Stochastic modelling of daily global irradiation

A statistical analysis of the solar daily global irradiation for Genoa, Italy, has been carried out using a 9 year time series. The frequency distribution of the fluctuations in the daily values of the time series about the mean, normalized by the standard deviation, has been transformed into a standard Normal distribution. An Autoregressive process of order 1 has been fitted to the transformed series. The daily means and the standard deviations have been estimated by two approaches, viz. (i) Fourier expansion of the daily means and standard deviations with one and two harmonics, respectively; (ii) smoothing of the daily values of these parameters by the “monthly averages method”. For both the approaches, the Autoregressive parameter has been estimated in two ways, viz. (i) keeping it time invariant; (ii) changing it day by day during a year. The fitted model has been used to generate synthetic sequences of daily solar irradiations. All the four methods produce synthetic series which almost satisfactorily match the empirical one without showing any appreciable superiority of the one over the other method.     

Statistical behaviour of solar irradiation over consecutive days

The statistical distributions of solar irradiation accumulated over periods of consecutive days were determined for Lisbon according to the number of days and the long-term clearness index. These distributions were found of interest in connection with the design of storage capacity associated with solar heat collecting systems to guarantee desired heat output rates. A statistical analysis of time series of daily solar irradiation sums observed in Lisbon was carried out. Evidence is obtained for a medium-term correlation (for at least 10–15 days long) and for short- and medium-term periodicities (of about 3 and 10–15 days). A zeroeth order Markov chain does not account for the observed behaviour and a first order Markov chain neither, except perhaps in the short term in July.     

Using probabilistic finite automata to simulate hourly series of global radiation

A model to generate synthetic series of hourly exposure of global radiation is proposed. This model has been constructed using a machine learning approach. It is based on the use of a subclass of probabilistic finite automata which can be used for variable-order Markov processes. This model allows us to represent the different relationships and the representative information observed in the hourly series of global radiation; the variable-order Markov process can be used as a natural way to represent different types of days, and to take into account the “variable memory” of cloudiness. A method to generate new series of hourly global radiation, which incorporates the randomness observed in recorded series, is also proposed. As input data this method only uses the mean monthly value of the daily solar global radiation. We examine if the recorded and simulated series are similar. It can be concluded that both series have the same statistical properties.     

Markov processes and Fourier analysis as a tool to describe and simulate daily solar irradiance

Time series of 20 years of daily solar irradiance data from four italian stations are analyzed on a statistical basis. It is shown that the irradiation sequences are not stationary, both in the mean and in the variance. They can be determined by three components: (a) a mean, well described by a Fourier series with only one coefficient; (b) a variance about the mean, well fitted by a Fourier expansion with two coefficients; (c) a stochastic component. The stochastic component follows a first order Markov model. Since it has a non-normal distribution, a normalizing transform has been introduced which does not affect its statistical properties.     

Analysis of the performances of the first-order two-state Markov model using solar radiation properties

The first-order two-state Markov chains have been used for modelling daily sunshine duration and global solar radiation data, recorded in Reading (United Kingdom), Kuwait-City (Kuwait) and various locations in Algeria (i.e., Algiers, Batna, Oran and Setif) during periods of 8–21 years. In a different way, these data were represented by a model based on the observation of two independent states, called “bad weather” and “fine weather”, respectively. In comparing both models, similarities are, respectively, found between the long-term Markovian and a priori probabilities of having one of both states, the Markovian and arithmetical computations of the mean, the one-day lag autocorrelation and persistence coefficients. The identity between the long-term Markovian and a priori probabilities means that the steady state of the first-order two-state Markov process is described by a Bernoulli random variable. If the identities mentioned above are proved for other locations in the world, they could supplement the statistical tests of validity of this type of Markov models. For instance, it is shown that the properties involved by the limiting distribution of first-order two-state Markov chains, are useful for sizing stand-alone photovoltaic systems in Algiers and for analyzing their performances.     

Mean daily averages of beam radiation received by tilted surfaces as affected by the atmosphere

A method to compare the values of the monthly average beam radiation tilt factor, , obtained if atmospheric effects are accounted for or not, is presented. It is found that the difference between the two determinations may vary greatly with declination, latitude and ratio of daily averaged diffuse to global irradiation. In cases when this ratio is not obtained via direct measurement and its value is large enough, the accuracy of the “atmospheric” may be seriously affected. By comparison with measured radiation data from Montreal, Canada, the “atmospheric” as calculated by the proposed method appears very accurate for south vertical surfaces, and of acceptable accuracy for east or west vertical surfaces. In the latter case, a part of the discrepancy is attributed to various experimental errors, which are discussed.     

Prediction of global daily solar radiation using higher order statistics

The main concern of the present paper is to present and to analyse two procedures for modelling daily global solar radiation. The first one uses the clearness index techniques and the second one uses a totally different type of approach for taking in consideration important properties of such data, including non-Gaussian shape and non-stationarity. This procedure uses the difference between the extraterrestrial and the observed daily global radiation denoted “lost solar component”. Both procedures are based on higher order statistics for generating the global solar radiation using mainly a random process. The prediction results show that the sequences of values generated have the same statistical characteristics as those of sequences observed. The comparison between the two methods used indicates that the developed model based on the “lost solar component” is better than the model obtained using the conventional procedure based on the clearness index.     

Performance assessment of global solar irradiation models under Romanian climate

Six global solar irradiation models were tested against data measured at three stations from Romania in the year 2000. One of the tested models, called EIM, is an empirical global solar irradiation model fitted by us, which requires as input a single meteorological parameter associated with the mean daily cloud amount. The accuracy of the EIM is acceptable and comparable to that of parametric models, which need more than one meteorological datum as input. The main conclusion is that such simple empirical “local-models” are a useful alternative for the more complicated one. In addition, the details of obtaining EIM are presented as a driven-tool, which may serve as a guide to elaborate similar simple solar irradiation models in any other location as well.     

The performance results of Trombe-Wall passive systems under Aegean Sea climatic conditions

Since 1977, at the “Solar Energy Thermal Applications Laboratory of M.T.A.” in Marmaris, Turkey, an experimental study of a “Trombe Thermal Storage Wall” passive system continues in operation; two analogous houses have been built, one of which is serving as a reference system. The incident solar radiation on a vertical surface, the temperature changes of the internal and external surfaces of the wall, at interval of thirty minutes, and the convective flow through the “Trombe-Wall” have been measured in order to determine the solar fractions transmitted by the processes of radiation, convection and thermocirculation. In the present study, the variation of the “Trombe-Wall” efficiency has been calculated during the period of a year using a computer evaluation model in which the half hour measurement data have been introduced. The experimental results show that, according to the months, 15–35 per cent of the incident solar radiation over the south facade is transmitted to the interior through the “Trombe-Wall”, while the reference house transmits from the south facade 8–19 per cent of solar radiation received. Finally, it can now be confirmed that the results of the “Trombe-Wall” system without any special type insulation, contributes to practically the total heating load of the village houses under the sunny Aegean Sea Climatic conditions.     

Multiplicative ARMA models to generate hourly series of global irradiation

A methodology to generate hourly series of global irradiation is proposed. The only input parameter which is required is the monthly mean value of daily global irradiation, which is available for most locations. The procedure to obtain new series is based on the use of a multiplicative autoregressive moving-average statistical model for time series with regular and seasonal components. The multiplicative nature of this models enables us to capture the two types of relationships observed in recorded hourly series of global irradiation: on the one hand, the relationship between the value at one hour and the value at the previous hour; and on the other hand, the relationship between the value at one hour in one day and the value at the same hour in the previous day. In this paper the main drawback which arises when using these models to generate new series is solved: namely, the need for available recorded series in order to obtain the three parameters contained in the statistical ARMA model which is proposed (autoregressive coefficient, moving-average coefficient and variance of the error term). Specifically, expressions which enable estimatation of these parameters using only monthly mean values of daily global irradiation are proposed in this paper.     

Convection film boiling on horizontal cylinders

Free, mixed and forced convection film boiling on a horizontal cylinder in a saturated or subcooled liquid is studied theoretically using a single model based on a two-phase laminar boundary layer integral method. The vapour flow is described accurately by including the inertia and convection terms in the momentum and energy equations, in order to study convection film boiling in the cases of very high superheat. Different film boiling cases are then analysed with this model. The case of high superheat and low subcooling was first analysed by comparing the model with an experiment consisting in the quenching of wires with very high superheat: the model was able to predict the measured heat transfer from the cylinder with errors less than 30%, performing better than previous models or correlations. Additional calculations in other high superheat conditions have also been performed and compared with a model which does not include the inertia and convection terms in order to have a more quantitative idea of their effects on the heat transfers. The case of low superheat and high subcooling is then analysed by comparing the model with other forced convection experiments with cylinders at lower temperatures. By analysing different experiments, it is found that there are in fact two different forced convection film boiling sub-regimes characterised by relatively “low” or “high” heat transfers, and that the existence of these sub-regimes is probably linked with the stability of the vapour film during film boiling. The model results compare quite well with the experimental data which belong to the “stable” sub-regime but, on the other hand, the model largely underestimates the heat transfer for experimental data which belongs to the “unstable” sub-regime. Finally, the model is compared to some free convection experimental data. The model was able to predict the measured heat transfers from the cylinder with errors less than 30% both in saturated and subcooled cases.     

Stochastic simulation model of hourly total solar radiation

A stochastic simulation model of hourly global solar radiation is presented in this paper. It is developed by introducing the concept of “time dependent frequency distribution” (TDFD) of hourly insolation values. In this model the two most critical aspects of time series simulation, i.e., the reproduced time series values which have the appropriate time dependent frequency distribution for the parameter being simulated and the correlation between successive values, are taken into account. The elimination of the TDFD of the data and the transformation of the data distribution to a Gaussian distribution (required for the stationary time series analysis) were carried out using a mapping technique. The autocorrelation function of the transformed data showed that the produced time series is stationary. Then, an antimapping coefficient matrix is developed, which provides a simple yet an effective simulation device. The described model has been applied in Athens (Greece) where hourly insolation data covering a period of two years are used. The theoretical results obtained using this simulation model, regarding both the TDFD and the correlation, are in agreement with the measured data.     

Evaluation of porous silicon carbide monolithic honeycombs as volumetric receivers/collectors of concentrated solar radiation

Porous monolithic multi-channeled silicon carbide (SiC) honeycombs employed as open volumetric receivers of concentrated solar radiation, were evaluated with respect to their porous structure and thermomechanical properties before and after long-time operation. Proper “tuning” of porosity, pore size distribution and microstructure can provide SiC honeycombs with improved mechanical properties (higher bending and compressive strength) in the “as-manufactured” state. Exposure under solar irradiation was found to affect both their pore structure and their mechanical characteristics. During the first stages of exposure, a re-structuring of the porous structure takes place shifting the mean pore size to higher values and slightly decreasing the total porosity; this re-structuring ceases after some “characteristic” exposure time. After solar exposure the honeycombs become harder and exhibit significantly higher compressive strength. Extension of anticipated lifetime can be achieved by materials with enhanced mechanical properties like silicon-infiltrated (siliconized) SiC.     

On the calculation of solar radiation in hazy atmospheres and on turbidity in Ibadan

A previous model is examined in greater detail for the harmattan haze periods (November to January) of 1975–1980 at Ibadan, when the atmosphere is often heavily overcast with dust. It is shown that for air mass (m) and turbidity (β) product values of mβ > 4.05 the scattering transmittance τ_scat, for the instantaneous direct solar radiation becomes negative, while for mβ < 0.22 τ_scat changes too rapidly, thus failing to give acceptable values of irradiation for such periods. This problem is overcome by employing the limits 0.22 mβ 3.07 in the calculations, which corresponds to β of 0.40 and 0.22 for “very hazy” and “hazy-hot clear” atmospheres, respectively. The resulting daily and monthly average total insolation are mainly within ± 15% and ± 10%, respectively, of the experimental data provided by the International Institute of Tropical Agriculture (I.I.T.A.), Ibadan.The variation of the turbidity in the city is also studied for all days of the six years. Daily average values 0.10 β 0.175 occur on about 60% of the days, while “clear” (β < 0.125) and “very hazy” (β 0.40) weather days have about 16% and 12.37%, respectively. Some 74.4% of the very hazy days fall within the harmattan period during which the monthly average turbidity is often higher than 0.35.     

A comparison between one year of daily global irradiation from ground-based measurements versus METEOSAT images from seven locations in Tunisia

Three numerical images from METEOSAT B2 per day have been processed over a period of 12 months, from October 1985 to September 1986, to estimate the daily values of available solar radiation in Tunisia. The methodology used, GISTEL, on the images of the “visible” channel of METEOSAT, is described. Results are compared with measured radiation values from seven stations of the “Institut de la Météorologie de Tunisie.” Among more than 2200 measured-estimated daily pairs, a high percentage, 89%, show a relative error of + or −10%. Many figures concerning Sidi-Bou-Saïd, Kairouan, Thala, and Gafsa are presented to show the capability of GISTEL to map the daily available solar radiation with a sufficient spatial resolution in countries where radiation measurements are too scarce.     

A wind turbine blade profile analysis code based on the singularities method

The aerodynamic characteristics of wind turbines are closely related to the geometry of their blade profiles. The innovation and the technological development of wind turbine blade profiles can be centred on two tendencies. The first is to improve the shape of the existing airfoils and the second is to design new shapes of airfoils in order to get some more ambitious aerodynamic characteristics and enhanced performance.The aim of this paper is to develop an accurate airfoil analysis lower order code, based on the singularities method, for wind turbine applications. The 2D incompressible potential flow model has been used. In the implementation of the singularities method, source–vortex distributions over the airfoil contour are used to compute the flow characteristics. The accuracy and the validity of the results have been tested using experimental data obtained from Wind Turbine Airfoil Catalogue “Risø National Laboratory, Roskilde, Denmark, August 2001” and have shown considerable agreement.     

An analytical method for reflected flux density calculations

Conception, evaluation and real time control of solar “power tower” systems require the use of fast and accurate computer programs for calculating the flux density distributions on the receiver. Since the classical methods of “cone optics” and “hermite polynomial expansion” have some limitations of speed and accuracy, we have built an analytical model for calculating the convolution of the solar brightness distribution with the principal image of a heliostat (i.e. the fictive image for a “point sun”). We first characterize a principal image of a focusing heliostat by its shape and its geometrical concentration factor. Then this image is projected back onto the central plane (which passes through the center of the mirror), and considered as a flat reflecting surface. And the problem is reduced to density calculation for a flat heliostat. For each point of the receiver, the density of flux reflected by a heliostat is obtained by direct resolution of a convolution integral. The different formulations used to express the density function correspond to the various types of intersections between the image of the solar disk for the considered point and the principal image of the heliostat. Confrontation of this method with a program based on “cone optics” shows a good concordance of results and a strong decrease of computation time. We want to apply this method to the existing “THEMIS” solar plant built in France and to compare our results with real observations. Our density calculation programs will help conceiving fields of focusing heliostats for a new generation of power systems (gaz turbine systems).     

About the minimum number of years required to stabilize the solar irradiation statistical estimates

Many researchers install Campbell-Stokes sunshine recorders to evaluate the global irradiation from heliophany measurements using the well-known Ångström linear regression. Because of the statistical nature of this relation, the actual solar irradiation values are determined with some given standard deviation, the fluctuation of which a poor empirical solar radiation database can contribute to. In this paper a procedure (“cross-validation”) has been devised, based on 8 years of daily sunshine and daily global solar radiation data, in order to investigate the dependence of the general standard deviation on the size of the database. Therefore one can infer that a database of at least 14 years is necessary and may be sufficient not only to obtain, by Ångström regression, a trustworthy and statistically stable estimation of the daily global solar irradiation but, further, to set up a “theoretical” and statistically stable distribution of any climatic quantity. Beyond this limit, the variability of the climatological quantities originates from only their intrinsic statistical nature, independently of the size of the database used     

Calcite deposition at Miravalles geothermal field Costa Rica

The calcite deposition problem at Miravalles has been studied since it was observed in the first three wells drilled on the slopes of the Miravalles Volcano. Long-term tests have been carried out to study reservoir characteristics. The change in the production behavior of the wells with the restriction imposed by the deposited calcite has been studied trying to evaluate and quantify the scaling problem. Work is being done on predictions of the deposition rate, location and distribution of the deposited mineral inside the wells. This work was compared with real data obtained from caliper logs of the wells before and after production. The feasibility of the first 55 MW power plant has been demonstrated. It was considered that the solution for the calcite problem is the “reaming during discharge of the wells” trying at the same time to minimize the cleaning interventions with a new well design. A long-term inhibitor test has been scheduled by the beginning of 1989. The economic evaluation of this test may affect the decision of reaming the wells as a solution for the feasibility of the first and subsequent power units. It is also believed, due to the thermodynamics and chemical characteristics of the extracted fluids, that it is possible to find a “non-deposition zone” which will permit the drilling of wells without a scaling problem.