A bi-variable probability density function for the daily clearness index

The assumption that probability density functions of daily clearness indices are unimodal was tested. Bimodal behaviour was observed in almost 60% of 600 monthly data sets and was shown to be the most usual shape at 50 locations between latitudes 18.43°N and 64.81°N and at between 2 and 2297 m above sea level. A bi-exponential probability density function was proposed that fits the observed behaviour of daily clearness indices. The proposed function uses the mean monthly clearness index and the mean monthly solar height at noon as inputs. The bi-exponential method for predicting daily distributions is shown to reduce the RMS error by over 20% compared with earlier methods. Parametric representations of monthly maximum and minimum clearness index using and are combined with the bi-exponential function to obtain a bi-variable probability density function and a cumulative distribution function , analogous to those of Liu and Jordan.     

A probability density function for the clearness index,with applications

This paper uses classical probability theory to derive expressions for the expected (or mean) value of quantities such as the irradiation on inclined surfaces, collector output, and net gain through windows. The random variables are the clearness index k_t and the diffuse fraction k, whose means are $\text{k}{}_{\mathrm{t}}$ and $\text{k}$, respectively. The probability function for k_t is assumed to depend only upon $\text{k}{}_{\mathrm{t}}$ and is represented by a simple function chosen so that its corresponding distribution function is a close fit to the one originally graphed by Liu and Jordan. The probability function for k is represented, for a given k_t, by an impulse function centred at the $\text{k}$ observed at that k_t; the Orgill-Hollands functional form is used for the function $\text{k}\text{(k}{}_{\mathrm{t}}\text{)}$. A general purpose integral expression is presented for the expected value of any quantity which can be expressed as a functionof k, k_t and time of day. The integral is evaluated for several such quantities: the solar irradiation on an inclined surface, the output of both flat plate and concentrating collectors, and the net heat gain through a window with an automatic shutter. Example calculations are included.     

Fluctuations in instantaneous clearness index: Analysis and statistics

Solar radiation is characterized by short fluctuations introduced by passing clouds. An analysis of these fluctuations with regard to solar energy applications should focus on the instantaneous clearness index. Its probability distribution for a given mean clearness index is, as a first approximation, independent from the season and partly also from the site. This is verified for four annual datasets from three different sites.An analysis of fluctuations in solar radiation must focus on their amplitude, persistence, and frequency of occurrence rather than their location in time. The Fourier analysis cannot satisfactorily provide this information since time series of the instantaneous clearness index exhibit no periodicity. Instead, a localized spectral analysis based on wavelet bases rather than on periodic-ones has been applied. This analysis allows the decomposition of the fluctuating clearness index signal into a set of orthonormal subsignals. Each of them represents one specific scale of persistence of the fluctuation.The annual mean square values of all subsignals have been analysed, permitting the allocation of the signal’s power content to the different scales of persistence of a fluctuation. These annual mean values agree well for the different datasets, indicating the existence of statistically significant mean square values of the fluctuations as a function of their persistence.The analysis offers a valuable tool for the estimation of power flow fluctuations introduced by direct solar energy systems. With further elaboration it may be applied by power system operators for network planning in distribution grids with a high density of embedded generation.     

Correlationships between clearness index and relative sunshine duration for Sudan

The clearness index for 16 meteorological stations in Sudan has been expressed in terms of the fraction of bright sunshine duration, applying a variety of regression forms, namely linear, exponential, power and second-order polynomial. The significance and performance characteristics of the relations have been viewed using several statistical procedures. Although all of the four relationships are empirical and exhibit relative discrepancies, it has been found that they can reasonably be used to represent the underlying effect of relative duration of sunshine on clearness index for mean values. The estimates of the relationships are statistically significant at the 99.9% confidence level. For around 85% of the cases, the absolute error is fairly less than 4%. No estimate is found to fall beyond about ±10% accuracy level. Unless otherwise investigated under extreme conditions, especially S/S_o = 0 and 1, the predictive value of the relations remains questionable for values of S/S_o outside the ranges covered herein. From the bulk of the data available, the important finding is that solar radiation is plentiful in this vast country and could be regarded as an essential, renewable source of energy for various applications.     

Making full use of the clearness index for parameterizing hourly insolation conditions

An enhanced parameterization of insolation conditions based only on the knowledge of global irradiance is presented. Two limitations associated with the current approach using the clearness index are pointed out: its dependence on solar elevation and its inability to differentiate between different conditions that produce the same global irradiance. Suggestions are provided which could overcome part of these limitations. Arguments are substantiated with solid experimental evidence. It is further shown that noticeable gains in accuracy for the decomposition of global into direct and diffuse irradiance are possible if one makes optimum use of the information available within a global irradiance time series.     

Estimation of global radiation using clearness index model for sizing photovoltaic system

A methodology for developing a simple theoretical model for calculating global insolation on a horizontal surface is described in this paper. The input parameters to the model are the latitude of the desired location and the amount of total precipitable water content in the vertical column at that location. Both the parameters are easily measurable with inexpensive instrument such as global positioning system (GPS). The principal idea behind the paper is to have a model that could be used for designing a photovoltaic system quickly and within reasonable accuracy. The model in this paper has been developed using measured data from 12 locations in India covering length and breadth of the country over a period of 9–22 years. The model is validated by calculating theoretical global insolation for five locations, one in north (New Delhi), one in south (Thiruvanandapuram), one in east (Kolkata), one in west (Mumbai) and one in central (Nagpur) part of India and comparing them with the measured insolation values for these five locations. The measured values of all these locations had been considered for developing the model. The model is further validated for a location (Goa) whose measured data is not considered for developing the model, by comparing the calculated and measured values of the insolation. Over the range of latitudes covering most parts of India, the error is within 20% of the measured value. This gives the credibility of the model and the methodology used for developing the model for any region in the world.     

Statistical distribution of the clearness index with radiation data integrated over five minute intervals

In this paper, the influence of the measurement interval of solar radiation data on the cumulative probability distribution of the clearness index is studied. The distribution observed in southern Spain is bimodal using 5 min data, and this property fades away as the data are aggregated over larger time intervals, and it also depends on the air mass. This is used to confirm the existence of two kinds of types of radiation associated with clear or cloudy skies. Also, with 5 min radiation data, a new statistical model is proposed, based on a mixture of two normal distributions, which provides a good fit for the data measured in Seville.     

Cumulative distribution curves of daily clearness index in a southern tropical climate

Most previous works, involving the analysis of the statistical properties of solar radiation, have considered places situated in the Northern Hemisphere, generally at latitudes above 30°. In this work, the cumulative distribution curves for 23 sites located in the Southern Hemisphere, in the tropical and inter-tropical regions of Brazil, between the Equator and 30° latitude were calculated, trying to enlarge the available information on the statistical properties of solar radiation towards locations in sub-equatorial regions of the world. Results are compared with four distinct models of daily cumulated distribution functions , proposed by Bendt et al., Hollands and Huget, Saunier et al., and Babu and Satyamurty. A first comparison shows that the function proposed by Saunier et al. is well adjusted to our experimental data for the greater part of the Brazilian locations, except for places with temperate climates, in which the Hollands and Huget model, with the maximum clearness index equal to 0.864, as proposed by these authors, is superior. By substituting the maximum clearness index 0.864 of the Hollands and Huget model for a local maximum, as obtained experimentally in this work, results improve considerably. This study ratifies the conclusions previously obtained by other authors such as Saunier et al., and Babu and Satyamurty on the non-universal character of the Liu and Jordan cumulative distribution functions (CDFs).     

Sky conditions at Ilorin as characterized by clearness index and relative sunshine

Clearness index (K_T) and relative sunshine (S) are used to characterize the sky conditions at Ilorin, Nigeria (latitude: 8°32′N, longitude: 4°34′E, elevation: 375 m), a tropical location. Daily analysis shows that the majority of the days are relatively cloudy with hardly any very clear or overcast days. Based on monthly averaged K_T, the monthly variations of cumulative frequency of K_T, f, and prevalent climatic conditions, the annual sky conditions have been classified into six patterns: three patterns each for the two broad seasons, dry and rainy. The trend in the seasonal variation of S lends further credence to this six seasonal pattern. Determined cumulative frequency curves further confirm the assertion that Liu and Jordan’s generalized cumulative distribution curves (CDC) are not applicable to tropical locations hence available design procedures based on these curves should be reconsidered [Liu, B. Y. U and Jordan, R. C. (1960). The Inter-relationaship and characteristic distribution of direct, diffuse and total solar radiation. Solar Energy 4, 1–19]. An empirical expression relating monthly maximum clearness index (K_max) and the monthly average clearness index (K_T) is presented and recommended for tropical locations. Comparative analysis shows that a similar expression by Saunier et al. is also appropriate to this location while that of Hollands and Huget is quite inadequate. [Saunier G. Y., Reddy T. A. and Kumar S. (1987). On monthly probability distribution function of daily global irradiation values appropriate for both tropical and temperate locations. Solar Energy 38, 169–177; Hollands K. G. T. and Huget R. G. (1983) A probability density function for clearness index, with applications. Solar Energy 30, 195]. Furthermore, the values of the minimum clearness index (0.12–0.24) for Ilorin are noticed to be much higher than the generally accepted value of 0.05 for overcast sky conditions. The results are further compared, especially with those of Ideriah and Suleman [Ideriah, F. J. K. and Suleman, S. O. (1989). Sky conditions at Ibadan during 1975–1980. Solar Energy 43, 325–330.] and Akuffo and Brew-Hammond [Akuffo, F. O. and Brew-Hammond, A. (1993). The frequency distribution of global solar radiation at Kumasi. Solar Energy 50, 145–154.] for Ibadan and Kumasi, all in the West African sub-region. Although the locations are slightly different in agro-climatic conditions, results are found to be very comparable.     

Diffuse solar radiation estimates from sunshine hours and clearness index for Karachi, Pakistan

Correlations for the estimation of monthly average daily diffuse solar radiation as a function of the sunshine hours and clearness index have been obtained from Karachi. Generally, two types of correlations are used: (a) diffuse radiation as a function of relative sunshine hours and extraterrestrial radiation, and (b) diffuse radiation as a function of global and extraterrestrial radiation. These correlations are mostly first- and second-order polynomials in the sunshine hours and clearness index, indicating the presence of the diffuse solar radiation component. The diffuse solar radiation shows a peak value during the monsoon months of July–August. The diffuse to global ratio is found to be 0.32 from the analysis, and the diffuse to extraterrestrial radiation ratio is nearly 0.19 throughout the year. Among the established relations, Iqbal and Stanhill overestimate the radiation value, while Liu and Jordan underestimate it.     

Assessment of the method used to construct clearness index maps for the new European Solar Radiation Atlas (ESRA)

We present an assessment of the methods used to construct maps for a new solar radiation atlas for Europe. For this atlas, station data and satellite-derived data are used in an interpolation/merging process to derive maps of the long-term monthly global radiation that cover an area ranging from 30° W to 70° E and from 25° to 75° N. Our focus is on the discussion of the accuracy of the method applied: a co-kriging technique. Special emphasis is put on a discussion as to whether the use of satellite-derived radiation maps with a low spatial resolution brings benefits.     

Comments on derivation of an index for evaluating economics of cogeneration systems and its applications

Industrial cogeneration systems usually must satisfy a power load and heat loads at different temperatures. The limitations of the economic index proposed by Pak and Suzuki for such cogeneration systems is discussed in this paper. The importance of a rational exergetic basis for evaluation of different grades of energy is emphasised. Thermodynamic criteria, e.g. the exergetic efficiency, relative fuel savings and fuel chargeable to power, are shown to provide useful information regarding cogeneration options. Any assessment scheme for cogeneration schemes must incorporate thermodynamic criteria in addition to economic criteria.     

The dependence on solar elevation of the correlation between monthly average hourly diffuse and global radiation

In the present work, the dependence on of the correlation between and is studied, , , and respectively being the monthly average hourly values of the global, diffuse, and extraterrestrial radiation, all of them on a horizontal surface, and the solar elevation at midhour. The dependence is studied for Uccle for the following sky conditions. Condition A: clear skies (fraction of possible sunshine = 1) and the maximum values of direct radiation measured during the period considered (each of the hours before or after the solar noon for which radiation is received); Condition B: corresponding to all the values of radiation measured when the sunshine fraction is 1 during the period considered; Condition C: corresponding to all the data collected, independently of the state of the sky; Condition D: corresponding to overcast skies ( ).From the available values of and (monthly average hourly direct radiation on a horizontal surface), values of and for 5° ≤ ≤ 45° and Δ = 5° are calculated using Newton's divided difference interpolation formula. The interpolated values occupy three clearly different regions in the plot, one for each of the conditions A,B, and C. For Condition A and each value of best linear fits with high correlation coefficients are obtained for the correlation. The influence of the Linke turbidity factor on the correlation for Condition A is studied for 5° ≤ ≤ 35°, Δ = 5°.     

Estimation of ambient temperature bin data from monthly average temperatures and solar clearness index. Validation of the methodology in two Greek cities

Ambient temperature bin data are used for estimating the energy consumption in HVAC systems with air-source heat pumps and cooling equipment. In this paper a methodology for estimating the ambient temperature bin data, based on monthly average outdoor temperatures and solar clearness index, is presented. For the two most populated cities of Greece, namely Athens and Thessaloniki, the estimated data are compared to the bin data produced by statistical analysis of 10 years hourly dry-bulb temperature measurements. Both data sets were also used for estimating the heating and cooling energy requirements of a case study building. The results obtained are similar, with very small differences, suggesting that the proposed methodology can be used for estimating bin data for other cities.     

Methodology for generating daily clearness index index values Kt starting from the monthly average daily value . Determining the daily sequence using stochastic models

The facility to generate weather data from limited inputs and independently of specific locations would allow simulations of energetic systems to be run at locations for which detailed weather records do not exist. This article presents a methodology to calculate synthetic daily solar radiation values and describes how sequences of daily global radiation can be generated using as input the monthly average radiation. A stochastic model, ARIMA(1,1,1) is presented, as well.     

A new approach to estimating the diffuse irradiance on inclined surfaces

The diffuse irradiance on an inclined surface is usually estimated from the hourly horizontal irradiance measurements with a slope irradiance model. It is also possible to calculate the slope irradiance by integrating the sky radiance distribution generated with a sky radiance model. In this paper, five slope irradiance models and six sky distribution models are compared with the hourly irradiance measurements on 24 inclined surfaces in Turku, Finland (60°27′N, 22°18′E). Of the sky distribution models, the Perez all-weather sky model agrees best with the measurements. Of the slope irradiance models, the Reindl model has the lowest average mean bias difference (MBD), but the Perez slope irradiance model gives the lowest root mean square difference (RMSD) for all but one of the 24 surface orientations. The average RMSD for the Perez all-weather sky model is 1.5 percentage points lower than for the Perez slope irradiance model.     

A note on diffuse solar radiation on a tilted surface

We are presenting a note on comparative study of diffuse component of solar radiation on tilted surface with different angle of inclination. The monthly mean daily diffuse radiation have been calculated on inclined surfaces from the data of horizontal surface using the Circumsolar and Isotropic model and the anisotropic model of Klucher and Hay at Lucknow (Latitude 26.75°, Longitude 80.85°), Uttar Pradesh, India. These calculated results are compared with our measured value. The comparison shows that Klucher model who considered the effect of cloudy sky conditions gives comparatively good estimations particularly at low inclination angles.     

Methodology for predicting sequences of mean monthly clearness index and daily solar radiation data in remote areas: Application for sizing a stand-alone PV system

In this paper, a suitable adaptive neuro-fuzzy inference system (ANFIS) model is presented for estimating sequences of mean monthly clearness index () and total solar radiation data in isolated sites based on geographical coordinates. The magnitude of solar radiation is the most important parameter for sizing photovoltaic (PV) systems. The ANFIS model is trained by using a multi-layer perceptron (MLP) based on fuzzy logic (FL) rules. The inputs of the ANFIS are the latitude, longitude, and altitude, while the outputs are the 12-values of mean monthly clearness index . These data have been collected from 60 locations in Algeria. The results show that the performance of the proposed approach in the prediction of mean monthly clearness index is favorably compared to the measured values. The root mean square error (RMSE) between measured and estimated values varies between 0.0215 and 0.0235 and the mean absolute percentage error (MAPE) is less than 2.2%. In addition, a comparison between the results obtained by the ANFIS model and artificial neural network (ANN) models, is presented in order to show the advantage of the proposed method. An example for sizing a stand-alone PV system is also presented. This technique has been applied to Algerian locations, but it can be generalized for any geographical position. It can also be used for estimating other meteorological parameters such as temperature, humidity and wind speed.     

Reply to comments on ‘derivation of an index for evaluating economics of cogeneration systems and its applications’

The Economics of a cogeneration system for district heating and cooling (CGS) depend not only on the exergetic efficiencies of generating heat and power, but also on various other factors such as the capacity factor, the construction cost of the CGS, and the annual cost rate for the construction cost. One of economic difficulties of a CGS constructed in Japan lies in the fact that capacity factor of the CGS is very low. A previously proposed profit index has the advantage that it makes it easily possible to evaluate the economic viability of a CGS based on its various fundamental economic and technological factors determining its economics. Sensitivity analyses of the various factors can also easily be performed.