average daily totals of the solar radiation incident on a horizontal surface are available. However, radiation data on tilted surfaces, required by the design procedures, are generally not available. A simple method of estimating the average daily radiation for each calendar month on surfaces facing directly towards the equator has been presented by Liu and Jordan [1]. This method is verified with experimental measurements and extended to allow calculation of monthly average radiation on surfaces of a wide range of orientations.     

Monthly average daily insolation on tilted collectors in Libya

This work presents monthly average total insolation values calculated for various locations in Libya. The insolation values are for actual sky conditions based on sunshine records. Latitudes ranging from 24 to 32°N and tilt angles from 0 to 90° are considered. The presented insolation values show that the optimum value of tilt angle for space heating systems is about latitude plus 15° while that for cooling systems is 0° A combination of horizontal roof and vertical south wall gives more or less uniform insolation throughout the whole year. The maximum annual insolation occurs for tilt angle nearly equal to the latitude.     

Comments on “Calculations of monthly average insolation on tilted surfaces” by S. A. Klein

The amount of solar energy that is intercepted by surfaces of any orientation is estimated from a new model of the clear sky, spatial distribution of solar radiation. The model was developed from measurements made during clear sky conditions and uses direct, isotropic reflected, and anisotropic diffuse radiation. The effects of azimuth, tilt, season, latitude, atmospheric turbidity, and reflectivity of the surroundings were computed using hourly measurements of normal beam and horizontal total radiation at four stations in the United States. A transformation of the co-ordinates of orientation produced a general relationship between orientation and intercepted energy. The general relationship was tested against measurements from six locations in the Northern Hemisphere and was found to be valid. The model is also a better estimator of energy intercepted by a tilted surface than are the more commonly used models.     

Calculation of monthly average global solar radiation on horizontal surfaces using daily hours of bright sunshine

Several statistical models calculating the monthly average global solar radiation on horizontal surfaces using the daily hours of bright sunshine have been extensively evaluated and compared for Canadian regions. Results show that Gariépy's model provides the best overall performance and Hay's model is rated next best, whereas Rietveld's model has been found to be the least accurate. Local performances of models have also been established throughout Canada that show that both Gariépy's and Hay's models perform quite well. Recommended models have been suggested for the studied regions.     

Calculation of monthly average insolation on a shaded surface at any tilt and azimuth

Estimation of the monthly average solar radiation on surfaces of arbitrary orientation is necessary for many solar performance prediction programs and is useful for other applications. For passive solar applications, especially, overhangs are often used to seasonally modulate the amount of radiation striking the surface.

Liu and Jordan[1] have developed a method for estimation of monthly average radiation on unshaded tilted surfaces based on horizontal surface data. This method has been extended to unshaded surfaces of arbitrary tilt and azimuth by Klein[2]. Utzinger and Klein[3] have presented a graphical method for estimating monthly average radiation on shaded vertical surfaces, while Jones [4]has offered an analytical method for the same configuration. This paper presents an analytical solution to the calculation of monthly average insolation on shaded surfaces at any tilt and azimuth. Results are comparable among the three methods when shaded vertical surfaces are analyzed.

This analytical method offers an alternative to slower and less accurate numerical integration and to less general regression of numerical integration results for use in solar performance prediction programs.     

Evaluation of models to predict insolation on tilted surfaces

An empirical study was performed to evaluate the validity of various insolation models which employ either an isotropic or an anisotropic distribution approximation for sky light when predicting insolation on tilted surfaces. Data sets of measured hourly insolation values were obtained over a 6-month period using pyranometers which received diffuse and total solar radiation on a horizontal plane and total radiation on surfaces tilted toward the equator at 37° and 60° angles above the horizon. Data on the horizontal surfaces were used in the insolation models to predict insolation on the tilted surface; comparisons of measured vs calculated insolation on the tilted surface were examined to test the validity of the sky light approximations. It was found that the Liu-Jordan isotropic distribution model provides a good fit to empirical data under overcast skies but underestimates the amount of solar radiation incident on tilted surfaces under clear and partly cloudy conditions. The anisotropic-clear-sky distribution model by Temps and Coulson provides a good prediction for clear skies but overstimates the solar radiation when used for cloudy days. An anisotropic-all-sky model was formulated in this effort which provided excellent agreement between measured and predicted insolation throughout the 6-month period.     

Hourly vs daily method of computing insolation on inclined surfaces

Insolation on south-facing inclined planes has been computed using hourly values of total and diffuse radiation, obtained from experimental data. Such a computation procedure is then compared with the widely used method by Liu and Jordan for obtaining daily insolation on surfaces tilted toward the equator. Very small differences are noted between the results obtained by the two methods. These differences are mainly due to three factors; (a) Liu and Jordan's formulation uses a theoretical day-length while the hourly method uses day-length as incicated by the radiation data; (b) hourly method takes into account the asymmetries of total and diffuse radiation around solar noon while the daily method implicitly assumes symmetry of the same; (c) the daily method assumes uniform atmospheric transmissivity to beam radiation throughout the day. On the other hand, the hourly method assumes constant atmospheric transmissivity for one hour only.     

Estimation of monthly average daily global solar irradiation using artificial neural networks

This study explores the possibility of developing a prediction model using artificial neural networks (ANN), which could be used to estimate monthly average daily global solar irradiation on a horizontal surface for locations in Uganda based on weather station data: sunshine duration, maximum temperature, cloud cover and location parameters: latitude, longitude, altitude. Results have shown good agreement between the estimated and measured values of global solar irradiation. A correlation coefficient of 0.974 was obtained with mean bias error of 0.059 MJ/m² and root mean square error of 0.385 MJ/m². The comparison between the ANN and empirical method emphasized the superiority of the proposed ANN prediction model.     

Two estimation methods for monthly mean hourly total irradiation on tilted surfaces from monthly mean daily horizontal irradiation from solar radiation data of Ajaccio, Corsica

The sizing of a photovoltaic or a thermal solar system is generally based on monthly mean values of daily solar radiation on tilted surfaces. Many authors have demonstrated that it will be better to use monthly mean values of hourly radiation, particularly taking into account the Sun's position and to predict long-term performances of solar systems. (Liu and Jordan, 1963; Clark et al., 1984). Moreover, for most of the sites around the world, only monthly mean values of daily horizontal total irradiation are available for use in such calculations. We propose, by using well-known correlations in the literature, to estimate these monthly mean values of hourly total irradiation on tilted planes from monthly mean values of daily total horizontal irradiation, using three steps:

• — determination of monthly mean value of hourly total horizontal irradiation;

• — determination of monthly mean value of hourly diffuse horizontal irradiation;

• — determination of monthly mean value of hourly total irradiation on tilted planes.

In the first step, using the Collares Pereira and Rabl correlation, the root mean square error (RMSE) between correlated and experimental calculated data is 8%. In the second step, we used two methods: the first one utilizes the Erbs correlation and the second one is based on a local correlation which has been developed in our centre. Both of them gave identical results with an RMSE lower than 9%. We calculated monthly mean values of hourly total irradiation on three tilted planes (30°, 45° and 60°) and we compared these results with the experimental ones, obtaining a RMSE respectively of less than 10%. The method is then validated by these results.     

Solar climates of the United States based on long-term monthly averaged daily insolation values

The purpose of this paper is to develop and analyze an objective classification and regionalization scheme of solar climates for the conterminous United States. The use of principal components analysis and Euclidean distance cluster analysis generates 18 solar climatic types based upon long-term mean monthly data. Two of the resultant regions contain 76 per cent of the 221 stations used in this study. The results are shown to vary significantly from the previously published maps of solar climates in the United States.     

Solar radiation on tilted south oriented surfaces: validation of transfer-models

Three transfer-models in use for estimating solar radiation on tilted surfaces are tested. A 12 year series of hourly global, diffuse, and reflected solar irradiation measured with horizontal pyranometers as well as hourly global solar irradiation measured with tilted south oriented pyranometers is available. One model uses daily irradiation, the other two use hourly irradiation. The models converting hourly solar irradiation on a horizontal surface to a tilted surface yield better results than that using daily irradiation. The best results are gained if pairs of hourly global and diffuse solar irradiation are available. The root mean square errors exceed 10% only if the sky is covered by more than 85% with clouds or if the solar elevation angle is less than 10°.     

Comparison between measurements and models for daily solar irradiation on tilted surfaces in Athens, Greece

This paper presents a comparison of several models against measurements of daily total solar radiation on surfaces of various tilts and azimuths in Athens. Such measurements are carried out at the Institute of Meteorology and Physics of the Atmospheric Environment, National Observatory of Athens and are the first of this kind in Greece. The models encounter a constant surface albedo of 0.2 and are of two categories. Category I includes models estimating hourly diffuse radiation on tilted planes (Bugler, Gueymard, Reindl and Temps-Coulson), which obtain the necessary global and diffuse horizontal radiations from either measurements or estimations from the Meterological Radiation Model (MRM) developed at the National Observatory of Athens. Category II includes Isotropic and Klein models for daily estimations; the models obtain their input from available measurements. Three types of diffuse radiation estimation in the second category are encountered. The purpose of the present paper is two-fold. First, it aims at showing the difference in estimating daily global irradiation between hourly models (category I) and daily models (category II). Second, it intends to give an alternative way to estimating daily irradiations on tilted surfaces using just MRM for engineering purposes.     

Determining seasonal optimum tilt angles,solar radiations on variously oriented,single and double axis tracking surfaces at Dhaka

The performance of a solar radiation conversion system is affected by its orientation and tilt angle with the horizontal plane. This is because both of these parameters change the amount of solar energy received by the surface of solar system. Three mathematical models- the Isotropic, the Klucher and the Perez model for the point source with parameters optimized for a variety of climatic conditions have been employed to determine hourly and seasonal optimum tilt angles. Theoretical optimum tilt angles (10° for Mar–Sep and 40° for Oct–Feb) were compared with measured data for Nov 2007 to Oct 2008 at Dhaka. The anisotropic Perez model showed least rmse of 0.09 for monthly tilt factor estimation. This model was also used for theoretical study of solar radiation on variously oriented, single axis and double axis tracking surfaces.     

Estimation of monthly average daily and hourly solar radiation impinging on a sloped surface using the isotropic sky model for Dhahran, Saudi Arabia

In this paper the isotropic sky method of Liu and Jordan is used to theoretically estimate the monthly average daily and hourly solar radiation impinging on an unshaded tilted surface in Dhahran, Saudi Arabia. The surface receiving solar radiation is assumed to be fixed at a tilt angle, β, equal to the latitude of Dhahran, Ø and oriented such that it is facing south with zero azimuth angle, γ. The calculation of total radiation on a sloped surface from measurements on a horizontal surface is discussed. Monthly average daily and hourly solar radiation values are then tabulated. The results obtained can be effectively employed in solar process design calculations.