Electricity from the Sun

Electricity from the Sun

1 Introduction

The sun is a yellow star at the center of our solar system, in fact is a magnetically active star which supports a strong and changing magnetic field that varies year to year and reverses directions about every eleven years. This magnetic field gives rise to many effects that are so called solar activities, including sunspots on the surface of the Sun, solar flares, and variations in the solar wind that carries material through the solar system. Effects of solar activity on the Earth include auroras at moderate to high latitudes and disruption of radio communications and electric power.

Heat and light from the Sun has supported almost all life on Earth.

The industrial ages gave us the understanding of sunlight as a energy source.This has never been more important than now when we realise that the exploitation of fossil energy sources may affect planets ambient.

Today, the direct conversion of light into electricity, or photovoltaic conversion is becoming accepted as an important form of power generation.

Photovoltaic power generation is reliable, involves no moving parts and the operation and the maintenance costs are very low. The operation of a photovoltaic system is silent and creates no atmospheric pollution. Photovoltaic systems are modular and can be quickly installed. Power can be generated where it is required without the need for transmission lines.

Solar electricity suffers one major drawback at present, and that is the high capital cost of photovoltaic installation. Nevertheless, as the production volumes continue to grow and research brings in new developments, the impact of photovoltaics on power generation is set to increase at a rapid rate. The economic comparison with conventional energy sources is certain to receive a further boost as the environmental and social costs of power generation are included in the picture.

2 Solar Radiation

Solar radiation has become increasingly appreciated because of its influence on living matter and the feasibility of its application for useful purposes. It is a source of natural energy that, along with other forms of renewable energy, has a great potential for a wide variety of applications, because it is abundant and accessible. Solar radiation is rapidly gaining ground as a supplement to the non-renewable sources of energy, which have a finite supply.

The spectrum of solar radiation extents from 200 to 3000 nm wavelengths. It is almost identical with the 6000 K black-body radiation spectrum.

The average energy flux incident on a unit area perpendicular to the beam outside the Earths atmosphere is known as the solar constant :

S= 1361 W/m²

In general, the total power from a radiant source falling on a unit area is called irradiance.

Figure 1 : The solar spectrum [2]

The radiation is distinguished as :

• ultra-violet radiation, 200 to 380 nm, producing photochemical effects, bleaching, sunburn, etc.
• visible light, 380 (violet) to 700 (red).
• infra-red radiation, 700 to 3000 nm, radiant heat, with some photochemical effects.

When the suns rays pass through the earths atmosphere :

a portion of solar radiation is scattered when striking on molecules of air, water vapour and dust particles (the process of scattering occurs when small particles and gas molecules diffuse part of the incoming solar radiation in random directions without any alteration to the wavelength of the electromagnetic energy, Figure 2). A significant proportion of scattered shortwave solar radiation is redirected back to space.

Figure 2 : Atmospheric scattering [3]

- another portion of solar radiation is absorbed ( the process of absorption is defined as one in which solar radiation is retained by a substance and converted into heat energy. The creation of heat energy also causes the substance to emit its own radiation, Figure 3)

Figure 3 : Atmospheric absorption

- the final process in the atmosphere that modifies incoming solar radiation is reflection( is a process where sunlight is redirected by 180 after it strickes an atmospheric particle. Most of the reflection in our atmosphere occurs in clouds when light is interecepted by particles of liquid and frozen water, Figure 4)

Figure 4 : Atmospheric reflection [3]

The sunlight that reaches the Earths surface unmodified by any of the above atmospheric

processes is termed direct solar radiation. Solar radiation that reaches the Earths surface after it was altered by the process of scattering is called diffused solar radiation.

The reflectivity or albedo of the Earths surface varies with the type of material that covers it.For instance fresh snow-can reflect up to 95% of the isolation that reaches it surface, dry sand up to 35-40%, grass type vegetation 15 to 25%. The Earths average albedo, reflectance from both the atmosphere and the surface is about 30%.

Figure 5 describes the modification of solar radiation by atmospheric and surface processes for the whole Earth over a period of one year. Of all the sunlight that passes through the atmosphere annually, only 51 % is available at the Earth's surface to do work. This energy is used to heat the Earth's surface and lower atmosphere, melt and evaporate water, and run photosynthesis in plants. Of the other 49 %, 4 % is reflected back to space by the Earth's surface, 26 % is scattered or reflected to space by clouds and atmospheric particles, and 19 % is absorbed by atmospheric gases, particles, and clouds.

Figure 5 The components of the incoming solar radiation [3]

To identify the position of the sun in the space it will be considered a spherical coordinating system where earth is considered a point-like entity (Figure 6 ).Are defined the following plans:the horizon( the horizontal plan ) and the equator .

Figure 6 : Reference system [4]

The following solar angles are defined:

• solar altitude is the angle between the suns rays and the horizontal plane on the earths surface
• - solar azimuth is the angle between the due-south direction line and the horizontal projection of the suns rays
• z zenith angle is the angle between the sun-earth ray and the zenith direction(point directly overhead)
• solar declination angle is the angle between the earth-sun line and the ecuatorial plane
• hour angle is the angle between the local solar noon and the horizontal projection of the suns rays
• latitude angle is the angle between the ecuatorial plane and the centre of the earth
• longitude angle is the angle between the meridian plan of the placement and the meridian plan of Greenwich

Another very important component is the so-called time equation as shown in Table 1.

Table 1 :Values corresponding to time equation for a bissextile year

Moon Model

The Moon Model supplies the theoretical instantaneous values of irradiance for a dear sky on surfaces oriented in any direction.

The irradiance can be subdivided in several positions

• The total horizontal irradiance : Ith measurable with a pyranometer (a sensor that is designed to measure the solar radiation flux density in watts per square meter or W/m² from field of view of 180 degrees ) .

• The normal direct irradiance : Ibn measurable with a radiometer ( a device used to measure the radiant flux or power in electromagnetic radiation )

• The horizontal diffuse irradiance : Idh

The input data to supply to the model are:

• The days and months of the year that will be converted in a number of progressive order g
• The total number of the days of the year for a bissextile(leap) year gtot
• The hour of the day h
• The latitude angle ( -lat) and longitude angle ( ζ-long) of the studied photovoltaic panel
• The solar altitude (β) and solar azimuth ) of the surface in examination
• The reflection coefficient of the area (surface) albedo ρ (ro)

There are calculated :

-the surface-sky view factor

F=(1+ cos β )/2  (1)

-the peak hour:

h_p=12 (ζ - ζ_ref )/15 τ  (2)

Also, the Moon Model fixes the values of some parameters :

_{A B C}

_{A B C}

_{A B} ψ_C=268;

These parameters are used to calculate the following constants :

Apparent solar constant A

A= _A sin (2π( _A+g )/gtot)

(3)

Atmospheric extinction coefficient B

B= μ_B + Δ_B sin (2 (ψ_B+ g )/gtot)  (4)

Diffuse irradiance coefficient C

C= μ_C + Δ_C sin (2 _C+ g )/gtot)  (5)

Solar declination angle :

π/180 sin (2π( + g )/gtot)  (6)

Solar declination angles for the northern hemisphere are:

spring equinox March 21/21 δ= 0

summer solstice June 21/22 - δ= +23.5

autumnal equinox September 21/21 δ= 0

winter solstice December 21/22 - δ= -23.5

The following values are determinated depending on the time of day .Time is given in solar time as the hour of the day from midnight.

Hour angle :

k= ( h - h_p )*15* π/180 (7)

Cosine of solar zenith is dependent upon latitude, solar declination angle and time of day :

cos z =sin β =cos ξ cos δ cos k + sin ξ sin δ  (8)

For cos z >0 , the following values of irradiance are calculated:

The normal direct irradiance

I_bn=A e^(-B/cos z ) 

The horizontal diffuse irradiance

I_dh=C I_bn  (10)

The total horizontal irradiance

I_th=(C+cos z) I_bn = I_bn cos z + I_dh (11)

The total irradiance is calculated depending on an auxiliary variable θ :

cos θ = sin δ sin ξ cos β - sin δ cos ξ sin β cos Φ +cos δ cos ξ cos β cos k +

+ cos δ sin ξ sin β cos Φ cos k + cos δ sin β sin Φ sin k  (12)

For cos θ >0 , the total irradiance value is:

I_t(θ) = I_bn cos θ + I_dh F + ro I_th (1-F)  (13)

3 An application of Moon Model on a photovoltaic panel in Turin

As an application of the Moon Model, we will take the example of a photovoltaic panel in Torino,Italy.

The placements coordinates are:

Latitude  45.1

Longitude  7.7

Albedo  0.2

Solar altitude  32

Solar azimuth  0

It will be considered a specific day, for example 30^th January :

month : 1

day : 30

With the help of the Moon Model Program we have

the sight factor surface-sky : Fss=0.924

the peak hour : 1265

The values for constants A,B,C depending on the day are:

A=1225 [W/m²]

B

C

0.315 rad

=0.22056 (from Tabel nr.1)

The next values for : k, cosz, cosθ, Ibn, Idh, Ith and It are defined in accordance with the hour (h=1:24)

With the Moon Model we have determinated an ideal characteristic of irradiance varying with time. Having some measurements we can also outline some real characteristics :

Figure : The Hour Angle varying with Time

Figure : The evolution of variables:cos(z) and cos(teta)

Figure : The evolution of the components of irradiance

Figure : The evolution of total irradiance during a day

Figure : The evolution of the total irradiance during the day for a whole year

Moon Model Program

tau=[

0.225833 0.209444 0.0691667 -0.0475 -0.040278 0.059444 0.105 0.0041667 -0.1675 -0.2725 -0.1875

0.228056 0.206111 0.0641667 -0.049722 -0.037778 0.062778 0.104167 -0.001111 -0.173056 -0.273056 -0.181389

0.07 0.23 0.202778 0.0591667 -0.051667 -0.035278 0.065833 0.102778 -0.006667 -0.178333 -0.273333 -0.175

0.077778 0.231944 0.199444 0.0541667 -0.053333 -0.0325 0.068889 0.101667 -0.011944 -0.183333 -0.273333 -0.168333

0.085278 0.233333 0.195556 0.0491667 -0.055 -0.029722 0.071944 0.1 -0.0175 -0.188611 -0.273333 -0.161667

0.092778 0.234722 0.191944 0.0444444 -0.056389 -0.026667 0.075 0.098333 -0.023056 -0.193611 -0.272778 -0.154722

0.235833 0.188056 0.0394444 -0.057778 -0.023889 0.077778 0.096667 -0.028611 -0.198611 -0.272222 -0.147778

0.107222 0.236667 0.183889 0.0347222 -0.059167 -0.020833 0.080278 0.094722 -0.034167 -0.203333 -0.271389 -0.140556

0.114167 0.2375 0.18 0.0302778 -0.06 -0.0175 0.083056 0.0925 -0.04 -0.208056 -0.270278 -0.133333

0.121111 0.238056 0.175556 0.0255556 -0.060833 -0.014444 0.085556 0.090278 -0.045833 -0.2125 -0.268889 -0.125833

0.127778 0.238056 0.171389 0.0211111 -0.061389 -0.011111 0.087778 0.087778 -0.051389 -0.216944 -0.267222 -0.118333

0.134444 0.238333 0.166944 0.0166667 -0.061944 -0.007778 0.090278 0.085278 -0.057222 -0.221389 -0.265278 -0.110556

0.140833 0.238056 0.1625 0.0122222 -0.0625 -0.004167 0.092222 0.0825 -0.063056 -0.225556 -0.263333 -0.102778

0.147222 0.237778 0.158056 0.0080556 -0.0625 -0.000833 0.094444 0.079444 -0.068889 -0.229444 -0.260833 -0.095

0.153333 0.237222 0.153333 0.0036111 -0.0625 0.0027778 0.096111 0.076389 -0.075 -0.233333 -0.258333 -0.086944

0.159167 0.236389 0.148611 -0.000278 -0.0625 0.0063889 0.098056 0.073333 -0.080833 -0.236944 -0.255556 -0.078889

0.165 0.235278 0.143889 -0.004444 -0.062222 0.01 0.099722 0.07 -0.086667 -0.240556 -0.2525 -0.070833

0.170556 0.234167 0.139167 -0.008056 -0.061667 0.0136111 0.101111 0.066389 -0.0925 -0.243889 -0.249444 -0.062778

0.175833 0.232778 0.134167 -0.011944 -0.061111 0.0172222 0.1025 0.062778 -0.098611 -0.247222 -0.245833 -0.054444

0.180833 0.231389 0.129444 -0.015556 -0.060278 0.0208333 0.103611 0.059167 -0.104444 -0.250278 -0.242222 -0.046389

0.185833 0.229444 0.124444 -0.019167 -0.059444 0.0244444 0.104722 0.055278 -0.110278 -0.253333 -0.238333 -0.038056

0.190556 0.2275 0.119444 -0.0225 -0.058333 0.0280556 0.105556 0.051111 -0.116111 -0.255833 -0.234167 -0.029722

0.195278 0.225556 0.114444 -0.025833 -0.057222 0.0316667 0.106389 0.046944 -0.121944 -0.258611 -0.229722 -0.021389

0.199444 0.223333 0.109444 -0.028889 -0.055833 0.0352778 0.106667 0.042778 -0.127778 -0.260833 -0.225278 -0.013056

0.203611 0.220833 0.104444 -0.031944 -0.054167 0.0388889 0.107222 0.038333 -0.133611 -0.263056 -0.220556 -0.005

0.2075 0.218333 0.099444 -0.035 -0.052778 0.0425 0.1075 0.033611 -0.139444 -0.265 -0.215556 0.0033333

0.211111 0.215556 0.094444 -0.037778 -0.050833 0.0461111 0.1075 0.029444 -0.145278 -0.266667 -0.210278 0.0116667

0.214444 0.2125 0.089167 -0.040278 -0.048889 0.0494444 0.107222 0.024167 -0.150833 -0.268333 -0.204722 0.0197222

0.217778 0 0.084167 -0.042778 -0.046944 0.0530556 0.106944 0.019167 -0.156389 -0.269722 -0.199167 0.0277778

0.220556 0 0.079167 -0.045278 -0.044722 0.0563889 0.106389 0.014167 -0.161944 -0.270833 -0.193333 0.0361111

0.223333 0 0.074167 0 -0.0425 0 0.105833 0.009167 0 -0.271944 0 0.0441667

[m,n]=size(tau)

no_days=[31 28 31 30 31 30 31 31 30 31 30 31];

%Input data for the selected day:

month=input('Input the month: '

day=input('Input the day: '

if (month==1)

g=day;

else

g=sum(no_days(1:month-1))+day;

end

An example for a photovoltaic cell in Torino- Moon Modell

%Latitude - lat

lat=45.1*pi/180;

%Longitude - long

long=7.7*pi/180;

%Reference longitude - longrif

longrif=15*pi/180;

%The total number of the days of the year-gtot=365 for bissextile year

gtot=365;

%albedo(terreno) - ro

ro=0.2;

%solar azimuth

fi=0;

%Solar altitude - beta

beta=32*pi/180;

%Hour of the day - h

Peak hour - hp

for j=1:n

for i=1:no_days(n)

hp(j,i)=12-(long-longrif)/15-tau(i,j);

end

end

%The Moon model fixes the values of some parameters:

delta0=23.45;

psi0=284;

miuA=1158;

deltaA=-73;

psiA=268;

miuB=0.17;

deltaB=0.035

psiB=268;

miuC=0.095;

deltaC=0.04;

psiC=268;

%There are defined:

for j=1:n

for i=1:no_days(n)

if (j==1)

ggg=i;

else

ggg=sum(no_days(1:j-1))+i;

end

%Appearing Solar Constante A

A(j,i)=miuA+deltaA*sin(2*pi*(psiA+ggg)/gtot);

%Coefficient of atmospheric extinction B

B(j,i)=miuB+deltaB*sin(2*pi*(psiB+ggg)/gtot);

%Coefficient of cancellation diffuza C

C(j,i)=miuC+deltaC*sin(2*pi*(psiC+ggg)/gtot);

%Solar declination - decl

decl(j,i)=delta0*pi/180*sin(2*pi*(psi0+ggg)/gtot);

end

end

%Hour angle k

for j=1:n

for i=1:no_days(n)

for h=1:24

k(j,i,h)=(h-hp(j,i))*15*pi/180;

end

end

end

%It's obtained the sight factor surface-sky (is the right translation?)

Fss=(1+cos(beta))/2;

%Solar zenit z

for j=1:n

for i=1:no_days(n)

for h=1:24

z(j,i,h)=acos(cos(lat)*cos(decl(j,i))*cos(k(j,i,h))+sin(lat)*sin(decl(j,i)));

end

end

end

%The normal direct irradiance Ibn

for j=1:n

for i=1:no_days(n)

for h=1:24

if (cos(z(j,i,h))<=0)

Ibn(j,i,h)=0;

else

Ibn(j,i,h)=A(j,i)*exp(-B(j,i)/cos(z(j,i,h)));

end

end

end

end

%Horizontal diffuse irradiance Idh

for j=1:n

for i=1:no_days(n)

for h=1:24

Idh(j,i,h)=C(j,i)*Ibn(j,i,h);

end

end

end

% The total horizontal irradiance Ith

for j=1:n

for i=1:no_days(n)

for h=1:24

if(cos(z(j,i,h))<=0)

Ith(j,i,h)=0;

else

Ith(j,i,h)=(C(j,i)+cos(z(j,i,h))).*Ibn(j,i,h);

end

end

end

end

teta=zeros(m,31,24);

for j=1:n

for i=1:no_days(n)

for h=1:24

teta(j,i,h) =acos(sin(decl(j,i))*sin(lat)*cos(beta)-sin(decl(j,i))*cos(lat)*sin(beta)*cos(fi)+cos(decl(j,i))*cos(lat)*cos(beta)*cos(k(j,i,h))+cos(decl(j,i))*sin(lat)*sin(beta**cos(fi)*cos(k(j,i,h))+cos(decl(j,i))*sin(beta)*sin(fi)*sin(k(j,i,h)));

end

end

end

%The total irradiance It

for j=1:n

for i=1:no_days(n)

for h=1:24

if(cos(teta(j,i,h))<=0&cos(z(j,i,h))<=0)

It(j,i,h)=0;

else

It(j,i,h)=(Ibn(j,i,h)).*cos(teta(j,i,h))+Idh(j,i,h)*Fss+ro*(1-Fss)*Ith(j,i,h);

end

end

end

end

%Characteristics :

% 1. Hour angle varying with time

for h=1:24

k_inter(h)=k(month,day,h);

end

figure;

plot(k_inter); grid;

% Cos(z) & cos(teta) varying with time

f=cos(z);

g=cos(teta);

for h=1:24

f_inter(h)=f(month,day,h);

g_inter(h)=g(month,day,h);

end

figure;

plot(f_inter,'-r');hold on

plot(g_inter,'-b');

% 3. The components of irradiance varying with time

for h=1:24

Ibn_inter(h)=Ibn(month,day,h);

Idh_inter(h)=Idh(month,day,h);

Ith_inter(h)=Ith(month,day,h);

end

figure;

plot(Ibn_inter,'-r');hold on

plot(Idh_inter,'-b'

plot(Ith_inter,'-g'

figure;hold on

% 4. The total irradiance varying with time

for h=1:24

It_inter(h)=It(month,day,h);

end

plot(It_inter,'-k'

figure;hold on

% 5. The evolution of the total irradiance during the year

for j=1:n

for i=1:no_days(n)

for h=1:24

It_inter(h)=It(j,i,h);

end

plot(It_inter,'k'

end

end

4. Measurements from Aosta, Italy during the year 2004

The measurements have been taken starting with 5 A.M. till 9 P.M., hour by hour.

Having the values for irradiance we can plot the Irradiance varying with Hour for a whole year.

The next step is to take from Moon Model program the values for total irradiance and as before, to plot a figure like the one above. The only difference is that the last one is representing the ideal shape of the irradiance.

TITLE???

The comparison between the ideal values and the real one:

References

1. T. Markwart(ed.) , Solar electricity , Wiley , Chichester , UK , 1994

https://www.csr.utexas.edu/projects/rs/hrs/pics/irradiance.gif

3. https://www.physicalgeography.net/fundamentals/7f.html

4. Analisi Sperimentale di impianti fotovoltaici connessi a rete , Gianfranco Chicco,Filippo Spertino