Category Archives: EuroSun2008-13


The extraterrestrial solar radiation on the horizontal surface is the function of Latitude of the location. As the solar radiation passes through the earth’s atmosphere, it is further modified by the processes of scattering and absorption due to the presence of the cloud and atmospheric particles. Hence, the solar radiation incident on the horizontal surface is to some extent locality-dependent and less than the extraterrestrial irradiation.

Several climatic parameters have been used to develop empirical relations to predict the solar radiation incident on the horizontal surface. Among these existing correlations, the Angstrom-Prescott type regression equation related monthly average daily radiation to clear-day radiation at the location in question and average fraction of possible sunshine hours is considered to widely accept by the scholars [9]:

h . S

= a+b (1)

H о ^

The Equation (1) has been proved to accurately predict the global solar radiation.

H : the monthly average daily global radiation on the horizontal surface(MJm-2 day-1)

H0: the monthly average daily extraterrestrial radiation on the horizontal surface (MJm-2day-1)

S : the monthly average daily number of hours bright sunshine

S0 : the monthly average daily maximum number of hours bright sunshine a, b : the regression constants to be determined

The extraterrestrial solar radiation [9] on a horizontal surface is determined by following equation.

24*3600 . 0 ■ п.

H0 = 10 f (cos л cos о sin a>s + a>s sin Asm0) (2)

n 180

Where I0 =1367 Wm 2 is the solar constant, Л is the latitude of the location, 0 is the declination angle.


and f = 1 + 0.033cos(360 ) (3)


Where n is the Julian number (for example Jan 1st is the number 1)




And a>s is the sunset hours angle given as

cos = cos 1 (- tan Л tan 0) The maximum possible sunshine duration

S 2

s0 =

0 15 s

H>, So in the equation are obtained from the equation(2) and (6).The regression coefficient a

Подпись:H S

and. The values of the monthly average daily H 0 S0

global radiation and the average number of sunshine duration are obtained from daily measurements for a period of ten years.

The regression coefficient (a, b) can be computed by the following formulas:







1 H bA S

a = z — z

n і=1 H 0 n і=1 S0

Where n is the number of measured data points.

Wavelet implementation of the MOS

The NWP uncertainties of the rainfall forecast is based on non-stationary, nonlinear and dynamic effects as stated by Todini [18]. As the solar radiation forecast is also a function of the cloud cover, it is probably subjected to the same underlying effects. For non-stationary signals the short-time Fourier transform, also named as Fast Fourier Transform (FFT) has the disadvantage that the information concerning the frequency content at a specific time interval can only be obtained with limited uncertainty. By the Heisenberg uncertainty theorem the method increases its uncertainty for the frequency, if the width of analyzing time window is small, and in the time location of a

particular shape if the windows width is large [19]. A high resolution in time and frequency is obtained by the wavelet convolution, also referred as mathematical microscope [20], where the analyzing time window with is variable in a single transformation. With digital computers the Discrete Wavelet Transform (DWT), has the advantage to reconstruct the decomposed signal with lower uncertainties than the Continuous Wavelet Transformation (CWT) [19]. Also the amount of convolutions is reduced with the DWT which increase the transform speed. This transform is based on the members of a family of functions [20]. One has to begin with the selection of the family of wavelets, e. g. the bi-orthogonal wavelet family, and one of the mother wavelets within the selected family. While the orthogonal DWT uses the inverse filters for the reconstruction of the signal, the bi-orthogonal transform introduced by Cohen [21] permits the utilization of distinct filters for the decomposition of the signal and its reconstruction (Souza [22] citing [21]) in order to obtain symmetric wavelet functions. The mother wavelet function determines the order and specifies the time window or support length of the convolution at the first time scale (m = 1). Also each mother wavelet has its own function shape and degrees of freedom [19]. A TDW transform is accomplished at different time scales (m = 1 … mx), using different functions, named by the members of a family, which are all specifically related to the mother wavelet function. If at a specific time location the signal shape is similar to the wavelet shape, one obtains high wavelet convolution coefficients. At each of the m time scales the signal is convoluted by the DWT with distinct wavelet functions. The daughter wavelet functions ym, n(t) (eqn.1) are equal to the expanded and translated mother wavelet functions у [19].

ym, n(t) = 2-m/2 y( 2-m t — n) ; m, n є Z; t є ^ (1)

Where m defines the scaling or expansion of the mother wavelet and n defines the translation of у, relatively to the time t of the time series values from the signal to be analyzed. Due to the expansion, the convolution support lengths are increased by the factor two from scale m to m+1. For the DWT, the wavelet convolutions are obtained by a filter bank of Finite Impulse Response (FIR) digital filters [19] (Figure 1a). The filter bank separate by low and high pass filters the signal to be analyzed in signals with distinct frequency bands. The mother wavelet (Figure 1a — first bk filter) represents the FIR high pass which separates the highest frequencies appearing within the bandwidth of (SL -1 … ГО). SL is the support length of the mother wavelet. The low pass filters ck, also named as scaling function, represent on its output the signal with the complementary low frequency band until to zero frequency. At m = 1, e. g. the complementary frequency bandwidth is (0 … SL-1) and for m = 2 its frequency content decrease to (0 .. .(2SL)’1). The frequency band of the high pass filter at this scale is ((2 SL)-1… SL-1) and from scale m to (m+1) its band width is reduced by the factor two. Where in the Fourier transform the frequency bins are hold constant, in DWT the energy is hold constant to obtain nearly complete reconstruction of the original time series signal. The signal details and approximations at distinct time scales or filter bands are obtained by the bk and ck filters (Figure 1a). The last scaling function is also known as father scaling function [20].

The downsampling function (2f) after each filter reduces the vector length by two, avoiding a redundant representation of the decomposed signal and due to the upsampling (2t) the signal is reconstructed to its original vector length. The decomposed signal can be represented by the wavelet and scaling coefficient vectors T(m, n) and S(m, n) (Figure 1), or by equal length partially reconstructed sub-signals. If during the reconstruction of the original signal, utilizing the inverse filter bank (Figure 1 b), only one of these vectors is supplied to its input, the signal which corresponds to the supplied vector, is reconstructed to the length of the original time series. This wavelet transform is also referred as Non Decimated Wavelet Transform (NDWT), or a trous WT and its partially reconstructed signal vectors are here named as sub-signals. Beside the



reconstruction based on the wavelet and scaling coefficients (Figure 1 b), with the NDWT one can reconstruct the original signal by the sum of the complete sub-signal set.

Подпись: S(3,n)Cf(1,k)


Figure 1 — Wavelet digital filter bank for the decomposition of a signal (a) and its reconstruction (b), where ( 2i ) stands for the downsampling process and ( 2t ) stands for the upsampling process

Residual kriging

In the residual kriging procedure, instead of kriging Z (x) directly, a regression analysis is first carried out between Z (x) and some external variables. Then, the residuals r (x) and the spatial field Z * (x) can be estimated for each point x generated from the regression results. The new variable r (x) retains the spatial variability of Z (x), but some of the variability has been removed, as a result of external information used in the regression model.

After this analysis, a kriging procedure is used for the residuals r(x) . As a results, a map of r(x), representing the corrections to apply to the regression model, is obtained. The final estimates Z(x) are obtained by combining both Z * (x) and r(x) estimates separately on the kriging grid:

Z( x) = Z * (x) + r( x).


Evaluation of a new model to calculate direct normal irradiance based. on satellite images of Meteosat Second Generation

A. Kemper, E. Lorenz, A. Hammer and D. Heinemann

Department of Energy and Semiconductor Research, Faculty V — Institute of Physics Oldenburg University, D-26111 Oldenburg, Germany Corresponding Author, axel. kemper@uni-oldenburg. de


We present a method to derive the direct normal irradiance DNI from MSG data. For this we apply the Heliosat method to extract cloudiness from the satellite images. Clouds are causing high fluctuations in the DNI. A new model for the direct fraction of the irradiance is introduced to calculate DNI. The clear sky irradiance is mainly determined by the aerosol optical depth (AOD) and water vapour content, which are taken from suitable climatologies. The accuracy of satellite derived DNI data is analyzed here for Spanish sites.

1. Introduction

Measurements of the direct normal irradiance DNI are needed for the planning of a solar thermal power plant at a given site. Direct solar irradiance is highly variable in space and time. As ground measurements are expensive, such data are rare.

Meteorological satellites operationally scan the Earth’s surface and clouds. So we can derive the direct normal irradiance from their data with a good spatial and temporal coverage. Since 2004, the satellites of the new generation MSG provide images of Africa and Europe every 15 minutes with a spatial resolution of approximately 1 km x 1 km at sub-satellite point.

In this document, we first present a method to derive the direct normal irradiance DNI from MSG data. A new model for the direct fraction of irradiance is used in combination with the Heliosat method applied to MSG data. In a second step we analyse the accuracy of the satellite derived DNI with ground measurements of six Spanish sites.

Experimental data. Ground measurements

In this study, we used one year (2005) of global irradiance measurements from 14 stations located in the Sierra Nevada Natural Park, in Hueneja (Granada, Spain). The stations are equipped with LICOR 200-SZ pyranometers, with a sample frequency of 2.5 minutes. In addition, we utilized a DTM with 100 m resolution containing the area where the stations are located, and provided by the local government Junta de Andalucia. Table 1 shows the characteristics of the stations and Figure 1 shows their location inside the studied area.

1.1. Meteosat Images

The satellite images used in this work cover a period of time convergent with the information from the pyrometers. These are images of high resolution and correspond to the visible channel of the satellite METEOSAT-7 (HRI-VIS) and they have been registered by the PDUS (Primary Data User Station) station located in the Department of Renewable Energies of the CIEMAT (DER-CIEMAT).

Station ID

Lat [°]

Long [°]
















































































37°08’31 "












Fig. 1 . Position of the stations in the studied area.


The METEOSAT is located in a geostationary orbit of longitude 0°. The principal element of detection of the satellite is the sensor MVIRI (Meteosat Visible and Infrared Imager), a radiometer of high


definition that constitutes the principal subsystem of the METEOSAT. In case of the images corresponding to the visible channel the field of instantaneous vision, according to the angular amplitude of the sensor and of the orbital distance of the satellite, is 2.5 km in the nadir or point of the subsatellite (latitude=0°, longitude= 0 °). Due to the terrestrial curvature this magnitude increases according to the distance to the point of the subsatellite (for the studied zone approximately 3.5 km).

Подпись: Longitude C) Fig. 2. A.Meteosat image (© 2005 EUMETSAT). B. Iberian Peninsula. C. Subimage used in this study and location of the radiometric stations.

Every pixel of the image has a value of brilliance, registered by the satellite, on the grey scale of 0-255 (8 bits). The 14 stations are located in four different pixels of the image, as shown in Figure 2.


Benchmarking is the largest activity within the MESoR project. The aim of the benchmarking exercise is to establish a coherent set of benchmarking rules and reference data sets to enable a transparent and comparable evaluation of the different solar radiation data sources. The rules are developed in conjunction with the IEA Task 36 on “Solar Resource Management” of the Solar Heating and Cooling Implementing Agreement and shall serve as a standard for benchmarking to make results comparable.

Analysed databases and integrated systems

Each of the databases analysed here is integrated within a system (software setup) that provides additional tools for search, query, maps display, and calculation of derived parameters. PVGIS (the European section), includes solar radiation database developed by combination of solar radiation model and interpolated ground observations. The datasets Satel-Light and HelioClim-2 (accessible through the SoDa web portal) are built from Meteosat and MSG satellite images, respectively. NASA SSE release 6 (accessible also through RETScreen software) combines results from ISCCP

satellite project with NCAR reanalysis products. Primary data incorporated inMeteonorm version

6.1 and ESRA are developed by interpolation of ground observed data with support of satellite images (MSG and SRB, respectively). In Tab. 1 we summarise the main characteristics of the databases/systems including the quality indicators Root Mean Square Difference (RMSD) and Mean Bias Deviation (MBD).

While NASA, Satel-Light, PVGIS, and partially also Meteonorm systems are available for free, the ESRA, HelioClim-2 and full version of Meteonorm are to be purchased.

Table 1. Technical parameters of the solar radiation databases.

Database & availability

Data inputs




Spatial resolution (in study region)


PVGIS Europe


~560 meteo stations




1 km x 1 km + on — fly disaggreg. by 100 m DEM



Meteonorm 6.1

(CDROM and internet)

Meteo stations + satellite data




Interpol. (on-fly)+ satellite; disaggreg. by 100 m DEM





~560 meteo stations + SRB satel. data




5 arc-minute x 5 arc-minute





Meteosat 5, 6, 7



4.6-6.2 km x 6.1-14.2 km





Meteosat 8 and 9 (MSG)

2004 — 2007


3.1-4.2 km x

4.1-9.6 km





GEWEX/SRB 3 + ISCCP satel. clouds + NCAR reanalysis



1 arc-degree x 1 arc-degree



Geographical extension of the spatial products differs: from global (NASA and Meteonorm) to cross-continental (HelioClim-2 covering Europe, Africa and Southwest Asia) and European (ESRA, PVGIS and Satel-Light). Here we focus on the subsection of the European continent (Fig. 1) where all the data sources overlap.

2. Method

2.1. Map comparison

Map-based comparison as performed here is a type of relative benchmarking of solar databases. It does not point to the “best” database, but it gives an indication of the user’s uncertainty at any location within the region. As the existing spatial products cover different periods of time, this comparison introduces also uncertainty resulting from the interannual variability of solar radiation. Here we perform a cross-comparison of maps of long-term average of yearly sum of global horizontal irradiation.

The maps from all data providers are harmonised and integrated into a geographical information system (GIS) with latitude/longitude spatial reference. The grid resolution of 5 arc minutes is chosen to provide a representative outlook at the regional (rather than local) differences within the continent. Choosing such a resolution, more detailed features that are present in the databases of HelioClim-2, Satel-Light, PVGIS and Meteonorm are suppressed (smoothed out), but the regional features are well pronounced. Integration of 6 data sources of yearly sums of global horizontal irradiation provides three results:

• Map of overall average gives the user an indication of spatial distribution of solar resource estimated by the simple averaging of the 6 datasets;

• Map of standard deviation provides information on magnitude of differences between the combined data sources, i. e. user’s uncertainty. If we assume that the estimates are normally distributed, from standard deviation a confidence interval can be calculated in which the value falls corresponding to a given probability. For example, a multiple of 1.95996 would give a range where the value from the average map falls with 95% probability.

• Maps of differences between individual databases and the overall average indicate deviation of the values in the particular dataset from the overall average.

Own measured data

A multicomponent solar thermal system was installed in Bishkek to preheat water for a district heating net in the context of a joint research project of Kassel University (Germany) and Kyrgyz State Technical University in Bishkek (Kyrgyzstan). For detailed investigations different parameters including solar radiation are measured since autumn 2004 with very accurate sensors every 15 seconds to generate one-minute mean values. Solar radiation is measured with pyranometer Kipp&Zonen CM11 with an accuracy of 1.5% of the measured values. For some periods, however, the measurement data is missing for different reasons (e. g. technical problems with the sensor power supply unit or no power supply at all). For this study mean values of monthly solar irradiation were generated from the available measurement data for the period 2005 — 2007.

2. Comparison of meteorological data

The focus of this study is to compare values of ambient temperature and solar irradiation from the mentioned sources because these two parameters are the most important for estimation of solar energy gains. The following abbreviations are applied for the sources of meteorological data:

• FrunzeM — measurements from weather station Frunze

• Met 5.1 and Met 6.0 — Meteonorm 5.1 and Meteonorm 6.0 respectively

• MD — own measurement data

Подпись: Fig. 2. Long-term mean temperatures and the change of annual ambient temperatures from different sources.

A fluctuation of mean annual ambient temperatures measured by the local meteorological station Frunze since 1928 and long term mean ambient temperatures are shown in Fig. 2.

A long term mean temperature from both versions of Meteonorm is about 0.2 K higher than that measured by the station Frunze. The difference results from different periods considered. In Meteonorm a long-term mean ambient temperature is calculated for the period from 1961 till 1990 (30 years), while a mean temperature from the station Frunze is calculated for the period from 1928 till 2003 (75 years). For the same period 1961-1990, mean temperatures from the station Frunze and Meteonorm are in agreement (see Table 1).

Table 1. Long-term mean ambient temperature in °C for Bishkek in the period 1961-1990. 1) Meteonorm 5.1, 2) Meteonorm 6.0, 3) Frunze meteostation.
























































A fluctuation of annual global solar irradiation and its mean values from different sources are shown in Fig. 3. The annual global solar irradiation from the station Frunze for the years 2003­2007 is estimated from 6 solar radiation measurements per day (see section 2.1).

Annual global solar irradiation from own measurements («1500 kWh/m2a) and the weather station Frunze (1572 kWh/m2a) are in a good agreement, while its values from Meteonorm 5.1 and 6.0 are approx. 20% lower (see Fig. 3) and even out of the fluctuation range.

FrunzeM /





As shown in Fig. 4, the solar irradiation from Meteonorm and other sources are in a good agreement in winter. In the period March — September the monthly diffuse solar irradiation values generated with Meteonorm are higher than the measured values, which leads to lower global solar irradiation values. The higher values of diffuse solar irradiation in Meteonorm are probably caused

by admitting higher cloudiness or air pollution. Furthermore, the concavity in summer of the monthly global solar irradiation generated with Meteonorm 5.1 is not typical for a continental climate. It is also inconsistent with the monthly sunshine duration for Bishkek (see Fig. 5), which has no concavity or only one maximum point.

kWh/m2month Monthly sum of global and diffuse solar radiation on horizontal surface



Sunshine duration

Fig. 4. Monthly global and diffuse solar irradiation for Bishkek from different sources.



If the latitude of Bishkek in Meteonorm 6.0 is changed from 42.8°N to 42.2°N, annual global irradiation increases to 1466 kWh/m2a (+20%) and annual diffuse irradiation decreases to

618 kWh/m2a (-7%). Thus, the radiation data for 42.2°N latitude is in the same range with the measurement data from the local weather station Frunze and the research project (MD). The same tendency occurs if the latitude of Bishkek is changed from 42.8°N to 43.4°N (1468 and 625 kWh/m2a or +20% and -6% respectively). This high change in the radiation data for a little change in the location is probably caused by the interpolation method of Meteonorm.

Подпись: kWh/m2aПодпись:Подпись:image150°C



12.5 12

11.5 11

10.5 10

9.5 9

As shown in Fig. 6, the annual global solar irradiation and the mean annual ambient temperature measured at the weather station Frunze in the period 1970 — 1991 are not positively correlated. The seasons (summer, autumn, winter, spring) and single months have the same tendency. The reason is that the weather and the ambient temperature in Kyrgyzstan are influenced not only by solar radiation but also by 17 weather processes [5], e. g. Siberian anticyclone brings cold air from Siberia.

3. Conclusion

Solar radiation and ambient temperature data from different sources (a local weather station, Meteonorm 5.1 and 6.0 and own measurements) have been compared with each other for Bishkek, Kyrgyzstan in this study. It was identified that the altitude of Bishkek city in Meteonorm is wrongly defined (2111 m instead of 760 m), which leads to significantly lower ambient temperatures. Therefore, it is necessary to define Bishkek manually as a new site with the correct coordinates. In this case the ambient temperatures are in agreement with those measured by the weather station Frunze.

Annual global solar irradiation from own measurements (1500 kWh/m2a) and weather station Frunze (1572 kWh/m2a) are in a good agreement, while its values from Meteonorm 5.1 and 6.0 are approx. 20% lower. Correspondingly, the ratio of the direct and diffuse solar irradiation is much lower in the Meteonorm data (approx. 1) than in the weather station Frunze data and own

measurement data (approx. 2). Furthermore, monthly sums of global solar irradiation generated with Meteonorm 5.1 have an untypical trend in summer, having two local maximum points. This is inconsistent with irradiation data from other sources and with the sunshine duration in the relevant period.

Such differences in solar radiation data can lead to significantly different solar gain predictions, especially if the solar irradiation on a tilted surface shall be calculated. Thus, the source of meteorological data shall be carefully selected. For sites, close to weather stations with relevant data available in the Meteonorm database, data generated with Meteonorm can be applied. If no weather station with relevant data close to the desired site available in Meteonorm database, other sources should be considered too, e. g. satellite-derived radiation rata, local meteorological stations or own measurements. In both cases, the data from Meteonorm should be proved on plausibility, particularly for sites or stations in developing countries.


The authors would like to express their gratitude to the Volkswagen Foundation, Germany for the financial support of the research project and the Central administrative board on hydrometeorology of Ministry of Emergency Measures of the Kyrgyz Republic for providing the meteorological data from the weather station “Frunze”.


[1] John A. Duffie, William A. Beckman, Solar Engineering of thermal processes, John Wiley & Sons, Inc., Hoboken, New Jersey, 2006

[2] J. C.McVeigh, Sun Power: an introduction to the Applications of Solar Energy, translated by Guhman G. A., Moscow Energoisdat, 1981

[3] Reference book on USSR climate, Leningrad 1989

[4] Handbook for hydro meteorological station on solar radiometry, Leningrad 1971

[5] Pavlova I. A., Changeability of synoptic processes in Kyrgyzstan, Metrology and Hydrology in Kyrgyzstan 2001

[6] Handbook Meteonorm 5.1

[7] Handbook Meteonorm 6.0

[8] E. Frank, K. Vajen, A. Obozov, V. Borodin (2006): Preheating for a District Heating Net with a Multicomponent Solar Thermal System, Proc. EuroSun 2006, Glasgow

[1] Introduction

Electric shower heads are presently installed in 73.1 % of the Brazilian houses. These devices accounts for around 60 % of the peak load in between 6:00 p. m. and 8:00 p. m., as is shown by reports of the Brazilian electric power system [1]. As demonstrated in a large scale experiment [2], Compact Solar Domestic Hot Water Systems (CSDHWS), conjugated to electric shower heads, are able to reduce the mentioned peak load due to shower heads by around 60 %. However the peak is expected to remain unchanged for those days of low solar radiation incidence. To further increase the peak power reduction and its confidence, an intelligent compact solar domestic hot water systems is under development and first simulation and optimization were carried out in [3] and [4]. According to the proposed system preheating by auxiliary energy should be done in order to provide preheated water at 6:00 a. m. and a specified storage water temperature at 6:00 p. m. Thus the preheating energy depends on daily available solar energy. Apart from the features solar energy use and peak reduction, additional advantage of this system is obtained by heating the water of the

[2] www. meteonorm. com

Linke’s Turbidity Factor Applied to Worldwide Global. Horizontal Irradiance Measurements

Frank Bason

SolData Instruments, Linabakken 13, DK-8600 Silkeborg, Denmark
Corresponding Author, soldata@soldata. dk


The data collection phase of the Danish Galathea III Expedition was conducted from August 2006 until April 2007 [1]. During this period the research vessel Vmdderen undertook a round the world voyage of nearly 100.000 kilometers while acting as a platform for scientific research in a range of disciplines. The researchers and instruments aboard the ship collected data from many locations around the world from 66.90 N to 67.50 S latitude. Among the experiments aboard the ship was an optics table sponsored by SolData Instruments and containing among other instrumentation three pyranometers for continuous measurement of global solar irradiance on a horizontal surface [2]. Knowledge of global solar irradiance is important for studies of the atmosphere and solar radiation and consequently for modeling the evolution of the Earth’s climate. We employ the Linke turbidity factor in our analysis, for this parameter is often referred to in the literature of atmospheric physics [3].

Keywords: Galathea Expedition, Linke turbidity factor, solar irradiance

1. Introduction

The purpose of this paper is to present an analysis of the global irradiance measurements. In particular we will focus on the computation of Linke’s turbidity factor TL at a wide range of locations and the implications of these results for a general description of solar resources around the world. The factor TL is closely related to the transmittance of the atmosphere on clear days [4]. As a general rule the atmosphere is clearer at higher latitudes, and the large amount of data available from the expedition has permitted the development of an algorithm to describe this relationship. Among the useful results obtained is a correlation between the Linke turbidity factor and the latitude. Secondary parameters such as the surface air water vapor content are also examined.

The turbidity factor was obtained from the horizontal radiation data using an incident direct plus diffuse rad­iation model, observed values of the solar irradiance and a numerical algorithm to determine TL. Based on these results it is possible to provide a highly realistic prediction of the global irradiance on a clear day in a maritime environment. The model developed also supplies information about the distribution of diffuse and direct irradiance, information which is important for the design of solar energy systems. Furthermore TL has implications for atmospheric visibility.

Data base and compare

1.1. Information of stations

The global solar radiation and sunshine duration data during the period from 1994 to 2003 reported in this paper were provided by the China Meteorological Administration (CMA). Professor Tien Shengyuan has divided climatic zone of China into eight large-scale climatic zones, and he also has advanced the representative city for every climatic zone in the year of 1989. The information of these cities is as follows [3]

Table. 1. The information of representative stations




longitude (°E)

altitude (m)

data duration