Literature Reviewshodhganga.inflibnet.ac.in/bitstream/10603/23550/12/12... · 2018. 7. 9. · 100 and worldwide the estimate is approximately 1:500 (Viorel Badescu[67]). Even in the

Development of a Site-independent Mathematical Model for the Estimation of Global Solar Radiation

21

Chapter 2

Literature Review

2.1 Introduction

Almost all the renewable energy sources originate entirely from the

sun. The sun’s rays that reach the outer atmosphere are subjected to

absorption, scattering, reflection and transmission processes through the

atmosphere, before reaching the earth’s surface.

Solar radiation data at ground level are important for a wide range of

applications in meteorology, engineering, agricultural sciences, particularly

for soil physics, agricultural hydrology, crop modeling and estimation of crop

evapo-transpiration, as well as in the health sector, in research and in many

fields of natural sciences. A few examples showing the diversity of

applications may include: architecture and building design (e.g. air

conditioning and cooling systems); solar heating system design and use; solar

power generation and solar powered car races; weather and climate

prediction models; evaporation and irrigation; calculation of water

requirements for crops; monitoring plant growth and disease control and

skin-cancer research.

The solar radiation reaching Earth’s upper atmosphere is rather

constant in time. But the radiation reaching some point on Earth is random in

nature due the gases, clouds and dust within the atmosphere which absorbs

and/or scatters radiation at different wavelengths. Obtaining reliable


22

radiation data at ground level requires systematic measurements. However,

in most countries the spatial density of actinometrical stations is inadequate.

For example, the ratio of weather stations collecting solar radiation data

relative to those collecting temperature data in the USA is approximately 1:

100 and worldwide the estimate is approximately 1:500 (Viorel Badescu[67]).

Even in the developed countries there is a dearth of measured long-term solar

radiation and daylight data.

2.2 Existing Estimation models

2.2.1 Radiative Transfer Model

In radiation, the energy is transmitted by electromagnetic waves

emitted by the atoms and molecules inside the hot body.

Stefan in 1879 found experimentally that, at a given temperature T ºK,

the total energy E, radiated by a body is given by,

E = σ T4 (2.1)

where σ is a constant (5.6687×10-8 Wm-2 K-4), called Stefan-Boltzmann

constant.

Sun is the star at the center of the solar system. Its surface temperature

is about 5778 ºK. For all theoretical purposes, sun is considered as a black

body radiating energy in all direction. As per Stefan’s relation this amounts to

6,31,82,037 Wm-2 at the sun’s surface.

With, radius of Sun, R= 6.96×105 km and of distance of Earth from Sun,

r=1.496×108 km, solar radiation striking the top of the earth’s atmosphere,

referred as solar constant Io would be given by the relation,


23

�� (2.2) which is equal to 1360 Wm-2. The solar constant is the amount of

energy received at the top of the Earth's atmosphere on a surface oriented

perpendicular to the Sun’s rays at the mean distance of the Earth from the

Sun.

The earth revolves around the sun in an elliptical orbit. This leads to

variation of extraterrestrial radiation flux. This value on any day of the year

can be calculated from the equation,

�� 1 � 0.033�� (2.3) where, n is the day of the year.

To specify the position of a point on the surface of the earth, one

should know the latitude λ (horizontal lines) and longitude φ (vertical lines)

of the point. Figure 2.1 shows various geometrical parameters related to sun-

earth relations.

The angular displacement of the sun from the plane of the earth’s

equator is termed as the declination of the sun, δ. This angle varies between

+23.45º to -23.45º as the earth performs its yearly circum-navigation around

the sun.

The hour angle, ω is an angular measure of time and is equivalent to

15º per hour. It varies from -180º to +180º.


24

Figure 2.1 Measuring longitude, declination and hour angle

(Courtesy: Duffie and Beckman[68])

Before the expression for global solar radiation is established, it is

necessary to understand the following parameters,

h = Elevation angle, measured up from horizon

θZ = Zenith angle, measured from vertical and

A = Azimuth angle, measured clockwise from north

The above useful angles are represented in figure 2.2.

λ


25

If θZ called as the zenith angle, is the angle between an incident beam

of flux I and the normal to a plane surface, then the equivalent flux falling

normal to the surface is given by,

�� (2.4)

Figure 2.2 Measuring elevation angle, zenith angle and azimuth angle

(Courtesy: Sukhatme[69])

For a horizontal surface, we can show that,

�� sin � sin � � cos � cos � cos � (2.5) The hour angle corresponding to sunrise or sunset, ωs on a horizontal

surface can be found from above equation by substituting the value of 90º for

the zenith angle. Thus,

cos �� tan � tan � (2.6) Then instantaneous global solar radiation on a horizontal surface is

computed as,


26

�� 1 � 0.033�� sin � sin � � cos � cos � cos �� (2.7) Thus, monthly averaged global radiation denoted by Ĥo, is obtained by

integrating over the day length as follows:

Ĥ� � �� 1 � 0.033�� sin � sin � � cos � cos � cos �� (2.8)

Now, � � �� (2.9) where, t is in hours and ω is in radians.

Hence, �� (2.10) Substituting in the above equation,

Ĥ� � �� 1 � 0.033�� sin � sin � � cos � cos � cos �� (2.11)

After simplification, we obtain,

Ĥ� � � ! � �� 1 � 0.033�� sin � sin � � cos � cos � sin �� (2.12)

Equation (2.12) could be used for calculating the monthly averaged

global radiation at extra-terrestrial plane called extra-terrestrial radiation

(ETR).

As shown in figure 2.3, the atmosphere scatters and absorbs some of

the Sun's energy that is incident on the Earth's surface. Scattering of radiation

by gaseous molecules (e.g. O2, O3, H2O and CO2), is called Rayleigh scattering.

Almost half of the radiation that is scattered is lost to outer space. The

remaining half is directed towards the Earth's surface from all directions as


27

diffuse radiation. Absorption of solar radiation is mainly by oxygen and

ozone molecules in the atmosphere.

Figure 2.3 Plane of earth receiving the component of beam, diffuse radiations from extraterrestrial radiation.

(Courtesy: Rai[70])

Clouds reflect a lot of radiation and absorb a little. The rest is transmitted

through atmosphere which helps regulating the surface temperature. The

fraction of the total solar radiant energy reflected back to space from

clouds, scattering and reflection from the Earth's surface is called the

albedo of the Earth-atmosphere system, is roughly 0.3 for the Earth as a

whole. Figure 2.3 also shows that a plane on the Earth's surface receives:

• Beam (or direct) radiation – coming straight through the

atmosphere to hit the plane (very directional);

• Diffused radiation – scattered in all directions in the atmosphere

and then some arrives at the plane on the Earth’s surface (not

directional);

Reflected


28

• Reflected radiation – beam and diffused radiation that hits the

Earth's surface and is reflected onto the plane.

The amount of solar radiation energy reflected, scattered and absorbed

depends on the condition of atmosphere that the incident radiation travels

through as well as the levels of dust particles and water vapor present in the

atmosphere. The latter is usually difficult to judge. The distance travelled

through the atmosphere by incident radiation depends on the angle of the

Sun.

2.2.2 Empirical Models

The utility of existing weather data sets is greatly expanded by including

information on solar radiation. Radiation estimates for historical weather can

be obtained by predicting it using either a site-specific radiation model or a

mechanistic prediction model. A site-specific model relies on empirical

relationships of solar radiation with commonly recorded weather station

variables. Although a site-specific equation requires a data set with actual

solar radiation data for determining appropriate coefficients, this approach is

frequently simpler to compute and may be more accurate than complicated

mechanistic models. These simple, site-specific equations, therefore, may be

very useful to those interested in sites near to where these models are

developed. In the following sections, various such site-specific models are

discussed.

2.2.2.1 Sunshine based models

The fundamental Angstrom-Prescott-Page[4,5,6] model, is the most

commonly used and is given by,


29

�� " � # � �� (2.13)

where H is the monthly averaged daily global solar radiation, Ho is the

monthly averaged daily extraterrestrial radiation; So is the day length, S is the

maximum sunshine duration. ‘a’ and ‘b’ are empirical coefficients which vary

depending upon the site.

Page[5] has given the coefficients a=0.23 and b=0.48, for Angstroms-

Prescott-Page model, which is believed to be applicable anywhere in the

world. For Turkey, Tiris et al.[71] gave a=0.18 and b=0.62. Bahel et al[72].,

suggested a=0.175 and b= 0.552 for Saudi Arabia. Louche et al.[73], presented

a model for French Mediterranean site with a=0.206 and b=0.546. Monthly

specific correlations with S/So and λ (latitude) are given by Dogniaux and

Lemoine[74] for Europe. Rietveld[75] examined several published values of a

and b coefficients and noted that a is related linearly and b hyperbolically to

the appropriate mean value of S/So. Soler[76] applied Rietveld’s model to 100

European stations and gave the specific monthly correlations.

Zabara[77] modified the Angstroms expression for Greece and

expressed the a and b coefficients as a third order function of (S/So), as under

–

" � 0.395 � 1.247 � �� 2.680 � �� 1.674 � ��

� (2.14) # � 0.395 � 1.384 � �� 3.249 � ��

� � 2.055 � �� (2.15)

Kilic and Ozturk[78] calculated the a and b empirical coefficients for

Istanbul as, a = 0.103 + 0.000017 Z + 0.198 cos (λ-δ), b=0.533 – 0.165 cos (λ-δ),

where Z is the altitude of the site.


30

For Spain, Almorox and Hontoria[19] proposed the exponential model

as,

�� " � # exp � �� (2.16) Togrul et al.[49], proposed variations in correlations for Ealzig, Turkey,

as a function of sunshine duration ratio.

Akinoglu and Ecevit[79] obtained the correlations in a second order

polynomial equation for Turkey as –

�� 0.145 � 0.845 � �� 0.280 � ��

� (2.17) Bahel[80] developed a worldwide correlation based on bright sunshine

hours and global radiation data of 48 stations around the world. The model is

given by,

�� 0.16 � 0.87 � �� 0.16 � ��

� � 0.34 � �� (2.18)

Bahel et al.[72] suggested the following model for Saudi Arabia,

�� 0.175 � 0.552 � �� (2.19)

Samuel’s[81] model for Srilanka is given as under,

�� 0.14 � 2.52 � �� 3.71 � ��

� � 2.24 � �� (2.20)

Raja and Twidell[82], for Pakistan offered the following model,

�� 0.388 cos � � 0.367 � �� (2.21)

Model including a logarithmic term is given by Newland[83]for South China

as,


31

�� 0.34 � 0.40 � �� 0.17 /�0 � �� (2.22)

2.2.2.2 Temperature based models

Bristow and Campbell[25] suggested the following model for India as –

�� "11 � exp�# � �� 2 (2.23)

Hargreaves et al.[26] reported a simple model based on temperatures as –

�� " � �� . � # (2.24) Allen[27] suggested self calibrating model that is function of the daily

extraterrestrial radiation, mean monthly maximum and minimum

temperatures as –

�� " � �� . (2.25)

For China, Chen at al.[30], presented the following model –

�� " /3� �� # (2.26)

For six stations in India, S.S. Chandel et al.[84] model is

� � ��7.9

�� sin�� exp��0.0001184��

(2.27)

2.2.2.3 Cloud observation based models

Black[33], using data from many parts of the world, proposed the following

quadratic equation –


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�� 0.803 � 0.340 4 � 0.458 4� (2.28)

where C is the monthly average fraction of the daytime sky obscured

by clouds.

Badescue[37] suggested the following models for Romania,

�� " � # 4 (2.29) �

�� " � # 4 � � 4� (2.30) �

�� " � # 4 � � 4� � � 4� (2.31) Supit and Kappel[85] proposed the following model –

�� " 5� �� # 6�1 � �� (2.32)

2.2.2.4 Multiple parameter based models

For various places in Sudan, Elagib and Mansell[86] have suggested

the following models

�� " 789 :# � ��; (2.33) �

�� " � # � ��

(2.34)

�� " � #� � �< � � � �� (2.35) �

�� " � #� � �< � � � �� (2.36) �

�� " � #� � � � �� (2.37)


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Abdalla[87] modified these equations for Baharin as,

�� " � # � �� != (2.38) �

�� " � # � �� != � 7>? (2.39) where PS is the ratio between mean sea level pressure and mean daily

vapor pressure.

Trabe and Shaltout [88]suggested the model for Egypt as,

�� " � # � �� @ � 7!= � A> (2.40) where V, is the water vapor pressure.

Gopinathan[14] introduced a multiple linear regression equation of the form,

�� " � # �� < � � � �� 7 � A!= (2.41)

Dogniaux and Lemoine[74] proposed the following correlation for Europe,

where the coefficients of the Angstrom-Prescott-Page model seem to be a

function of the latitude of the site,

�� 0.37022 � B0.00506 � �� 0.00313C � � 0.32029 � �� (2.42)

For South Western Nigeria, Ojosu and Komolafe[89] proposed the following

equation,

�� " � # � �� (2.43)

Ododo et al.[90] proposed two new models for Nigeria as under –

�� " � ��

! �� != (2.44)


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�� 7 � A � �� 0 �� D!= � E �� (2.45)

Garg and Garg[91] proposed the model for India as below –

�� " � #� � �F (2.46)

where F � 0.0049!= G"#$ %��.�� & H For Zimbabwe, Lewis [46] gave following models

log = � 7 � A log != (2.47) log = � 0 � D log ? � E log != (2.48) ln = � K � L ? (2.49) ln = � / � M != (2.50) ln = � 3 � N�? � !=� (2.51)

Ertekin and Yaldiz[48] estimated the monthly average daily global solar

radiation by multiple linear regression model based on nine variables, as

follows –

= � " � #=� � �� != � 7 � �� A � 0 ? � D 4 � E> � K� (2.52)

where TS – Soil Temperature, P – Precipitation and E – Evaporation.

Chen et al., [30] presented the following models for China,

= � " � # � �� sin � � � �� (2.53) = � " � #=� � � � �� sin � � A!= � 0 �� (2.54)


35

= � " � #=� � � � �� != � A ? � 0 �� (2.55) = � " � #=� � � � �� sin � � 7!= � A ? � 0 �� (2.56)

Nadir Ahmed Elagib, Sharief Fadul Babiker and Shamsul Haue

Alvi[47], have given new empirical models for estimating the monthly

averaged daily global solar radiation from commonly measured

meteorological parameters such as relative humidity and temperature, at

Bahrain. These models are,

Ĝ � 76.0527 � 0.8790 != (2.57) Ĝ � 31.2510 � 0.3764 �!= � P�� (2.58) Ĝ � 27.0682 � 0.3866 �!= � Q � P�� (2.59)

2.2.3 Satellite Observation Model

The accurate knowledge of solar radiation at the earth’s surface is of

great interest in solar energy, meteorology, and many climatic applications.

Ground solar irradiance data is the most important data required for

characterizing the solar resource of a given site but the spatial density of such

measuring meteorological stations is far low because of economic reasons. In

this context, satellite-derived solar radiation estimation has become a valuable

tool for quantifying the solar irradiance at ground level for a large area. Thus

derived hourly values have proven to be at least as good as the accuracy of

interpolation from ground stations at a distance of 25 km (Zelenka et al.[92]).

Several algorithms and models have been developed during the last

two decades for estimating the solar irradiance at the earth surface from


36

satellite images (Gautier et al.[93]; Tarpley[94]; Hay[95]). All of them can be

generally grouped into physical and pure empirical or statistical models

(Noia et al.[96]). Statistical models are simpler, since they do not need

extensive and precise information on the composition of the atmosphere, and

rely on simple statistical regression between satellite information and solar

ground measurements. On the contrary, the physical models require input of

the atmospheric parameters that model the solar radiation attenuation

through the earth’s atmosphere. On the other hand, the statistical approach

needs ground solar data and such models suffer from lack of generality.

Satellites observations of the earth can be grouped, according to its

orbit. In polar orbiting satellites, with an orbit of about 800 km have high

spatial resolution but a limited temporal coverage. The geostationary

satellites, orbiting at about 36000 km, can offer a temporal resolution of up to

15 minutes and a spatial resolution of up to 1 km. Most of the methods

(Shafiqur Rehman and Saleem Ghori, [97]) for deriving solar radiation from

satellite information make use of geostationary satellite images.

The solar radiation absorbed at the earth surface may be expressed as a

function of surface albedo (ρ) and incident solar irradiation (IG).

�� R �'�1 � S� (2.60) Therefore, the solar radiation on the earth surface may be expressed as:

�' � �(��)* 1�� 2 (2.61) where,

Io is the extraterrestrial irradiation ETR= 1360 Wm-2

Ea is the absorbed energy


37

IS is the radiation measured by the satellite’s radiometer.

Above equation is thus the fundamental equation for all the models aiming at

deriving solar radiation from satellite images.

The use of satellite images to estimate the solar radiation has, in fact,

noticeable advantages, in particular the following are worth to mention:

• Satellites collect information for large extensions of ground at the same

time, which allows identifying the spatial variability of solar radiation

at ground level.

• When the relevant information is available from satellite images, they

can be superimposed on the corresponding images of that area. It is

possible to study the time evolution of values in an image pixel or in a

certain geographic area.

• Satellites images allow the analysis of the solar resource in a potential

emplacement that has no previous ground measurements.

2.2.4 ANN Model

An Artificial Neural Network (ANN) is an interconnected structure of

simple processing units. The functionality of ANN can graphically be shown

to resemble that of the biological processing elements called the neurons.

Neurons are organized in such a way that the network structure adapts itself

to the problem being considered. The processing capabilities of this artificial

network assembly are determined by the strength (weightage factor) of the

connections between the processing units.

Haykin[98] states that: “A neural network is a massively parallel

distributed processor that has a natural propensity for storing experiential


38

knowledge and making it available for use. It resembles the brain in two

respects:

1. Knowledge is acquired by the network through a learning process;

2. Interconnection strengths between neurons, known as synaptic

weights or weights, are used to store knowledge”.

During the last two decades, ANN has proven to be excellent tools for

research, as they are able to handle non-linear interrelations (non-linear

function approximation), separate data (data classification), locate hidden

relations in data groups (clustering) or model natural systems (simulation).

Naturally, ANN found a fertile ground in solar radiation research. A detailed

survey about the applicability of ANN to various Solar Radiation topics is

given below.

Mohandes et al.[59] performed an investigation for modeling monthly

mean daily values of global solar radiation on horizontal surfaces; they

adopted a back-propagation algorithm for training several multi-layer feed-

forward neural networks. Data from 41 meteorological stations in Saudi

Arabia were employed in this research: 31 stations were used for training the

neural network models; the remaining 10 stations were used for testing the

models. The input nodes of the neural networks are: latitude (in degrees),

longitude (in degrees), altitude (in meters) and sunshine duration.

The output of the network is the ratio of monthly mean daily value of

the global solar radiation divided by extraterrestrial radiation received at the

top of the atmosphere. The results from the 10 test stations indicated a

relatively good agreement between the observed and predicted values. Along

the same line is the research by Mohandes et al.[99], in another research for

simulating monthly mean daily values of global solar radiation (the output of


39

the model is the ratio of monthly mean daily value of the global solar

radiation divided by extraterrestrial radiation outside the atmosphere). They

retained the same input parameters as measured above (latitude, longitude,

altitude, sunshine duration) but they added a new one, namely, the month

number. They made use of the same data sets, which were also separated into

the same training and testing sub-sets. In this research, they use Radial Basis

Functions (RBF) neural networks technique (Wassereman[100]; Bishop[101])

and compare its performance with that of the MLP as used in their previous

study (Mohandes et al. [59]).

The comparative performance of both the RBF and MLP networks was

tested against the independent set of data from 10 stations by using the mean

absolute percentage error as the testing statistic. The test has indicated mixed

results for individual stations but, overall, RBF performs better than MLP.

In a more recent endeavor, Mellit et al.[102] studied wavelet network

architecture and its suitability in the prediction of daily total solar radiation.

Wavelet networks are feed-forward networks using wavelets as activation

functions and have been used successfully in classification and identification

problems. This architecture provides a double local structure which results in

an improved speed of learning. The objective of this research was to predict

the value of daily total solar radiation from preceding values; in this respect,

five “structures” were studied involving as input various combinations of

total daily solar radiation values. The meteorological data that have been used

in this work are the recorded solar radiation values during the period

extending from 1981 to 2001 from a meteorological station in Algeria. Two

datasets have been used for the training of the network. The first set includes

the data for 19 years and the second dataset comprises data for one year (365

values) which is selected from the database. In both cases, the data for the


40

year 2001 are used for testing the network. The validation of the model was

performed with data which the model had not seen before and predictions

with a mean relative error of 5% were obtained. This is considered as an

acceptable level for use by design engineers.

2.3 Study of drawbacks, limitations and ideas to overcome

In this section, limitations on the existing empirical models (detailed in

2.2.2) are discussed.

2.3.1 Sunshine Based Models

All these models contain the term S/So. Linear models, polynomial

models, exponential models, trigonometric models and logarithmic models

presented and as given in 2.2.2.1 for different parts of the world which need

the measured data of S/So. Estimation models are developed with known

values of S/So . Sunshine duration can be recorded by an instrument called

‘sunshine recorder’. The instrument is needed to be kept on the horizontal

earth surface under the sun. As the instruments are costly; measurements and

recording is laborious, generally such facility is made available with primary

meteorological stations of any country. For example, India has only 18

primary stations and many secondary stations. The main limitation is actual

measurement by instruments. If the arrangements for measurement of

sunshine duration are being made, it is possible to measure the global solar

radiation with pyranometer instrumentation set-up at the station.

Therefore it is necessary to look for other meteorological parameters,

which are easily and economically measurable. Such parameters could be -

maximum ambient temperature, minimum ambient temperature, relative

humidity, cloud coverage, precipitation, wind speed etc.


41

2.3.2 Ambient Temperature Based Models

These models, few of them are given in 2.2.2.2, generally use the data

Tmax and Tmin. Quadratic, exponential and logarithmic models have been

proposed by various researchers for different parts of the world. Measuring

and recording ambient temperature is an easy task. In India, there are more

than 451 stations recording the daily maximum and minimum temperatures

along with other meteorological data. Measuring and recording ambient

temperature is easy and economical; it serves as an important parameter, on

which the solar radiation estimation models could be developed.

Global solar radiation comprises of two components – direct radiation

and diffuse radiation. Maximum ambient temperature is not the indicator of

maximum solar energy received and vice versa. The ambient temperature

might be high due to green house effect on a cloudy sun day. Thus the

temperature based models may suffer from under such condition which is a

major drawback. Hence dependability on only this parameter is in question.

2.3.3. Cloud Observation Based Models

Few researchers have presented the estimation models based on the

cloud observations as discussed in 2.2.2.3. Cloud coverage (C) is done in eight

stages. For a non-cloudy clear sun day, C=0. A fully cloudy day counts C = 8.

It is difficult to derive the value of C for a partially cloudy day. Hence the

models based on cloud observations heavily depend on the value of C.

Judgment of C calls for expertise in the weather technology field. An

experienced weather specialist will be able to judge the suitable value of C, on

the cloud observations.


42

2.3.4. Multiple Parameter Based Models

Researchers around the world have presented the multiple parameter

estimation models. Few such models are discussed in 2.2.2.4. These

parameters include – sunshine duration, mean ambient temperature, relative

humidity, longitude, latitude, mean sea level and soil temperature etc. The

models with these parameters developed for specific area (region) cannot be

extended to the other areas, without relaxing on estimation accuracy.

Hence a need arises to choose the parameters which are constant with

time and all seasons, but represent the global radiation variation on earth

surface. Geographical parameters such as – longitude, latitude, mean sea

level are constants for a site. A model integrating these geographical

parameters and few meteorological parameters such as temperature and

relative humidity would be an ideal solution to estimate global solar

radiation.

2.4 Model Validation and Comparison

As outlined in the Handbook of Methods of Estimating Solar Radiation

(1984), a dataset to be used for the validation and comparison must:

• be randomly selected;

• be independent of models being evaluated;

• span all seasons;

• be selected from various geographical regions;

• be sufficiently large to include a spectrum of weather.


43

Considering the above points in mind, generally the researchers

investigate the goodness of the model based on a set of statistical parameters

such as MPE, MAPE, MBE and MABE, RMSE, r and R2. These statistical

testing terms are defined in the following sections.

2.4.1 Statistical Errors

In statistical literature, the performance of model is generally evaluated

in terms of statistical errors, such as – the mean percentage error (MPE), mean

absolute percentage error (MAPE), mean bias error (MBE), mean absolute bias

error (MABE) and root mean square error (RMSE). These errors are defined

below (Grewal [103]):

�� ∑ ��

�� 100�� (2.62)

�� ∑ ��

�� 100�� (2.63)

�� ∑ �� (2.64)

�� ∑ �|�� |�� (2.65)

�� ∑ ��

(2.66)

where Him is the ith measured value, Hie is the ith estimated value and k is the

total number of observations.

In calculating the MPE values, the percentage errors in individual

estimates are summed to calculate the mean. The MAPE gives the absolute

value of the percentage errors. The MBE provides information on long term

performance. A low MBE is always desirable. A positive value gives the


44

average amount error of over estimation of an individual observation, which

will cancel an underestimation in a separate observation.

The RMSE test gives information on the short term performance of the

correlations by allowing a term by term comparison of the actual deviation

between the estimated and measured values. The smaller the value of RMSE,

the better is model’s performance. Thus RMSE is a meaningful measure to

compare the selected estimation models.

2.4.2 Correlation Coefficient, r

Correlation coefficient indicates the strength and direction of a linear

relationship between two random variables. In general, its statistical usage

refers to departure of two variables from independence. The Pearson’s

correlation coefficient r, of series X and Y is

( )( )( ) ( )∑ ∑∑

−−

−−=

22aa

aa

yyxx

yyxxr

(2.67)

where, x and y are the series elements while xa and ya are the series averages.

The correlation coefficient is interpreted as low, medium or high

depending on the value of ‘r’, as given in the Table 2.1.

Table 2.1

Interpretation of Correlation Coefficient

Correlation Coefficient, r Low Medium High

Positive 0.10 to 0.29 0.30 to 0.49 0.50 to 1.00

Negative −0.29 to −0.10 −0.49 to −0.30 −1.00 to −0.50


45

2.4.3 Coefficient of Determination, R2

Coefficient of determination is computed as the square of correlation

coefficient i.e. r2. Once r has been estimated for any fitted model, its numeric

value may be interpreted as follows. For instance, if for a given regression

model r = 0.9, it means that R2 = 0.81. It may be concluded that 81% of the

variation in Y has been explained by the model under discussion, leaving 19%

to be explained by other factors.

2.4.4 Measure of Uncertainty

Standard deviation is a widely used measurement of variability or

diversity used in statistics and probability theory and serve as a measure of

uncertainty. It shows how much variation or 'dispersion' there is from the

'average' (mean or expected value). A low standard deviation indicates that

the data points tend to be very close to the mean, whereas high standard

deviation indicates that the data (x1, x2, x3, x4, x5, ….) is spread out over a

large range of values. The two variations of standard deviations are given

below,

• To estimate the standard deviation from the sample of entire

population of data, is given by,

� � �∑��

��

(2.68)

• To estimate the standard deviation from the entire population of the data, is given by,

�� ∑�� (2.69)

The above methods of model validation and comparison are widely

used to benchmark the performance of the estimate models under discussion.


46

In the next section, the basis of benchmark of performance of models is

discussed.

2.5 Benchmark of Performance

Design of solar thermal and PV conversion systems require several

types of data. The main categories of data often requested by users are shown

in table 2.2. It is known fact that, uncertainty in economic analysis of solar

energy systems is directly proportional to the uncertainty in solar resource

data. Researchers[3] show that the relative uncertainty in life cycle savings is

especially sensitive in cases of high capital cost or low auxiliary energy cost.

Many technologies depend on resources of global solar radiation data on a

tilted surface. However, tilt conversion models generally begin with resources

on a horizontal surface; the most commonly measured and modeled

parameter.

Keeping in the mind that the difficulties and involved costs to have the

measured data of global solar radiation for large number of locations for a

country, International Energy Agency (IEA) [104] report on the validation of

solar radiation models declared, “…. There is little to recommend sunshine

based models. Even though the Angstrom equation can be easily tuned to a

location’s climatic conditions by simple regression, it requires the existence of

radiation data in the first place to produce the prediction equation …”.

Further it concluded that, “… present solar radiation estimation models and

measurements are rather comparable, with absolute measurement

uncertainties in the order of 25-100 W/m2 (2.5% to 10.0%) in hemispherical

measured global solar radiation data.”


47

Therefore any model capable of estimating the global solar radiation

with estimate errors within 10% could be acceptable.

Few reported model uncertainties surveyed by different researchers

around the globe are summarized in table 2.3. It is observed that the errors are

in the range of 5% to 20%. Mainly these models are developed for few

numbers of stations, hence the inherent property of site-specific nature is

observed. The increase in errors indicates the limiting capability of the model.

Hence for the model to become really global, the errors in estimates will rise.

Therefore IEA states that, “the challenge for the solar radiation

measurements and estimation models in the 21st century is to reduce the

uncertainty in measured data, as well as develop more robust models (i.e.,

fewer input parameters and smaller residuals, under a wide variety of

conditions).”

Table 2.2

Radiation data formats required by solar energy system designers and planners

Type of data Time resolution Application

Hemispherical (Global) Seasonal/ daily Glazing energy balance

Illuminance (Sunshine) Seasonal/ daily Day-lighting

Hemispherical tilt (Global) Monthly/ annual Fixed flat plate

Hemispherical tracking (Global) Monthly/ annual Tracking flat plate

Direct normal (Beam) Monthly/ annual Focusing/ concentrating system

Monthly mean daily total (Global) Monthly/ daily Sizing and design specifications,

economics

Monthly mean (Global) Monthly Sizing and design specifications,

economics

Daily profiles (Global) Hourly System simulation, modeling and

rating

10-30 year hourly power (Global) Hourly System lifetime performance and

economics

Daily profiles power (Global) Sub-hourly System responses to clouds, etc.


48

Table 2.3

Summary of quoted uncertainties for various solar radiation models

Radiation component Reference/ Model RMSE Comments

Direct and

hemispherical all sky

Maxwell 1998[105] 5.2 % (direct)

3.0 % (hemi)

Annual mean daily total; 33 US

measurement data

Direct, clear sky Gueymard,

1995[106]

±10.0% Mean of 17 best of 22 models for

Canada

Direct from

hemispherical, all sky

Perez 1992[107] 8.5% Five models; 18 US and European

sites

All sky hemispherical Skartvei et al.

1997[108]

11.0% Five models; 4 European sites

All sky hemispherical Gul et al. 1998[16] 8.0% Three models; 12 UK stations

All sky hemispherical

from satellite

Zelenka et al.

1999[92]

20% 31 Swiss, 12 US measurement

stations

2.6 Layout of the Thesis

Chapter 1 presents the motivation and background behind the present

research work detailing the global energy concerns and discussing diverse

efforts being taken towards solution. Solar energy option and available

assessment methodologies are studied in short. The limitations and

drawbacks of existing methods are briefly discussed. The chapter ends with

the concept statement of the present thesis.

In chapter 2, the detailed theory of various methods of assessment of

solar radiation is discussed. The radiative transfer model explains how the

extra-terrestrial radiation is computed. Typical models based on sunshine

duration, temperature, cloud observations and multi-parameter inputs are

detailed in the empirical models section. Discussions on how the solar

radiation is estimated using satellite data is done. How artificial neural


49

network (ANN) technique is used for assessing the solar radiation is

discussed. The detailed study of drawbacks and limitations of the above four

methodologies is made, indicating the ideas to overcome these drawbacks.

Methods of model validation and comparison and benchmarking the

performance of estimation models are detailed in brief.

Chapter 3 covers the study and analysis of the existing empirical

models. Two existing models are presented with the results, tests against site

variations and establishing their limiting capabilities. The limiting capabilities

of these existing models lead to the necessity of a global or site-independent

model for estimating the solar radiation.

In chapter 4, new models are proposed, with several possible variants.

These models are implemented and studied for their effectiveness. Identifying

the competent model(s) among various proposed models is the objective of

this chapter. The chapter concludes by giving the validation analysis of

models which are identified.

In chapter 5, the proposed models are evaluated and revalidated with

artificial neural network (ANN) model and the comparative study of

proposed numerical models and ANN models is carried out.

Salient observations are recorded in the chapter 6 as conclusions. This

chapter also discusses the scope for research that could be carried out further.

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