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 diffuse fraction, with applications

A revised model relating the diffuse fraction k of solar radiation to the clearness index k_t is presented. The model extends an earlier model by Hollands, to account for the effect of multiple interreflections between the atmosphere and the ground (atmospheric back-scattering). The revised model, which now includes the ground reflectance (albedo) as a parameter, is shown to be in reasonable agreement with experimental data. The implications of this new model for the design of solar systems is shown to be of possible importance.     

A probability density function for daily insolation within the temperate storm belts

Long-term insolation data are frequently obtainable either from recordings or from various empirical formulae. Many radiation-driven processes are, however, highly non-linear. The ability to model the correct long-term frequency distribution of a sequence of radiation events having a specified long-term mean insolation, is, therefore, valuable. This paper emphasizes that the “U-shaped” cloud cover frequency distribution, characteristic of the temperate storm belts, is “mapped” into the irradiation domain in a way that augments the number of “poor” and “excellent” days of sunshine at the expense of “average” days. An analytical and adjustable model is proposed, correlating the probability density of daily clearness index with the monthly average clearness index. This model is tuned to the two longest records of solar radiation in Norway, and it is verified against independent observations from the northern temperate storm belt.     

A derivation of the diffuse fraction's dependence on the clearness index

When averaged over many events having the same k_t, the average diffuse fraction carries an important quasi-universal functional dependence on the clearness index k_t. This paper presents a theoretical derivation of k(k_t). The atmosphere is modeled as having two homogeneous, nonselectively absorbing layers: an upper layer (roughly, the ozonosphere) with zero scattering and beam transmittance τ_u, and a lower layer with isotropic, nonselective scattering characterized by an albedo ω_l. After an analysis which traces generations of scattered photons, this model predicts a function which, upon adjusting ω_l and τ_u, fits the measured data closely.     

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.     

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.     

Progress in probability density function methods for turbulent reacting flows

Probability density function (PDF) methods offer compelling advantages for modeling chemically reacting turbulent flows. In particular, they provide an elegant and effective resolution to the closure problems that arise from averaging or filtering the highly nonlinear chemical source terms, and terms that correspond to other one-point physical processes (e.g., radiative emission) in the instantaneous governing equations. This review is limited to transported PDF methods, where one models and solves an equation that governs the evolution of the one-point, one-time PDF for a set of variables that determines the local thermochemical and/or hydrodynamic state of a reacting system. Progress over the previous 20–25 years (roughly since Pope's seminal paper [24]) is covered, with emphasis on developments over the past decade. For clarity and concreteness, two current mainstream approaches are adopted as baselines: composition PDF and velocity–composition PDF methods for low-Mach-number reacting ideal-gas mixtures, with standard closure models for key physical processes (e.g., mixing models), and consistent hybrid Lagrangian particle/Eulerian mesh numerical solution algorithms. Alternative formulations, other flow regimes, additional physics, advanced models, and alternative solution algorithms are introduced and discussed with respect to these baselines. Important developments that are discussed include velocity–composition–frequency PDF's, PDF-based methods as subfilter-scale models for large-eddy simulation (filtered density function methods), PDF-based modeling of thermal radiation heat transfer and turbulence–radiation interactions, PDF-based models for soot and liquid fuel sprays, and Eulerian field methods for solving modeled PDF transport equations. Examples of applications to canonical systems, laboratory-scale flames, and practical combustion devices are provided to emphasize key points. An attempt has been made throughout to strike a balance between rigor and accessibility, between breadth and depth of coverage, and between fundamental physics and practical relevance. It is hoped that this review will contribute to broadening the accessibility of PDF methods and to dispelling misconceptions about PDF methods. Although PDF methods have been applied primarily to reacting ideal-gas mixtures using single-turbulence-scale models, multiple-physics, multiple-scale information is readily incorporated. And while most applications to date have been to laboratory-scale nonpremixed flames, PDF methods can be, and have been, applied to high-Damköhler-number systems as well as to low-to-moderate-Damköhler-number systems, to premixed systems as well as to nonpremixed and partially premixed systems, and to practical combustion devices as well as to laboratory-scale flames. It is anticipated that PDF-based methods will be adopted even more broadly through the 21st century to address important combustion-related energy and environmental issues.     

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.     

A joint probability density function of wind speed and direction for wind energy analysis

A very flexible joint probability density function of wind speed and direction is presented in this paper for use in wind energy analysis. A method that enables angular–linear distributions to be obtained with specified marginal distributions has been used for this purpose. For the marginal distribution of wind speed we use a singly truncated from below Normal–Weibull mixture distribution. The marginal distribution of wind direction comprises a finite mixture of von Mises distributions. The proposed model is applied in this paper to wind direction and wind speed hourly data recorded at several weather stations located in the Canary Islands (Spain). The suitability of the distributions is judged from the coefficient of determination R².

The conclusions reached are that the joint distribution proposed in this paper: (a) can represent unimodal, bimodal and bitangential wind speed frequency distributions, (b) takes into account the frequency of null winds, (c) represents the wind direction regimes in zones with several modes or prevailing wind directions, (d) takes into account the correlation between wind speeds and its directions. It can therefore be used in several tasks involved in the evaluation process of the wind resources available at a potential site. We also conclude that, in the case of the Canary Islands, the proposed model provides better fits in all the cases analysed than those obtained with the models used in the specialised literature on wind energy.     

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.     

Large eddy simulation of autoignition with a subgrid probability density function method

This paper applies the Eulerian stochastic field method to the solution of the subgrid joint probability density function (PDF) of the reacting scalars in a large eddy simulation (LES) of a jet of hydrogen issuing into a co-flow of vitiated air. The hot co-flow induces autoignition of the mixture and a lifted flame results downstream of the nozzle exit. The simulations were performed using a detailed H₂-air mechanism. The results were found to be sensitive to the co-flow temperature even with temperatures varied within the experimental uncertainty. The results obtained were in excellent agreement with the experimental data, both quantitatively and qualitatively. The method was able to capture partially premixed and partially extinguish zones with a relatively small number of stochastic fields. The radical HO₂ was found to be the trigger for autoignition. The fact that no large-scale premixed flame propagation was observed suggests that the stabilization mechanism is associated mainly with the chemistry.     

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.     

Large eddy simulation/probability density function simulations of bluff body stabilized flames

This work presents large eddy simulation/probability density function (LES/PDF) simulation results for the Sandia/Sydney series of bluff-body stabilized CH₄/H₂${\text{CH}}_4/{\text{H}}_2$ flames. Results are presented for the flames HM1, HM2 and HM3, using the 19-species ARM2 reduced chemical mechanism, and comparison is made with previous numerical simulations of the same flames. When compared to previous numerical studies of these bluff-body flames, the present simulation shows considerable improvement, particularly in the downstream regions of the flow. The simulations are shown to be sensitive to the treatment of heat transfer to the bluff-body face, with better agreement in the temperature profiles achieved with the addition of a Dirichlet temperature boundary condition.     

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.     

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.     

One-minute global irradiance probability density distributions conditioned to the optical air mass

Knowledge of the temporal variability of the solar irradiance is important to study solar energy systems involving thermal and photovoltaic processes. The differences between hourly and instantaneous values of the clearness index considerably affect the utilizability of photovoltaic systems. In this work, we analyzed the probability density distributions of one-minute values of global irradiance, conditioned to the optical air mass, considering those as an approximation to the instantaneous distributions. The study reveals that the bimodality that characterizes these distributions increases with optical air mass. We propose the use of a functional form based on Boltzmann's statistics in order to describe these distributions. This function can be used for the generation of synthetic radiation data. Expressing the distribution as a sum of two functions provides an appropriate modeling of the bimodality feature that can be associated with the existence of two levels of irradiation corresponding to two extreme atmospheric situations, cloudless and cloudy conditions.     

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.     

Probability density function method for turbulent hydrogen flames

Hydrogen combustion attracted much attention recently because of the need for a clean alternative energy. For the theoretical/numerical study of hydrogen combustion, there is a need for modeling capabilities for turbulent hydrogen flames. The present work examines the applicability of probability density function (pdf) turbulence models. For the purpose of accurate prediction of turbulent combustion, an algorithm that combines a conventional CFD flow solver with the Monte Carlo simulation of the pdf evolution equation has been developed. The algorithm is validated using experimental data for a heated turbulent plane jet. A study of H₂–F₂ diffusion flames has been carried out using this algorithm. Numerical results show that the pdf method is capable of correctly simulating turbulence effects on hydrogen combustion. © 1999 International Association for Hydrogen Energy. Published by Elsevier Science Ltd. All rights reserved.     

A monthly probability distribution function of daily global irradiation values appropriate for both tropical and temperate locations

In this paper it is shown that the generalized cumulative distribution curves of Liu and Jordan are not suitable for tropical locations. However, the Bendt et al. probability density function, originally proposed to predict the cumulative distribution curves of North American locations, can still be used for tropical locations provided the value of the maximum clearness index is suitably adjusted. But this probability density function is unrealistic at high irradiation values, which may result in an overestimation of calculated solar system performance. This shortcoming can be overcome by using a higher order probability density function proposed by Saunier. This function is a generalised single parameter function, namely the ratio of the monthly mean clearness index to the maximum clearness index, and seems to be suitable without ad hoc modification for any type of climate. Finally, the problem of estimating the value of the maximum clearness index has also been addressed and a simple model is proposed to evaluate it.