Jager Et Al. - 2014 - A Student Introduction To Solar Energy PDF
Jager Et Al. - 2014 - A Student Introduction To Solar Energy PDF
Jager Et Al. - 2014 - A Student Introduction To Solar Energy PDF
Klaus Jäger
Olindo Isabella
Arno H.M. Smets
René A.C.M.M. van Swaaij
Miro Zeman
iv
Preface
v
vi
Contents
I PV Fundamentals 1
2 Electrodynamic basics 7
2.1 The electromagnetic theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2 Electromagnetic waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.3 Optics of flat interfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.4 Optics in absorptive media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.5 Continuity and Poisson equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
3 Solar Radiation 15
3.1 Radiometric properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.2 Blackbody radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.3 Wave-particle duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.4 Solar spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
vii
viii Contents
II PV Technology 25
III PV Systems 27
4 Introduction to PV systems 29
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
4.2 Types of PV systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
4.3 Components of a PV system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
5 Location issues 35
5.1 The position of the sun . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
5.2 The sun path at different locations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
5.3 The equation of time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
5.4 Irradiance on a PV module . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
5.5 Direct and diffuse irradiance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
5.6 Shadowing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
6 Components of PV Systems 61
6.1 PV modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
6.2 Maximum power point tracking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
6.3 Photovoltaic Converters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
6.4 Batteries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
6.5 Cables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
Appendix 129
A Derivations in Electrodynamics 131
A.1 Basics of Vector Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
A.2 The Maxwell equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
A.3 Derivation of the electromagnetic wave equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
A.4 Properties of electromagnetic waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
Bibliography 137
x Nomenclature
Part I
PV Fundamentals
1
The Working Principle of a Solar Cell
1
In this chapter we present a very simple model of a the light consists of well defined energy quanta, called
solar cell. Many notions presented in this chapter will photons. The energy of such a photon is given by
be new but nonetheless the great lines of how a solar
cell works should be clear. All the aspects presented
in this chapter will be discussed in larger detail in the
following chapters. E = hν, (1.1)
3
4 1. The Working Principle of a Solar Cell
bine, i.e. the time it requires the charge carriers to reach the maximal energy conversion efficiency of a single-
the membranes must be shorter than their lifetime. This junction solar cell is considerably below the thermody-
requirement limits the thickness of the absorber. namic limit. This single bandgap limit was first calculated
by Shockley and Queisser in 1961 [5].
We will discuss generation and recombination of elec-
trons and holes in detail in chapter ??. A detailed overview of loss mechanisms and the result-
ing efficiency limits is discussed in chapter ??.
Loss mechanisms
7
8 2. Electrodynamic basics
frame of reference into another one that moves with light is emitted and absorbed by matter. For this pur-
respect to the first one with a constant velocity, elec- pose, quantum mechanics is required.
tric fields are transformed into magnetic fields and vice
versa.
Between 1861 and 1862, the Scottish physicist James 2.2 Electromagnetic waves
Clerk Maxwell published works in that he managed
to formulate the complete electromagnetic theory by a As shown in Appendix A.3, electromagnetic waves are
set of equations, the Maxwell equations. A modern for- described by
mulation of these equations is given in Appendix A.2. 2
∂2 ∂2 n2 ∂2 E
The transformation of the electric and magnetic fields ∂
+ + E − =0 (2.1a)
between different frames of reference is correctly de- ∂x2 ∂y2 ∂z2 c20 ∂t2
scribed by Albert Einstein’s theory of special relativity,
published in 1905. for the electric field E(r, t), where c0 denotes the speed
of light in vacuo and n is the refractive index of the
One of the most important prediction of the Maxwell material. In a similar manner we can derive the wave
equations is the presence of electromagnetic waves. equation for the magnetic field,
A derivation is given in Appendix A.3. Maxwell soon 2
∂2 ∂2 n2 ∂2 H
realised that the speed of these waves is (within exper- ∂
+ + H − = 0. (2.1b)
imental accuracy) the same as the speed of light that ∂x2 ∂y2 ∂z2 c20 ∂t2
then already was known. He brilliantly concluded that
light is an electromagnetic wave. The simplest solution to the wave equations (2.1) is
the plane harmonic wave, where light of constant
In the 1880s the German physicist Heinrich Hertz could wavelength λ propagates in one direction. Without loss
experimentally confirm that electromagnetic waves can of generality, we assume that the wave travels along the
be generated and have the same speed as light. His z direction. The electric and magnetic fields in this case
work laid the foundation for modern radiocommunic- are
ation that has shaped the modern world.
E(r, t) = E0 · eikz z−iωt , (2.2a)
The electromagnetic theory can perfectly describe how
light propagates. However, it fails in explaining how H(r, t) = H0 · eikz z−iωt , (2.2b)
2.3. Optics of flat interfaces 9
The intensities are proportional to the square of the 2.4 Optics in absorptive media
electric field, I ∝ E2 . For unpolarised light, we have
to take the mean values of the two polarisations. For
Let us recap what we have seen that far in this chapter:
the reflectivity R we thus obtain
Starting from the Maxwell equations we derived the
wave equations and looked at their properties for the
1 2
R= rs + r2p . (2.14) special case of plane waves. After that we looked at the
2 behaviour of electromagnetic waves at the interfaces
For normal incidence this leads to between new media. For the whole discussion so far
we implicitly assumed that the media is non-absorbing.
2
n1 − n2
The working principle of solar cells is based on the fact
R ( θ i = 0) = . (2.15)
n1 + n2 that light is absorbed in an absorber material and that
the absorbed light is used for exciting charge carriers
Because of conservation of energy the sum of R and the that can be used to drive an electric circuit. Therefore
transmittance T must be 1, we will use this section to discuss how absorption of
light in a medium can be described mathematically.
R + T = 1. (2.16)
In general, the optical properties of an absorbing me-
dium are described by an complex electric permittivity ẽ,
By combining Eqs. (2.14) with (2.16) and doing some
calculations we find ẽ = e0 + ie00 . (2.19)
Let us now substitute Eq. (2.21) into Eq. (2.2a), 2.5 Continuity and Poisson equa-
Ex (z, t) = Ex,0 · eik̃z z−iωt = Ex,0 ·
00 0
e−kz z eikz z−iωt . (2.22) tions
We thus see that the electric field is attenuated expo- At the end of this chapter we want to mention two
nentially, exp(−k00z z) when travelling through the ab- equations that are very important for our treatise of
sorbing medium. The intensity of the electromagnetic semiconductor physics in Chapter ??.
field is proportional to the square of the electric field,
κω 4πκ ∂Ey
α = 2k00z = 2
∂Ex ∂Ez ρ
= , (2.25) + + = , (2.26)
c λ0 ∂x ∂y ∂z ee0
where λ0 = 2πc/ω is the wavelength in vacuo.
where Ex , Ey and Ez are the components of the electric
Equation (2.25) is known as the Lambert-Beer law. In field vector, E = ( Ex , Ey , Ez ). Further, we here assume
general, the complex refractive index and hence the ab- that we are in an electrostatic situation, i.e. there are
sorption coefficient are no material constants but vary no moving charges . From the second Maxwell equa-
with the frequency. Especially α may change across sev- tion (A.1b) we know that in that case the electric field
eral orders of magnitude across the spectrum, making is rotation free. Vector calculus teaches us that then the
the material very absorptive at one wavelength but al- electric field is connected to the electric potential via
most transparent at other wavelength. Absorption spec-
tra will be discussed thoroughly later on when looking
∂U ∂U ∂U
at various photovoltaic materials in Part II. E=− , , . (2.27)
∂x ∂y ∂z
2.5. Continuity and Poisson equations 13
By combining Eqs. (2.26) with (2.27) we find the Poisson mulation that is given by
equation.
2 ∂ρ ∂Jx ∂Jy ∂Jz
∂2 ∂2
∂
+ + U=−
ρ
. (2.28) + + + = 0, (2.32)
∂x ∂y ∂z ee0 ∂t ∂x ∂y ∂z
In Chapter ?? we only will use the one-dimensional where Jx , Jy and Jz are the components of the current
form given by density vector, J = ( Jx , Jy , Jz ).
∂2 U ρ
=− . (2.29)
∂z ee0
For our discussion we assume a surface A that is irra- The quantity Le is called the radiance and it is one of
diated by light, as illustrated in Fig. 3.1 (a). For obtain- the most fundamental radiative properties. Its physical
15
16 3. Solar Radiation
dimension is
[ Le ] = W·m−2 ·sr−1 .
The factor cos θ expresses the fact that not the surface
element dA itself is the relevant property but the pro-
jection of dA to the normal of the direction (θ, φ). This
(a) (b) is also known as the Lambert cosine law.
We can express Eq. (3.2) as integrals of the surface co-
A
ordinates (ξ, η ) and the direction coordinates (θ, φ),
which reads as
dΩ
Z Z
P= Le (ξ, η; θ, φ) cos θ sin θ dθ dφ dξ dη. (3.3)
A 2π
θ
Since sunlight consists of of a spectrum of different fre-
quencies (or wavelengths), it is useful to use spectral
dA
properties. These are given by
η
dP dP
Pν = , Pλ = , (3.4)
ξ dν dλ
dLe dLe
Leν = , Leλ = , (3.5)
dν dλ
Figure 3.1: (a) Illustrating a surface A irradiated by light from
etc. Their physical dimensions are
various directions and (b) a surface element dA that receives
radiation from a solid angle element dΩ under an angle θ with [ Pν ] = W·Hz−1 = W·s, [ Pλ ] = W·m−1 ,
respect to the surface normal.
[ Leν ] = W·m−2 ·sr−1 ·s, [ Leλ ] = W·m−2 ·sr−1 ·m−1 ,
and similarly for Leν and Leλ . The − sign is because of and similarly for Ieν and Ieλ . Irradiance refers to ra-
the changing direction of integration when switching diation, that is received by the surface. For radiation
between ν and λ and usually is omitted. emitted by the surface, we instead speak of radiant emit-
tance, Me , Meν , and Meλ .
The spectral power in wavelength thus can be obtained
via Z Z As we discussed earlier, the energy of a photon is pro-
Pλ = Leλ cos θ dΩ dA, (3.7) portional to its frequency, E ph = hν = hc/λ. Thus, the
A 2π spectral power Pλ is proportional to the spectral photon
and analogously for Pν . The radiance is given by flow Nph, λ ,
hc
1 ∂4 P Pλ = Nph, λ , (3.11)
Le = , (3.8) λ
cos θ ∂A ∂Ω
and similarly for Pν and Nph, ν . The total photon flow N
and similarly for Leν and Leλ . is related to the spectral photon flow via
Another very important radiometric property is the ir- Z ∞ Z ∞
radiance Ie that tells us the power density at a certain Nph = Nph, ν dν = Nph, λ dλ. (3.12)
0 0
point (ξ, η ) of the surface. It often also is called the
(spectral) intensity of the light. It is given as the integ- The physical dimensions of the (spectral) photon flow
ral of the radiance over the solid angle, are
By comparing Eqs. (3.10) and (3.13) and looking at Eq. Wilhelm Wien empirically derived the following expres-
(3.11), we find sion for the spectral blackbody radiance:
hc
Ieλ = Φph, λ , (3.14) C1
C2
λ LW
eλ ( λ; T ) = exp − , (3.15)
λ5 λT
and analogously for Ieν and Φph, ν .
where λ and T are the wavelength and the temperature,
respectively. While this approximation gives good res-
ults for short wavelengths, it fails to predict the emitted
3.2 Blackbody radiation spectrum at long wavelengths, thus in the infrared.
Two approximations for the blackbody spectrum were In 1900, Max Planck found an equation, that interpol-
presented around the turn of the century: First, in 1896, ates between the Wien approximation and the Rayleigh-
3.2. Blackbody radiation 19
Jeans law,
BB 2hc2 1
Leλ (λ; T ) = , (3.17a)
λ5 exp hc − 1
λk T B
Spectral radiance W/ m2 · nm · sr
Planck constant. Via Eq. (3.6) we find the Planck law ex-
Planck
pressed as a function of the frequency ν,
30000
BB 2hν3 1
Leν (ν; T ) = , (3.17b)
c2 exp hν − 1
k T 20000
B
Spectral radiance W/ m2 · nm · sr
Wien approximation with C1 = 2hc2 and C2 = hc/k B .
6000 K
For long wavelength we can use the approximation 30000
hc hc
exp −1 ≈ , 20000
λk B T λk B T
5000 K
which directly results in the Rayleigh-Jeans law.
10000
The total radiant emittance of a black body is given by 4000 K
Z ∞
0
Z
MeBB ( T ) = BB
Leλ (λ; T ) cos θ sin θ dθ dφ dλ = σT 4 , 0 500 1000 1500 2000 2500
2π 0
(3.18) Wavelength (nm)
where
Figure 3.3: The blackbody spectrum at three different temper-
2π 5 k4B atures
σ= ≈ 5.670 · 10−8 J s−1 m−2 K−4 (3.19)
15c2 h3
is the Stefan-Boltzmann constant. Equation (3.18) is
known as the Stefan-Boltzmann law. As a matter of fact, wavelength of maximal radiance is indirectly propor-
it already was discovered in 1879 and 1884 by Jožef tional to the temperature,
Stefan and Ludwig Boltzmann, respectively, i.e. about
twenty years prior to the derivation of Planck’s law. λmax T = b ≈ 2.898 · 10−3 m · K. (3.20)
This law is very important because it tells us that if
the temperature of a body (in K) is doubled, it emits
16 times as much power. Little temperature variations
Figure 3.3 shows the spectra for three different temper-
thus have a large influence on the total emitted power.
atures. Note the strong increase in radiance with tem-
Another important property of blackbody radiation perature and also the shift of the maximum to shorter
is Wien’s displacement law, which states that the wavelengths.
3.4. Solar spectra 21
In Planck’s law, as stated in Eqs. (3.17), the constant As we already mentioned in chapter 1, only photons of
h appeared for the first time. Its product with the fre- appropriate energy can be absorbed and hence generate
quency, hν = hc/λ has the unit of an energy. Planck electron-hole pairs in a semiconductor material. There-
himself did not see the implications of h. It was Ein- fore, it is important to know the spectral distribution of
stein, who understood in 1905 that Planck’s law ac- the solar radiation, i.e. the number of photons of a par-
tually has to be interpreted such that light comes in ticular energy as a function of the wavelength λ. Two
quanta of energy with the size quantities are used to describe the solar radiation spec-
trum, namely the spectral irradiance Ieλ and the spectral
Eph = hν. (3.21) photon flux Φph (λ). We defined these quantities already
Nowadays, these quanta are called photons. In terms in section 3.1.
of classical mechanics we could say that light shows the The surface temperature of the sun is about 6000 K. If
behaviour of particles. it would be a perfect black body, it would emit a spec-
On the other hand, we have seen in Chapter 2 that light trum as described by Eqs. (3.17), which give the spec-
also shows wave character which becomes obvious when tral radiance. For calculating the spectral irradiance a
looking at the propagation of light through space or at blackbody with the size and position of the sun would
reflection and refraction at a flat interface. It also was have on earth, we have to multiply the spectral radi-
discovered that other particles, such as electrons, show ance with the solid angle of the sun as seen from earth,
wave-like properties. BB
Ieλ BB
( T; λ) = Leλ ( T; λ)Ωsun . (3.22)
This behaviour is called wave-particle duality and is a
We can calculate Ωsun with
very intriguing property of quantum mechanics that was
discovered and developed in the first quarter of the
Rsun
2
twentieth century. Many discussion was held on how Ωsun = π . (3.23)
AU − Rearth
this duality has to be interpreted - but this is out of the
focus of this book. So we just will accept that depend- Using Rsun = 696 000 km, an astronomical unit AU =
ing on the situation light might behave as wave or as 149 600 000 km, and Rearth = 6370 km, we find Ωsun ≈
particle. 68 µsr.
22 3. Solar Radiation
6000 K blackbody radiation
Spectral irradiance W/ m2 nm
of the earth, it is attenuated. The most important para- 2.0 AM0 radiation
meter that determines the solar irradiance under clear
AM1.5 radiation
sky conditions is the distance that the sunlight has to
1.5
travel through the atmosphere. This distance is the
shortest when the sun is at the zenith, i.e. directly over-
head. The ratio of an actual path length of the sun- 1.0
light to this minimal distance is known as the optical
air mass. When the sun is at its zenith the optical air 0.5
mass is unity and the spectrum is called the air mass
1 (AM1) spectrum. When the sun is at an angle θ with 0.0
the zenith, the air mass is given by 0 500 1000 1500 2000 2500
Wavelength (nm)
1
AM := . (3.24)
cos θ Figure 3.4: Different solar spectra: the blackbody spectrum of
a blackbody at 6000 K, the extraterrestrial AM0 spectrum and
For example, when the sun is 60° from the zenith, i.e. the AM1.5 spectrum.
30° above the horizon, we receive an AM2 spectrum.
Depending on the position on the earth and the posi-
tion of the sun in the sky, terrestrial solar radiation var-
ies both in intensity and the spectral distribution. The
attenuation of solar radiation is due to scattering and
absorption by air molecules, dust particles and/or aer-
3.4. Solar spectra 23
osols in the atmosphere. Especially, steam (H2 O), oxy- ticular place on the earth is extremely variable. In ad-
gen (O2 ) and carbon dioxide (CO2 ) cause absorption. dition to the regular daily and annual variation due
Since this absorption is wavelength-selective, it results to the apparent motion of the sun, irregular variations
in gaps in the spectral distribution of solar radiation have to be taken into account that are caused by local
as apparent in Fig. 3.4. Ozone absorbs radiation with atmospheric conditions, such as clouds. These condi-
wavelengths below 300 nm. Depletion of ozone from tions particularly influence the direct and diffuse com-
the atmosphere allows more ultra-violet radiation to ponents of solar radiation. The direct component of
reach the earth, with consequent harmful effects upon solar radiation is that part of the sunlight that directly
biological systems. CO2 molecules contribute to the ab- reaches the surface. Scattering of the sunlight in the at-
sorption of solar radiation at wavelengths above 1 µm. mosphere generates the diffuse component. A part of
By changing the CO2 content in the atmosphere the ab- the solar radiation that is reflected by the earth’s sur-
sorption in the infrared, which has consequences for face, which is called albedo, may be also present in the
our climate. total solar radiation. We use a term global radiation to
refer to the total solar radiation, which is made up of
Solar cells and photovoltaic modules are produced by these three components.
many different companies and laboratories. Further,
many different solar cell technologies are investigated The design of an optimal photovoltaic system for a par-
and sold. It is therefore of utmost importance to define ticular location depends on the availability of the solar
a reference solar spectrum that allows a comparison of all insolation data at the location. Solar irradiance integ-
the different solar cells and PV modules. The industrial rated over a period of time is called solar irradiation.
standard is the AM1.5 spectrum, which corresponds to For example, the average annual solar irradiation in the
an angle of 48.2°. While the “real” AM1.5 spectrum cor- Netherlands is 1 000 kWh/m2 , while in Sahara the av-
responds to a total irradiance of 827 W·m−2 , the indus- erage value is 2 200 kWh/m2 , thus more than twice as
trial standard corresponds to Ie (AM1.5) = 1000 W·m−2 high. We will disuss these issues in more detail in in
and is close to the maximum received at the surface of Chapter 4.
the earth. The power generated by a PV module under
this conditions is thus expressed in the unit Watt peak,
Wp.
PV Technology
25
Part III
PV Systems
27
Introduction to PV systems
4
4.1 Introduction 4.2 Types of PV systems
29
30 4. Introduction to PV systems
(a) (b)
PV Modules
PV Modules
Charge controller
DC
=
AC
~
Inverter Loads
DC
Water reservoir
Water pump
Batteries
Figure 4.1: Schematic representation of (a) a simple DC PV system to power a water pump with no energy storage and (b) a com-
plex PV system including batteries, power conditioners, and both DC and AC loads.
4.2. Types of PV systems 31
35
36 5. Location issues
ls
Merid
is a ∈ [−90◦ , 90◦ ], where negative angles correspond
ph
ere
to the object being below the horizon and thus not vis-
ian
Object S
180° ible. The azimuth A that is the angle between the line of
sight projected on the horizontal plane and due North.
Altitude a
coordinates via
ξ cos a cos A
υ = cos a sin A . (5.1)
ζ sin a North North
ecliptic Axial celestial
Note that ξ 2 + υ2 + ζ 2 = 1 for all points on the celestial pole pole
tilt
sphere. ε
Earth orbits the sun in an elliptic orbit at an average
distance of about 150 million kilometres. Due to the stial equator 23 Sep
Cele rc
Ecliptic
a
elliptic orbit the speed of Earth is not constant. This tic c
irc l e
is because of Kepler’s second law that states that “A ec un
line joining a planet and the Sun sweeps out equal 21 D EA 21 J
RTH
areas during equal intervals of time.” On the celestial
sphere the sun seems to move on a circular path with 21 Mar
one revolution per year. This path is called the ecliptic
and illustrated in Fig. 5.2. For describing the apparent
movement of the sun on the celestial sphere is is con- ox
uin
e
venient to use coordinates in that the ecliptic lies in the eq
er
l
ph
a s
fundamental plane. These coordinates are called the ec- rn al
Ve ti
liptic coordinates. As principal direction, the position of les
Ce
the sun at the spring (vernal) equinox (thus around 21
March) is used, which is indicated by the sign of Aries,
à. As obvious from Fig. 5.2, à lies both in the ecliptic
plane as well as in the equatorial plane. The ecliptic co- Figure 5.2: Illustrating the ecliptic, i.e. the apparent movement
ordinate system is sketched in Fig. 5.3 (a). The two an- of the sun around earth. Further, the celestial equator and the
gular coordinates are called the ecliptic longitude λ and direction of the vernal equinox are indicated. The sizes of Sun
and Earth are not in scale.
the ecliptic latitude β. Note, that in this coordinate sys-
tem the rotation of the earth around its axis is not taken
into account.
38 5. Location issues
In ecliptic coordinates approximate position the pos- It may be convenient to normalise q and g to the range
ition of the sun can be expressed easily. The approx- [0◦ , 360◦ ) by adding or subtracting multiples of 360◦ .
imation presented here has an accuracy of about 1 ar-
Now the ecliptic longitude of the sun is given by
cminute within two centuries of 2000 and is published
by the the Astronomical Applications Departement of the λS = q + 1.915◦ sin g + 0.020◦ sin 2g. (5.5)
U.S. Naval Observatory.
The ecliptic latitude can be approximated by
To express the position of the sun we first have to ex-
press the time D elapsed since Greenwich noon, Ter- β S = 0. (5.6)
restrial Time, on 1 January 2000, in days. For astro-
nomic purposes it may be convenient to relate D to the For estimating the radiation it might also be convenient
Julian date JD via to approximate the distance of the Sun from the Earth.
In astronomical units (AU) this is given by
D = JD − 2451545.0. (5.2)
R = 1.00014 − 0.01671 cos g − 0.00014 cos 2g. (5.7)
The Julian Date1 that is defined as the number of days
since 1 January 4713 BC in a proleptic2 Julian calendar
or since 24 November 4717 BC in a proleptic Gregorian As stated above, for PV applications it is convenient to
calendar. use horizontal coordinates. We therefore have to trans-
form from ecliptic coordinates into the horizontal co-
Now, the mean longitude of the sun corrected to the ab- ordinates. This is done via three rotations that are to be
erration of the light is given by performed after each other consecutively:
q = 280.459◦ + 0.98564736◦ D (5.3) First, we have to transform from ecliptic coordinates
into equatorial coordinates. As illustrated in Fig. 5.2,
Because of the elliptic orbit of earth and hence a vary-
the fundamental plane of these coordinates is tilted to
ing speed throughout the year, we have to correct with
the ecliptic with an angle e,
the so-called mean anomaly of the Sun,
e = 23.429◦ − 0.00000036◦ D (5.8)
g = 357.529◦ + 0.98560028◦ D. (5.4)
1 Calculating the Julian Date is implemented in MatLab. The principal direction is again given by the vernal
2 Proleptic means that a calendar is applied to dates before its introduction. equinox à. In Fig. 5.3 (b) the equatorial coordinate
5.1. The position of the sun 39
e e
er Object er Object
ph ph
S
De
l
l
tia
tia
clina
les
les
alt
Ecliude β
Ce
Ce
it
ptic
tion δ
Eclip Righ
longitic ascent
tude sion
λ α
Centre of Centre of
the Earth Cel the Earth
Eclip estia
tic l equa
tor
Figure 5.3: Illustrating (a) the ecliptic coordinate system and (b) the equatorial coordinate system.
40 5. Location issues
L MST
Those two angles are connected to each other via
IC H
θL
h = θ L − α, (5.10) GREEN W
OBSER VER
M
G
where θ L is the local mean sidereal time, i.e. the angle ST
between the vernal equinox and the meridian. All
these angles are illustrated in Fig. 5.4. A sidereal day λ0
l o n git u d e
is the duration between two passes of the vernal equi-
α
nox through the meridian and it is slightly shorter r i g ht a s c e n si o n
than a solar day. We can understand this by realising
that the earth has to rotate by 360◦ and approximately
360◦ /365.25 between two passes through the meridian. Figure 5.4: Illustrating the right ascension α, the local hour
The duration of mean sidereal day is approximately 23 angle h, the Greenwich Mean Sidereal Time GMST and the local
h, 56 m and 4 s. mean sidereal time θ L .
ξ −1 0 0 sin φ0 0 − cos φ0 cos θ L sin θ L 0 1 0 0 cos β cos λ
υ = 0 −1 0 0 1 0 sin θ L − cos θ L 0 0 cos e − sin e cos β sin λ . (5.17)
ζ 0 0 1 cos φ0 0 sin φ0 0 0 1 0 sin e cos e sin β
Please note that matrix multiplications do not commute, i.e. the order in which the rotations are applied must not be
altered. Now, we apply that the ecliptic latitude of the sun β S = 0. By calculating Eq. (5.32) we find
ξ S = cos aS cos AS = − sin φ0 cos θ L cos λS − (sin φ0 sin θ L cos e − cos φ0 sin e) sin λS , (5.18a)
υS = cos aS sin AS = − sin θ L cos λS + cos θ L cos e sin λS , (5.18b)
ζ S = sin aS = cos φ0 cos θ L cos λS + (cos φ0 sin θ L cos e + sin φ0 sin e) sin λS , (5.18c)
where we also used the the relationship between Cartesian and spherical horizontal coordinates from Eq. (5.1). Di-
viding Eq. (5.18b) by Eq. (5.18a) and leaving Eq. (5.18c) unchanged leads to the final expressions for the solar posi-
tion,
υS − sin θ L cos λS + cos θ L cos e sin λS
tan AS = = , (5.19a)
ξS − sin φ0 cos θ L cos λS − (sin φ0 sin θ L cos e − cos φ0 sin e) sin λS
sin aS = ζ S = cos φ0 cos θ L cos λS + (cos φ0 sin θ L cos e + sin φ0 sin e) sin λS . (5.19b)
AS and aS now can be derived by applying inverse trigonometric functions. While arcsin uniquely delivers an alti-
tude in between −90◦ and 90◦ , applying arctan leads to ambiguities. For deriving an azimuth in between 0◦ and
360◦ , we have to look in which quadrant is lying. Therefore we use ξ and υ from Eqs. (5.18a) and (5.18b), respect-
ively. We find
f (. . .) denotes the function at the right hand side of Eq. (5.19a). Note that an altitude aS < 0◦ corresponds to the sun
being below the horizon. This means that the Sun is not visible and no solar energy can be harvested.
The approximations presented on the previous pages are accurate within arcminutes for 200 centuries of 2000. Sev-
eral years ago, NREL presented a much more complicated model, the so-called Solar Position Algorithm (ASP), with
uncertainties of only ±0.0003◦ in the period from 2000 BC to 6000 AD [7].
Example
As an example we will calculate the position of the Sun in Delft on 14 April 2014 at 11:00 local time.
For determining the solar position we need next to date and time (in UTC) the latitude and longitude. Since the time zone in Delft on 14
April is the CEST, the Central European Summer Time, the time difference with UTC is +2 hours, such that 11:00 CEST corresponds to
9:00 UTC. According to Google Maps, the latitude and longitude of the Markt in the centre of Delft are given by
φ0 = 52.01◦ N = +52.01◦ ,
λ0 = 4.36◦ E = + 4.36◦ .
For the calculation we first have to express date and time as the time elapsed since 1 January 2000 noon UTC.
9
D = 4 · 366 + 10 · 365 + 2 · 31 + 28 + 13 − 0.5 + 24 = 5216.875.
Now we can calculate the mean longitude q and the mean anomaly g of the sun according to Eqs. (5.3) and (5.4),
where the values were normalised to [0◦ , 360◦ ). From Eq. (5.5) we thus obtain for the latitude of the Sun in ecliptic coordinates
For the axial tilt e of the Earth we obtain from Eq. (5.8)
where we used T = D/36525 and normalised to [0 h, 24 h). We then find for the local mean sidereal time θ L
15◦
θ L = GMST + λ0 = 341.8197303◦ ,
hour
Now we have all variables required to calculate the solar position. From Eqs. (5.19) and (5.20) we thus find
which leads to the solar altitude aS =36.1◦ and the solar azimuth AS =127.2◦ .
5.3. The equation of time 45
5.2 The sun path at different loca- year, such that it seems to run along the shape of an
Eight. This closed curve is called the analemma.
tions
The difference between the apparent solar time (AST),
i.e. the timescale where the sun really is highest at noon
In this section we discuss the solar paths throughout every day, and mean solar time is described by the so-
the year at several locations around the earth. Figures called equation of time, which is defined as
5.5-5.8 shows four examples: Delft, the Netherlands
(φ0 = 52.01◦ N), the North Cape, Norway (φ0 = 71.17◦ EoT = AST − MST. (5.21)
N), Cali, Colombia (φ0 = 3.42◦ N), and Sydney, Aus-
tralia (φ0 = 33.86◦ S). Note that all the times are given The equation of time is given as the difference between
in the apparent solar time (AST). While in Delft and on the mean longitude q, as defined in Eq. (5.3), and the
the North Cape, the Sun at noon always is South of the right ascendent αS of the sun in the equatorial coordin-
zenith, in Sydney it is always North. In Cali, close to ate system,
the equator, the Sun is either South or North, depend-
ing on the time of the year. Since the North Cape is 1 hour
EoT( D ) = [q( D ) − αS ( D )] . (5.22)
north of the arctic circle, the Sun does not set around 21 15 deg
June. This phenomenon is called the midnight sun. On
The right ascendent is connected to the ecliptic longit-
the other hand, the sun always stays below the horizon
ude of the sun λS , as given in Eq. (5.5) via
around 21 December - this is called the polar night.
tan αs = cos e tan λS , (5.23)
Figure 5.5: The sun path in apparent solar time in Delft, the Netherlands (φ0 = 52.01◦ N). The sun path was calculated with the
Sun path chart program by the Solar Radiation Monitoring Lab. of the Univ. of Oregon [8].
5.3. The equation of time 47
Figure 5.6: The sun path in apparent solar time on the North Cape, Norway (φ0 = 71.17◦ N). The sun path was calculated with the
Sun path chart program by the Solar Radiation Monitoring Lab. of the Univ. of Oregon [8].
48 5. Location issues
Figure 5.7: The sun path in apparent solar time in Cali, Colombia (φ0 = 3.42◦ N). The sun path was calculated with the Sun path
chart program by the Solar Radiation Monitoring Lab. of the Univ. of Oregon [8].
5.3. The equation of time 49
Figure 5.8: The sun path in apparent solar time in Sydney, Australia (φ0 = 33.86◦ S). The sun path was calculated with the Sun
path chart program by the Solar Radiation Monitoring Lab. of the Univ. of Oregon [8].
50 5. Location issues
Equation of Time
15
12:00 0
60◦ 13 Jun
-5
50◦ 15 Apr 1 Sep effect of anomaly
08:30 15:30 -10
Solar Altitude
40◦
-15
30◦
11 Feb 3 Nov 01 Feb 01 Apr 01 Jun 01 Aug 01 Oct 01 Dec
20◦
25 Dec Figure 5.10: The effect of the anomaly of the terrestrial or-
10◦ bit and the axial tilt of the Earth rotation axis on the difference
between apparent and mean solar time. The equation of time
0◦
E SE S SW W nearly is the sum of these two effects.
Solar Azimuth
Figure 5.9: The analemma, i.e. the apparent curve of the sun
throughout the year when observed at the same mean solar
time every day. The analemma is shown for Delft (52◦ N latit- We see that the largest negative shift is on 11 February,
ude) at three points in time during the day. where, the apparent noon is about 14 min 12 s prior to
the mean solar noon. The largest positive shift is on 3
November, when the apparent solar noon is about 16
min 25 s past the mean solar noon. These points are
also marked in Fig. 5.9. There, also the zeros of the EoT
are shown, which are on 15 April, 13 June, 1 September,
and 25 December.
5.4. Irradiance on a PV module 51
Zenith then can describe the position of the module by the dir-
Ce ection of the module normal in horizontal coordinates
les
θM tia ( A M , a M ), where the altitude is given by a M = 90◦ − θ.
Let now the sun be at the position ( AS , aS ). Then the
ls
ph
direct irradiance on the module G M is given by the
ere
M equation
od
no
ul n M
r
e Horizo
m
M
ea
n Gdir dir
al
M = Ie cos γ, (5.24)
Altitud
ξM cos a M cos A M
After having discussed how to calculate the position n M = υ M = cos a M sin A M , (5.26)
of the Sun everywhere on the Earth and having looked ζM sin a M
at several examples, it now is time to discuss the im-
plications for the irradiance present on solar modules. ξS cos a S cos A S
For this discussion we assume that the solar module is nS = υ S = cos a S sin A S . (5.27)
mounted on a horizontal plane and that it is tilted un- ζS sin a S ,
der an angle θ M , as illustrated in Fig. 5.11. The angle
between the projection of the normal of the module where we used the relationship between Cartesian
onto the horizontal plane and due north is A M . We and spherical horizontal coordinates given in Eq. (5.1).
52 5. Location issues
Gdir dir As illustrated in Fig. 5.12 (a), the orientation of the roof
M = Ie [cos a M cos aS cos ( A M − AS ) + sin a M sin aS ]
in horizontal coordinates is characterised by the azi-
= Iedir [sin θ cos aS cos ( A M − AS ) + cos θ sin aS ] . muth A R and the altitude a R of its normal n R . The
(5.29) module is installed on the roof, and its orientation with
Note that this equation only holds when the sun is respect to the roof is best described in the roof coordin-
above the horizon ( aS > 0) and the azimuth of the sun ate system, where the fundamental plane is parallel to
is within ±90◦ of A M , AS ∈ [ AS − 90◦ , AS + 90◦ ]. Oth- the roof and the principal direction is along the gradi-
erwise, Gdir
M = 0. ent of the roof, as illustrated in Fig. 5.12 (b). In this sys-
If the azimuth of the solar position is the same as the tem, the module normal is given by the azimuth φ M
azimuth of the module normal A M = AS , Eq. (5.29) and the altitude is given by δM . The coordinate trans-
becomes form itself is transformed by combining two rotations:
First, we rotate with the angle 90◦ − a R around the axis
Gdir dir
M = Ie [cos a M cos aS + sin a M sin aS ]
(5.30) that is perpendicular to both n R and the gradient dir-
= Iedir cos ( a M − aS ) . ection of the roof. Secondly, we rotate with the angle
A R + 180◦ along the zenith. We thus obtain
When using the tilt angle θ = 90◦ − a M we find
Gdir dir
M = Ie sin ( θ + aS ) . (5.31)
5.4. Irradiance on a PV module 53
ξM cos a M cos A M − cos A R sin A R 0 sin a R 0 − cos a R cos δM cos φ M
υ M = cos a M sin A M = − sin A R − cos A R 0 0 1 0 cos δM sin φ M . (5.32)
ζM sin a M 0 0 1 cos a R 0 sin a R sin δM
The coordinates of the module in the horizontal coordinate system then are given by
ξ M = cos a M cos A M = − cos A R sin a R cos δM cos φ M + sin A R cos δM sin φ M + cos A R cos a R sin δM , (5.33a)
υ M = cos a M sin A M = − sin A R sin a R cos δM cos φ M − cos A R cos δM sin φ M + sin A R cos a R sin δM , (5.33b)
ζ M = sin a M = cos a R cos δM cos φ M + sin a R sin δM . (5.33c)
Dividing Eq. (5.33b) by Eq. (5.33a) and leaving Eq. (5.33c) unchanged leads to the final expressions for the module
orientation in horizontal coordinates,
− sin A R sin a R cos δM cos φM − cos A R cos δM sin φM + sin A R cos a R sin δM
tan A M = , (5.34a)
− cos A R sin a R cos δM cos φM + sin A R cos δM sin φM + cos A R cos a R sin δM
sin a M = cos a R cos δM cos φM + sin a R sin δM . (5.34b)
Finally, the cosine of the angle between the module orientation and the solar position is given by
ls
tilt angle θ R . Then, a R = 90◦ − θ R and A R = 90◦ .
ph
ere
On this roof a solar module is installed under a tilt-
Ro
of al n R
ing angle θ M with respect to the roof. The modules are
no
rm
Horizo
R
mounted parallel to the gradient of the roof. We thus
ea
n
have δM = 90◦ − θ M and φ M = 270◦ . From Eqs. (5.34)
Altitud
we thus obtain Ro t N
sin a M = cos a R cos δM · 0 + sin a R sin δM .
of
dien 0°
a
(5.36a) Gr θR
th A R
= sin a R sin δM Azimu
−0 − 0 + 1 · cos a R sin δM
tan A M =
0 + sin A R cos δM sin ·1 + 0 (5.36b)
(b) Roof normal nR
= cos a R tan δM .
θM
In the second example, the roof is facing southwards
and tilted under an angle θ R . We thus have a R =
M
90◦ − θ R and A R = 180◦. Now the module tilted un-
od
no
ul n M
r
e
der an angle θ M with respect to the roof and moun- Roof
m
M
eδ
al
ted perpendicular to the gradient of the roof. Hence,
Altitud
δM = 90◦ − θ M and φ M = 180◦ . Using Eqs. (5.34) we
M Roof gradient
find od
ule θM
sin a M = cos a R cos δM · (−1) + sin a R sin δM . th φ M
(5.37a) Azimu
= cos( a R + eta M ) = sin( a R − θ M ).
−0 − 0 + 0 Figure 5.12: Illustrating the angles used to describe (a) the
tan A M = = 0.
− sin a R cos δM + 0 + cos A R sin δM orientation of a roof on a horizontal plane and (b) the orienta-
(5.37b) tion of a module mounted on a roof.
5.5. Direct and diffuse irradiance 55
5.5 Direct and diffuse irradiance with the constant c = 0.14. The solar constant is given
as Ie0 = 1361 Wm-2 . In a first approximation, the diffuse
irradiance is about 10% of the direct irradiance. For the
As sunlight traverses the atmosphere, it is partially
global irradiance we hence obtain [11]
scattered, leading to an attenuation of the direct beam
component. On the other hand,the scattered light also global
Ie ≈ 1.1 · Iedir . (5.41)
partially will arrive at on the earths surface as diffuse
light. For PV applications it is important to be able to
estimate the strength of the direct and diffuse compon- A more accurate model was developed in the frame-
ents. work of the European Solar Radiation Atlas [12]. In that
model, the direct irradiance on a horizontal surface for
First, we discuss a simple model that allows to estimate clear sky is given by
the irradiance on a cloudless sky in dependence of the
air mass and hence the altitude of the sun. As we have Iedir = Ie0 ε sin aS exp [−0.8662 TL (AM2) m δR (m)] . (5.42)
seen in section 3.4, the air mass is defined as
I0 is the solar constant that takes a value of 1361 W/m2 .
1 1
AM = = , (5.38) The factor ε allows to correct for deviations of the sun-
cos θ sin aS earth distance from its mean value. aS is the solar alti-
where we used that the angle between the sun and the tude angle. TL (AM2) is the Linke turbidity factor with
zenith θ is connected to the solar altitude via θ = 90◦ − that the haziness of the atmosphere is taken into ac-
aS . This equation, however, does not take the curvature count. In this equation its value at an Air Mass 2 is
of the earth into account. If the curvature is taken into used. m is the relative optical air mass, and finally
account, we find [9] δR (m) is the integral Rayleigh optical thickness. The
different components can be evaluated as follows:
1
AM( aS ) = . (5.39)
sin aS + 0.50572(6.07995 + aS )−1.6364 The correction factor ε is given by
To estimate the direct irradiance at a certain solar alti- Ie ( R) R2
tude aS and altitude of the observer h, we can use the ε= = . (5.43)
Ie0 AU2
following empirical equation [10]
h 0.678
i The distance between earth and sun as a multiple of
Iedir = Ie0 (1 − ch) · 0.7(AM ) + ch , (5.40) astronomic units (AU) is given in Eq. (5.7). We thus ob-
56 5. Location issues
which leads to annual variations of about ±3.3%. They also present and expression for the diffuse irradi-
ance of the light, which is given by
The Linke Turbidity factor approximates absorption and
scattering in the atmosphere and takes both absorption Iedif = Ie0 εTrd [ TL (AM2)] Fd [ aS , TL (AM2)], (5.47)
by water vapour and scattering by aerosol particles into
account. It is a unit-less number and typically takes val- where Trd is the diffuse transmission function at zenith,
ues between 2 for very clear skies and 7 for heavily pol- which is a second-order polynomial of TL (AM2) Trd
luted skies. has typical values in between 0.05 for very clear skies
and 0.22 for a very turbid atmosphere. Fd is a diffuse
The relative optical air mass m expresses the ratio of angular function, given as a second-order polynomial of
the optical path length of the solar beam through the sin aS . For more details we refer to the paper [12].
atmosphere to the optical path through a standard at-
mosphere at sea level with the Sun at zenith. In can be
approximated as a function of the solar altitude aS by
5.6 Shadowing
exp(−z/zh )
m( aS ) = . (5.45)
sin aS + 0.50572( aS + 6.07995)−1.6364 An aspect that has to be kept in mind when planning a
PV system that consists of several rows of PV modules
Here, z is the site elevation and zh is the scale height of that are placed behind each other is shadowing. In this
the Rayleigh atmosphere near the Earth surface, given section we will determine how far behind the module
by 8434.5 m. the shadow reaches in dependence of the solar position,
Finally, the Rayleigh optical thickness δR (m) is given by the module orientation and the height of the module.
Figure 5.13 shows the important notions that we use in
1 this derivation.
=6.62960 + 1.75130 m − 0.12020 m2
δR ( m ) (5.46) For the determination we look at a module that is tilted
3 4 at an angle θ M . Its normal angle has an azimuth A M .
+ 0.00650 m − 0.00013 m .
5.6. Shadowing 57
For calculating the position of P0 we define the line s, where the direction vector r0 is given as
which goes through P and points to the sun,
s(t) = P + tnS . (5.52) cos A M
r0 = sin A M . (5.58)
When the position of the sun is described by its altitude
0
aS and azimuth AS , we find
− cos θ M cos A M cos aS cos AS
d is the distance between P0 and the intersection of g
s(t) = l − cos θ M sin A M + t cos aS sin AS (5.53)
with g0 . At this intersection we have
sin θ M sin aS ,
We find the shadow P0 of the point P as the intersection g ( u ) = g 0 ( v ),
of the line s with the horizontal plane, z = 0,
sin A M cos A M
l sin θ M + t sin aS = 0. (5.54) u − cos A M = P0 + v sin A M
Hence, 0 0
sin θ M u sin A M = Px0 + v cos A M
t = −l . (5.55)
sin aS
−u cos A M = Py0 + v sin A M
The coordinates of P0 = ( Px0 , Py0 , 0) then are given as
u sin A M cos A M = Px0 cos A M + v cos2 A M
Px0 = −l (cos θ M cos A M + sin θ M cot aS cos AS ) , (5.56a)
−u cos A M sin A M = Py0 sin A M + v sin2 A M
Py0 = −l (cos θ M sin A M + sin θ M cot aS sin AS ) . (5.56b)
As stated already earlier, the length of the shadow d is By adding the last two equations we find
given as the shortest distance between P0 and the line g
connecting E and F. Let g0 be the line perpendicular to
g, g ⊥ g0 , which goes through P0 . Since E = (0, 0, 0) we Px0 cos A M + v cos2 A M + Py0 sin A M + v sin2 A M = 0,
find for g and g0 (5.59)
and hence
g(u) = ur, (5.57a)
0 0 0
g (v) = P +vr , (5.57b) v = − Px0 cos A M − Py0 sin A M . (5.60)
5.6. Shadowing 59
61
62 6. Components of PV Systems
solar cell
PV panel PV array
Figure 6.1: Illustrating (a) a solar cell, (b) a PV module, (c) a solar panel, and (d) a PV array.
6.1. PV modules 63
Current I (A)
6
5
0.6 V
4
I sc series
3
2
(b)
1
Voc 2Voc 3Voc
0 0.3 0.6 0.9 1.2 1.5 1.8
Voltage V (V)
0.6 V
Figure 6.2: Illustrating (a) a series connection of three solar cells and (b) realisation of such a series connection for cells with a clas-
sical front metal grid. (c) Illustrating a parallel connection of three solar cells. (d) I-V curves of solar cells connected in series and
parallel.
64 6. Components of PV Systems
shown in Fig. 6.2 (a). In a series connection the voltages The reader may have noticed that we used I-V curves,
add up. For example, if the open circuit voltage of one i.e. the current-voltage characteristics, in the previous
cell is equal to 0.6 V, a string of three cells will deliver paragraphs. This is different to Parts I and II, where we
an open circuit voltage of 1.8 V. For solar cells with a used I-V curves instead, i.e. the current density - voltage
classical front metal grid, a series connection can be es- characteristics. The reason for this switch from J to I is
tablished by connecting the bus bars at the front side that on module level, the total current that the module
with the back contact of the neighbouring cell, as il- can generate is of higher interest than the current dens-
lustrated in Fig. 6.2 (b). For series connected cells, the ity. As the area of a module is a constant, the shapes of
current does not add up but is determined by the pho- the I-V and J-V curves of a module are similar.
tocurrent in each solar cell. Hence, the total current in a
string of solar cells is equal to the current generated by
one single solar cell.
For a total module, therefore the voltage and current
Figure Fig. 6.2 (d) shows the I-V curve of solar cells output can be partially tuned via the arrangements of
connected in series. If we connect two solar cells in the solar cell connections. Figure 6.3 (a) shows a typ-
series, the voltages add up while the current stays the ical PV module that contains 36 solar cells connected
same. The resulting open circuit voltage is two times in series. If a single junction solar cell would have a
that of the single cell. If we connect three solar cells in short circuit current of 5 A, and an open circuit voltage
series, the open circuit voltage becomes three times as of 0.6 V, the total module would have an output of
large, whereas the current still is that of one single solar Voc = 36 · 0.6 V = 21.6 V and Isc = 5 A. However,
cell. if two strings of 18 series-connected cells are connec-
ted in parallel, as illustrated in Fig. 6.3 (b), the output
Secondly, we can connect solar cells in parallel as il- of the module will be Voc = 18 · 0.6 V = 10.8 V and
lustrated in Fig. 6.2 (c), which shows three solar cells Isc = 2 × 5 A = 10 A. In general, for the I-V characterist-
connected in parallel. If cells are connected in paral- ics of a module consisting of m identical cells in series
lel, the voltage is the same over all solar cells, while and n identical cells in parallel the voltage multiplies
the currents of the solar cells add up. If we connect e.g. by a factor m while the current multiplies by a factor n.
three cells in parallel, the current becomes three times Modern PV modules often contain 60 (10 × 6), 72 (9 × 8)
as large, while the voltage is the same as for a single or 96 (12 × 8) solar cells that are usually all connected
cell, as illustrated in Fig. 6.2 (d). in series in order to minimise resistive losses.
6.1. PV modules 65
non-uniformly, the module performance reduces even current generated in the shaded cell is significantly
further. reduced. In a series connection the current is limited
by the cell that generates the lowest current, this cell
Often, differences between cell and module perform-
thus dictates the maximum current flowing through the
ance are mentioned in datasheets that are provide by
module.
the module manufacturers. For example, the datasheet
of a Sanyo HIT-N240SE10 module gives a cell level ef- In Fig. 6.4 (b) the theoretical I-V curve of the five un-
ficiency of 21.6%, but a module level efficiency of only shaded solar cells and the shaded solar cell is shown. If
19%. Despite all the technological advancements being the cells are connected to a constant load R, the voltage
made at solar cell level for improving the efficiency, still across the module is dropping due to the lower current
a lot must be done at the PV systems level to ensure a generated. However, since the five unshaded solar cells
healthy PV yield. For the performance of a PV system, are forced to produce high voltages, they act like a re-
not only the module performance is important, but also verse bias source on the shaded solar cell. The dashed
the yield of the PV system. line in Fig. 6.4 (b) represents the reverse bias load put
on the shaded cell, which is the I-V curve of the five
cells, reflected across the vertical axis equal to 0 V.
6.1.3 Partial shading and bypass diodes Hence, the shaded solar cell does not generate energy,
but starts to dissipate energy and heats up. The temper-
PV modules have so-called bypass diodes integrated. To ature can increase to such a critical level, that the en-
understand the reason for using such diodes, we have capsulation material cracks, or other materials wear out.
to consider modules in real-life conditions, where they Further, high temperatures generally lead to a decrease
can be partially shaded, as illustrated in Fig. 6.4 (a). The of the PV output as well.
shade can be from an object nearby, like a tree, a chim-
ney or a neighbouring building. It also can be caused These problems occurring from partial shading can be
by a leaf that has fallen from a tree. Partial shading prevented by including bypass diodes in the module,
can have significant consequences for the output of the as illustrated in 6.4 (c). As discussed in Chapter ??,
solar module. To understand this, we consider the situ- a diode blocks the current when it is under negative
ation in which one solar cell in the module shaded for voltage, but conducts a current when it is under posit-
a large part shaded. For simplicity, we assume that all ive voltage. If no cell is shaded, no current is flowing
six cells are connected in series. This means that the through the bypass diodes. However, if one cell is (par-
6.1. PV modules 67
(a) (b)
1
Current
current from 5Voc
R 0
shaded
Dissipated 6th cell
Current (A)
(c) -1 energy
J < Jsc
-2
Current
Jsc from 5 cells
-3
current
R
-5 -4 -3 -2 -1 0 1 2 3 4 5
Voltage (V)
Figure 6.4: Illustrating (a) string of six solar cells of which one is partially shaded, which (b) has dramatic effects on the I-V curve
of this string. (c) Bypass diodes can solve the problem of partial shading.
68 6. Components of PV Systems
• The back layer acts as a barrier against humidity and the temperature is increased to about 150°C. Now
and other stresses. Depending on the manufac- the curing process starts, i.e. a curing agent, which is
turer, it can be another glass plate or a composite present in the EVA layer, starts to cross-link the EVA
polymer sheet. A material combination that is often chains, which means that transverse bonds between the
used is PVF-polyester-PVF, where PVF stands for EVA molecules are formed. As a result, EVA has elasto-
polyvinyl fluoride, which is often known by its brand meric, rubberlike properties.
name Tedlar® . PVF has a low permeability for va-
pours and is very resistive against weathering. A The choice of the layers that light traverses before en-
typical polyester is polyethylene terephthalate (PET) tering the solar cell is also very important from an op-
tical point of view. If this layers have an increasing
• A frame usually made from aluminium is put refractive index, they act as antireflective coating and
around the whole module in order to enhance the thus can enhance the amount of light that is in-coupled
mechanical stability. in the solar cell and finally absorbed, which increases
the current produced by the solar cell.
• A junction box usually is placed at the back of the
module. In it the electrical connections to the solar
cell are connected with the wires that are used to
connect the module to the other components of the 6.1.5 Lifetime testing of PV Modules
PV system.
The typical lifetime of PV systems is about 25 years.
One of the most important steps during module pro- In these as little maintenance as possible should be re-
duction is laminating, which we briefly will explain for quired on the system components, especially the PV
the case that EVA is used as encapsulant [13]. For lam- modules are required to be maintenance free. Further-
ination, the whole stack consisting of front glass, the more, degradation in the different components of that
encapsulants, the interconnected solar cells, and the the module is made should be little: manufacturers typ-
back layer are brought together in a laminator, which ically guarantee a power between 80% and 90% of the
is heated above the melting point of EVA, which is initial power after 25 years. During the lifetime of 25
around 120°C. This process is performed in vacuo in years or more, PV modules are exposed to various ex-
order to ensure that air, moisture and other gasses are ternal stress from various sources [14]:
removed from within the module stack. After some
minutes, when the EVA is molten, pressure is applied • temperature changes between night and day as well
70 6. Components of PV Systems
as between winter and summer, • Humidity freeze testing in order to test delamination,
adhesion of the junction box, . . .
• mechanical stress for example from wind, snow and
hail, • UV testing, because UV light can lead to delamin-
ation, loss of adhesion and elasticity of the encap-
• stress by agents transported via the atmosphere such
sulant, ground fault due to backsheet degradation.
as dust, sand, salty mist and other agents,
Mainly, UV light can lead to a discoloration of the
• moisture originating from rain, dew and frost, encapsulant and back sheet, which means that they
get yellow. This can lead to losses in the amount of
• humidity originating from the atmosphere, light that reaches the solar cells.
• irradiance consisting of direct and indirect irradi- • Static mechanical loads in order to test whether
ance from the sun; mainly the highly-energetic UV strong winds or heavy snow loads lead to struc-
radiation is challenging for many materials. tural failures, broken glass, broken interconnect rib-
Before PV modules are brought to the market, they are bons or broken cells.
usually tested extensively in order to assure their stabil-
• Dynamic mechanical load, which can lead to roken
ity against these various stresses. The required tests are
glass, broken interconnect ribbons or broken cells.
extensively defined in the standards IEC 61215 for mod-
ules based on crystalline silicon solar cells and in IEC • Hot spot testing in order to see whether hot spots
61646 for thin-film modules. Since the modules cannot due to shunts in cells or inadequate bypass diode
be tested during a period of 25 years, accelerated stress protection are present.
testing must be performed. The required tests are [15]:
• Hail testing to see whether the module can handle
• Thermal cycles for studying whether thermal stress the mechanical stress induced by hail.
leads to broken interconnects, broken cells, elec-
trical bond failure, adhesion of the junction box,. . . • Bypass diode thermal testing to study whether over-
heating of these diodes causes degradation of the
• Damp heat testing to see whether the modules suf- encapsulant, backsheet or the junction box.
fer from corrosion, delamination, loss of adhesion
and elasticity of the encapsulant, adhesion of the • Salt spray testing to see whether salt that is present
junction box,. . . in salty mist or that is used in salty water for snow
6.1. PV modules 71
and ice removal leads to corrosion of PV module Such a stack of layers deposited onto a large glass
components. plate in principle forms one very large solar cell that
will produce a very high current. Since all the current
How these tests are to be performed is defined in other would have to be transported across the front and back
standards, for example IEC 61345 for UV testing and contacts, which are very thin, resistive losses in the
IV 61701 for salt-mist corrosion testing Usually these module is even a bigger problem than for c-Si mod-
tests are carried out by organisations like TÜV Rhein- ules. Therefore, the module is produced such that it
land. Refining the test requirements and understand- consists of many very narrow cells of about 1 cm width
ing which accelerated tests are required to guarantee a and the length being equal to the module length. These
lifetime of 25 years and more is subject to ongoing re- cells then are connected in series across the width of
search and development. the module. On the very left and right of the module
metallic busbars collect the current and conduct it to
the bottom of the module where they are connected
6.1.6 Thin-film modules with external cables.
does not contribute to the the current generated by the hind these processes is valid in general.
module. Therefore, the ratio between this width and
the total cell width (including the scribes) has to be
as small as possible. Another issue is the fact that the
three laser scribes are performed in different steps of 6.1.7 Some examples
production and thus often in different machines. Fur-
ther, the distance between the scribes might be different Table 6.1 shows some parameters of PV modules using
at the different processes when they are performed at different PV technologies:
different temperatures. This, aligning the glass plates
in all the production steps is extremely important for • A SunPower module made based on monocrystal-
manufacturing high-quality thin-film modules. line silicon solar cells,
Current I
equally valid for cells, modules, and arrays, although
MPPT usually is employed at PV module/array level.
As discussed earlier, the behaviour of an illuminated
solar cell can be characterised by an I-V curve. Inter-
connecting several solar cells in series or in parallel
merely increases the overall voltage and/or current, but
does not change the shape of the I-V curve. Therefore, Voltage V
for understanding the concept of MPPT, it is sufficient
to consider the I-V curve of a solar cell. The I-V curve Figure 6.7: Effect of increased temperature T or irradiance
is dependent on the module temperature on the irra- G M on the I-V curve.
diance, as we will discuss in detail in Section 7.1. For
example, an increasing irradiance leads to an increased
current and slightly increased voltage, as illustrated in
Fig. 6.7. The same figure shows that an increasing tem-
perature has a detrimental effect on the voltage.
Now we take a look at the concept of the operating
point, which is the defined as the particular voltage and
6.2. Maximum power point tracking 75
P = I · V. (6.1)
Pmpp
The operating point ( I, V ) corresponds to a point on Isc
the power-voltage (P-V) curve, shown in Fig. 6.8. For Impp
generating the highest power output at a given irradi- MPP
ance and temperature, the operating point should such
Current I
Power P
correspond to the maximum of the (P-V) curve, which
is called the maximum power point (MPP).
If a PV module (or array) is directly connected to an
electrical load, the operating point is dictated by that
load. For getting the maximal power out of the module,
it thus is imperative to force the module to operate at
the maximum power point. The simplest way of forcing Voltage V Vmpp Voc
the module to operate at the MPP, is either to force the
voltage of the PV module to be that at the MPP (called
Figure 6.8: A generic I-V curve and the associated P-V curve.
Vmpp ) or to regulate the current to be that of the MPP The maximum power point (MPP) is indicated.
(called Impp ).
However, the MPP is dependent on the ambient con-
ditions. If the irradiance or temperature change, the I-
V and the P-V characteristics will change as well and
hence the position of the MPP will shift. Therefore,
changes in the I-V curve have to be tracked continu-
ously such that the operating point can be adjusted to
76 6. Components of PV Systems
be at the MPP after changes of the ambient conditions. solar module is adjusted only on a seasonal basis. This
model is based on the assumption that for the same
This process is called Maximum Power Point Tracking or
level of irradiance higher MPP voltages are expected
MPPT. The devices that perform this process are called
during winter than during summer. It is obvious that
MPP trackers. We can distinguish between two categor-
this method is not very accurate. It works best at loc-
ies of MPP tracking:
ations with minimal irradiance fluctuations between
• Indirect MPP tracking, for example performed with different days.
the Fractional Open Circuit Voltage method.
• Direct MPP tracking, for example performed with Fractional open circuit voltage method
the Perturb and Observe method or the the Incre-
mental Conductance method. One of the most common indirect MPPT techniques is
All the MPPT algorithms that we discuss in this section the fractional open circuit voltage method. This method
are based on finding the and tuning the voltage until exploits the fact that – in a very good approximation –
VMPP is found. Other algorithms, which are not dis- the Vmpp is given by
cussed in this section, work with the power instead and
Vmpp = k · Voc , (6.2)
aim to find IMPP .
where k is a constant. For crystalline silicon, k usu-
ally takes values in between 0.7 and 0.8. In general,
6.2.1 Indirect MPPT k of course is dependent on the type of solar cells.
As changes in the open circuit voltage can be easily
First, we discuss indirect MPP Tracking, where simple tracked, changes in the Vmpp can be easily estimated
assumptions are made for estimating the MPP based on just by multiplying with k. This method thus can be im-
a few measurements. plemented easily. However, there are also certain draw-
backs.
Fixed voltage method First, using a constant factor k only allows to roughly
estimate the position of the MPP. Therefore, the oper-
For example, in the fixed voltage method (also called ating point usually will not be exactly on the MPP but
constant voltage method), the operating voltage of the in its proximity, with is called the MPP region. Secondly,
6.2. Maximum power point tracking 77
every time the system needs to respond to a change in Table 6.2: A summary of the possible options in the P&O al-
illumination conditions, the Voc must be measured. For gorithm.
this measurement, the PV module needs to be discon-
nected from the load for a short while, which will lead Prior Change in Next
to a reduced total output of the PV system. The more Perturbation Power Perturbation
often the Voc is determined, the larger the loss in output Positive Positive Positive
will be. This drawback can be overcome my slightly Positive Negative Negative
modifying the method. For this modification a pilot Negative Positive Negative
PV cell is required, which is highly matched with the Negative Negative Positive
rest of the cells in the module. The pilot cell receives
the same irradiance as the rest of the PV module, and
a measurement of the pilot PV cell’s Voc also gives an Perturb and observe (P&O) algorithm
accurate representation of that of the PV module, hence
it can be used for estimating Vmpp . Therefore, the op- The first algorithm that we discuss is the Perturb and
erating point of the module can be adjusted without Observe (P&O) algorithm, which also is known as "hill
needing to disconnect the PV module. climbing" algorithm. In this algorithm, a perturbation
is provided to the voltage at that the module is cur-
rently driven. This perturbation in voltage will lead to a
change in the power output. If an increasing voltage
leads to an increasing in power, the operating point
6.2.2 Direct MPPT is at a lower voltage than the MPP, and hence further
voltage perturbation towards higher voltages is re-
quired to reach the MPP. In contrast, if an increasing
voltage leads to a decreasing power, further perturb-
Now we discuss direct MPP tracking, which is more in- ation towards lower voltages is required in order to
volved than indirect MPPT, because current, voltage reach the MPP. Hence, the algorithm will converge to-
or power measurements are required. Further, the sys- wards the MPP over several perturbations. This prin-
tem must response more accurately and faster than in ciple is summarised in Table 6.2.
indirect MPPT. We shall look at a couple of the most
popular kind of algorithms. A problem with this algorithm is that the operat-
78 6. Components of PV Systems
Current I
Power P
consequently decremented. This process is iterated until
the incremental conductance is the same as the negative
instantaneous conductance, in which case Vref = Vmpp .
The incremental conductance algorithm can be more ef-
ficient than the P&O algorithm as it does not meander
around the MPP under steady state conditions. Fur-
ther, small sampling intervals make it less susceptible Voltage V
to changing illumination conditions. However, under
conditions that are strongly varying and under partial Figure 6.11: The P-V curve of partially shaded system that
shading, the incremental conductance method might exhibits several local maxima.
also become less efficient. The major drawback of this
algorithm is the complexity of its hardware implement-
ation. Not only currents and voltages must be meas- that will be discussed in section 6.3.2.
ured, but also the instantaneous and incremental con-
ductances must be calculated and compared. How such In modern PV systems, the MPPT is often implemented
a hardware design can look like however is beyond the within other system components like the inverters or
scope of this book. charge controllers. The list of techniques presented in
this section not exhaustive, we just discussed the most
common ones. The development of more advanced
MPPT techniques is going on rapidly and many sci-
6.2.3 Some remarks
entific papers as well as patents are published in this
area. Furthermore, manufacturers usually use propriet-
While a MPPT is used to find the MPP by changing the ary techniques.
voltage, it does not perform changes of the operating
voltage. This is usually done by a DC-DC converter Up to now we only looked at situations the total I-V
6.3. Photovoltaic Converters 81
curve is similar to that of a single cell. Let us now con- 6.3.1 System configurations
sider a system that is partially shaded, as illustrated in
Fig. 6.11. Then, the P-V curve will have different local Before digging into details about different converter to-
maxima. Depending on the used MPPT algorithm, it is pologies used for power conversion in PV systems, a
not sure at all that the algorithm finds the global max- general overview of different system architecture will
imum. Different companies use proprietary solutions be presented. The system architecture determines how
to tackle this issue. Alternatively, each string can be PV modules are interconnected and how the interface
connected to a separate MPPT. There are inverters that with the grid is established. Which of these system ar-
have connections for several strings (usually two). chitectures will be employed in a particular PV plant
depends on many factors such as environment of the
plant (whether the plant is situated in an urban envir-
onment or at an open area), scalability, costs etc. Fig-
ure 6.12 an overview of different system architectures is
given. The main advantages and disadvantages of the
different architectures are discussed below.
6.3 Photovoltaic Converters
In general solar inverters should have the following
characteristics [16]:
• Inverters should be highly efficient because the
A core technology associated with PV systems is the owner of the solar system requires the absolute
power electronic converter. An ideal PV converter maximum possible generated energy to be de-
should draw the maximum power from the PV panel livered to the grid/load.
and supply it to the load side. In case of grid connected
systems, this should be done with the minimum har- • Special demands regarding the potential between
monic content in the current and at a power factor close solar generator and earth (depending on the solar
to unity. For stand-alone systems the output voltage module type).
should also be regulated to the desired value. In this
• Special safety features like active islanding detec-
section a short review of different topologies often as-
tion capability.
sociated with PV systems is given. The semiconductor
switches in the following are assumed to be ideal. • Low limits for harmonics of the line currents. This
82 6. Components of PV Systems
requirement is enforced by law in most countries leading to several problems in the electricity grid.
since the harmonic limits of both sources and loads
Note that the term inverter is often used for two differ-
connected to the grid are regulated.
ent things: First, it is used for the actual inverter, which
• Special demands on electromagnetic interference is the electronic building block that performs the DC-to-
(EMI), which are regulated by law in most coun- AC conversion, as described in section 6.3.3. Secondly,
tries. The goal of these minimise the unwanted the term inverter also is used for the total unit pro-
influence of EMI on other equipment in the vicin- duced by manufacturers, that nowadays usually con-
ity or connected to the same supply. Think for ex- tains, an MPP tracker, a DC-DC converter, an DC-AC
ample of the influence of a mobile phone on an old converter and possibly also a charge controller of also a
radio. battery is connected.
all the PV modes connected in a single array in such a series with each string to prevent current circula-
centralised configuration offers the lowest specific cost tion inside strings.
(cost per kWp p of installed power). Since central invert-
ers only use a few components, they are very reliable
what makes them the preferred option in large scale PV Module Integrated or module oriented inverters
power plants.
In spite of their simplicity and low specific cost, central A very different architecture is that of the module in-
inverters suffer from the following disadvantages: tegrated inverters, as shown in 6.12 (b). These inverters
operate directly at one or several PV modules and have
1. Due to the layout of the system, a large amount of power ratings of several hundreds of watts. Because
power is carried over considerable distances using of the low voltage rating of the PV module, these in-
DC wiring. This can cause safety issues because verters require often require a two stage power conver-
fault DC currents are difficult to interrupt. Spe- sion. In a first stage boosts the DC voltage is boosted
cial precaution measures must to be taken such to the required value while it is inverted to AC in the
as ticker insulation on the DC cabling and special second stage. Often, a high frequency transformer is
circuit breakers, which can increase the costs. incorporated providing full galvanic isolation, which
enhances the system flexibility even further. As their
2. All strings operate at the same maximum power
name suggests, these inverters are usually integrated
point, which leads to mismatch losses in the mod-
with PV panel (so called ‘AC PV panels’). In this way
ules. This is significant disadvantage. Mismatch
the highest flexibility and the expandability of system
losses increase even more with ageing and with
is obtained. One of the most distinguishing features of
partial shading of sections of the array. Mismatch
this system is the “plug and play” characteristic, which
between the different strings may significantly re-
allows to build a complete (and readily expandable) PV
duce the overall system output.
system at a low investment cost. Another advantage of
3. Low flexibility and expandability of the system. these inverters, is minimisation of the mismatch losses
Due to the high ratings a system is normally de- that can occur because of non-optimal MPPT.
signed as a unit and hence difficult to extend. In
All these advantages come at certain expenses. Because
other words the system design is not very flexible.
these inverters are be mounted on a PV module, they
4. Power losses in the string diodes, which are put in must operate in harsh environment such as high tem-
6.3. Photovoltaic Converters 85
perature and large daily and seasonal temperature vari- Although partial shading of the string will influence
ations. Also, the specific are the highest of all the in- the overall efficiency of the system, each string can in-
verter topologies. Many topologies for module integ- dependently be operated at its MPP. Also, because no
rated inverters have been proposed, with some of them strings are connected in parallel, there is no need for
being already implemented in commercially available series diodes as in the case of PV arrays with many
inverters. parallel strings. This reduces losses associated with
these diodes. However, it still is a risk that within a
string hot-spot occurs because of unequal current and
String Inverters power sharing inside the string.
− −
Step-Down (Buck) Converter
(b)
vo Ts
Figure 6.14 (a) illustrates the simplest version of a
t on t off
buck DC-DC converter. The unfiltered output voltage
waveform of such a converter operated with pulse-
width-modulation (PWM) is shown in Fig. 6.14 (b). If
the switch is on, the input voltage Vd is applied to the
load. When the switch is off, the voltage across the load Vd
is zero. From the figure we see that the average DC Vo
output voltage is denoted as Vo . From the unfiltered
voltage, the average output voltage is given as 0
t
1 Ts 1 ton
Z
Vo = vo (t) dt =(ton Vd + toff · 0) = V. Figure 6.14: (a) A basic buck converter without any filters
Ts 0 Ts Ts d and (b) the unfiltered switched waveform generated by this
(6.9)
converter.
The different variables are defined in Fig. 6.14 (b). To
simplify the discussion we define a new term, the duty
88 6. Components of PV Systems
cycle D, as
ton
D := (6.10) (a) id
Ts
+
and hence
Vo = D · Vd . (6.11)
iL io
L
In general the output voltage with such a high har- Vd +
monic content is undesirable, and some filtering is re- + vL −
quired. Figure 6.15 (a) shows a more complex model of Vo
a step-down converter that has output filters included C Load
and supplies a purely resistive load. As filter elements − −
an inductor L and a capacitor C are used. The relation
between the input and the output voltages, as given in (b)
io
Equ. 6.11, is valid in continuous conduction mode, i.e.
when the current through the inductor never reaches +
zero value but flows continuously. We can change the
ratio between the voltages on the input and output side iL
by changing the duty cycle D. A detailed discussion L
about different modes of operation of a buck converter + vL − Vo
can be found in Reference [20]. C Load
Vd
In steady-state operation the time integral of the voltage
across the inductor v L taken during one switching cycle
is equal to zero. If this is not the case, the circuit is not − −
in steady state. Thus, in steady state, we obtain the fol-
lowing inductor volt-second balance: Figure 6.15: (a) A buck converter with filters and (b) a boost
converter.
Z Ts Z ton Z Ts
v L dt = v L dt + v L dt = 0. (6.12)
0 0 ton
6.3. Photovoltaic Converters 89
Vo = DVd , (6.13) +
Vd −
Step-Up (Boost) Converter
+
iL
In a boost converter, illustrated in Fig. 6.15 (b), an input vL Vo
DC voltage Vd is boosted to a higher DC voltage Vo . By C Load
− − L
applying the inductor volt-second balance across the +
inductor as explained in Eq. 6.12, we find io
Vd ton + (Vd − V0 )toff = 0. (6.14)
Figure 6.16: A buck-boost converter.
Using the definition for the duty cycle we find
V0 1
= . (6.15) schematic of a buck-boost converter is depicted in Fig.
Vd 1−D
6.16. Using inductor volt second balance as in Eq. 6.12,
The above relation is valid in the continuous conduc- we find
tion mode. The principle of operation is that energy
Vd ton + (−Vo )toff = 0 (6.16)
stored in the inductor (during the switch is on) is later
released against higher voltage Vo . In this way the and hence
energy is transferred from lower voltage (solar cell Vo D
voltage) to the higher voltage (load voltage). = (6.17)
Vd 1−D
Rmeas
−
V1 V2 R
Example
+
Assume a PV module has its MPP at VPV = 17 V and
Figure 6.17: A combination of a unit performing an MPPT IPV = 6 A at a given level of solar irradiance. The module
algorithm and a DC-DC converter (adapted from [21]). has to power a load with a resistance R L = 10 Ω. Calcu-
late the duty cycle of the DC-DC converter, if a buck-boost
converter is used.
MPP Tracking The maximum power from the module is PMPP = VMPP ·
IMPP = 102 W. If this power should be dissipated at the res-
istor, we have to use the relation
In section 6.2 we extensively discussed Maximum
PR = UR2 /R
Power Point Tracking. Or – more specific we discussed
different algorithms that are used for performing MPPT. and hence find for the voltage at the resistor
In these algorithms, usually the operating point of the p
module is set such that its power output becomes max- VR = PMPP R L = 31.94 V
imal. However, the MPPT algorithm itself cannot actu-
Using Eq. (6.17),
ally adjust the voltage or current of the operating point. Vo D
For this purpose a DC-DC converter is needed. Figure =
Vd 1−D
6.17 shows such a combination of the unit performing
with Vo = VR and Vd = VPV we find D = 0.65.
the MPPT and DC-DC converter. As illustrated in the
figure, this MPPT unit measures the voltage or the cur-
6.3. Photovoltaic Converters 91
-VDC
Mean voltage output
Figure 6.20: The unfiltered PWM signal and the sine signal
that is obtained with a low-pass filter. Figure 6.21: Illustrating an inverter that contains a trans-
former and MOSFETS as switches [21].
+ +
S2
L1
VDC L2 VDC VAC
L3 Load
S4
− −
Figure 6.22: Illustrating a three-phase inverter. Figure 6.23: Illustrating a half-bridge inverter.
Switches
compatible with the electricity grid. Nowadays, thyris- Potential induced degradation (PID)
tors only are used for inverters with a power of 100 kW
and higher. As mentioned earlier, in PV systems that have a
transformer-less inverter, no galvanic separation
All other inverters are self-commutated inverters that between the DC- and the AC-parts of the system is
generate an output with very little harmonic content given. Because of this lack of galvanic isolation, a po-
as described above and in Fig. 6.20. The switches there tential of -500 V or more between the PV modules and
are fully-controllable such that pulse-width modula- the ground can occur, which can lead to potential in-
tion becomes possible. As switches, GTOs (gate turn-off duced degradation (PID). Many thin-film contain a TCO
thyristors), IGBTs (insulated-gate bipolar transistors) or front contact that is deposited in superstrate configura-
MOSFETs (metal-oxide-semiconductor field-effect tran- tion on the glass top plate. Positively charged sodium
sistors) are used. More information can be found for ions than can travel into the TCO because of this po-
example in Reference [2]. tential. This leads to corrosion and consequently to per-
formance loss of the module. Also, for crystalline mod-
ules, PID can be a problem [21]. Therefore, for systems
containing thin-film modules the inverter must have a
Overall configuration transformer.
Figure 6.24: Illustrating an example of a transformer-less inverter unit as it is sold for residential PV systems [21]. As switches,
MOSFETs are used.
6.3. Photovoltaic Converters 97
must be prevented.
The inverter therefore must be able to detect, when the
electricity grid is shut-down. If this is the case, also the
inverter must stop delivering power to the grid.
For planning a PV system it is very important to know Figure 6.25: The power-dependent efficiency of several com-
the efficiency of the power converters. This efficiency of mercially available inverters. [21].
DC-DC and DC-AC converters is defined as
Po
η= , (6.20) For a complete inverter unit it is convenient to define
Po + Pd the efficiency as
i.e. the fraction of the output power Po to the sum of P
ηinv = AC , (6.23)
Po with the dissipated power Pd . For estimating the PDC
efficiency we thus must estimate the dissipated (lost) which is the ratio of the output AC power to the DC
power, which can be seen as a sum of several compon- input power. Figure 6.25 shows the efficiency of several
ents, commercially available inverters [21]. As we can see,
Pd = PL + Pswitch + other losses, (6.21) in general the lower the output power, the less efficient
i.e. the power lost in the inductor PL and the power lost the inverter. This efficiency characteristic must be taken
in the switch. PL is given as into account when planning a PV system.
2
PL = IL, rms R L , (6.22)
where IL, rms is the mean current flowing through the
inductor. The losses in the switch strongly depend on
the type of switch that is used. Other losses are for ex-
ample resistive losses in the circuitry in-between the
switches.
98 6. Components of PV Systems
Lead-acid batteries and LIB, the two main storage op- 1000
smaller
electrodes undergo chemical conversion during char-
ging and discharging, which makes their electrodes to
degenerate with time, leading to inevitable “ageing” of 600
the battery. In contrast, redox flow batteries combine
400 Li-ion
properties of both batteries and fuel cells, as illustrated NiMH Li-polymer
in Fig. 6.27. Two liquids, a positive electrolyte and a neg- 200
ative electrolyte are brought together only separated by lighter
Pb-acid
a membrane, which only is permeable for protons. The
cell thus can be charged and discharged without the re- 0 100 200 300 400
actants being mixed, which in principle prevents the li- Gravimetric energy density (Wh/kg)
quids from ageing. The chemical energy in a redox flow
battery is stored in its 2 electrolytes, which are stored Figure 6.28: A Ragone chart for comparing different second-
ary battery technologies with each other.
in two separate tanks. Since it is easy to make the tanks
larger, the maximal energy that can be stored in such
a battery thus is not restricted. Further, the maximal
output power can easily be increased by increasing the ured in Wh/kg. The volumetric energy density is the
area of the membrane, for example by using more cells amount of energy stored per volume of battery; it is
at the same time. The major disadvantages is that such given in Wh/l. The higher the gravimetric energy dens-
a battery system requires additional components such ity, the lighter the battery can be. The higher the volu-
as pumps, which makes them more complicated than metric energy density, the smaller the battery can be.
other types of batteries.
Figure 6.28 shows that lead acid batteries have both
Figure 6.28 shows a Ragone plot as it is used to com- the lowest volumetric and gravimetric energy densit-
pare different battery technologies. In contrast to Fig. ies among the different battery technologies. Lithium
6.26, here the the gravimetric energy density and volu- ion batteries show ideal material properties for using
metric energy density are plotted against each other. them as storage devices. Redox-flow batteries are very
The gravimetric energy density is the amount of energy promising. However, both LIB and redox-flow batter-
stored per mass of the battery; it typically is meas- ies are still in a development phase which makes these
6.4. Batteries 101
technologies still very expensive. Thus, because of to a source with a voltage higher than that of the battery
the unequalled maturity and hence low cost of the lead- enables the reverse reaction. In the PV systems, this
acid batteries, they are still the storage technology of source is nothing but the PV module or array provid-
choice for PV systems. ing clean solar power.
When the battery is discharged, electrons flow from the When talking about batteries, the term capacity refers
negative to the positive electrode through the external to the amount of charge that the battery can deliver
circuit, causing a chemical reaction between the plates at the rated voltage. The capacity is directly propor-
and the electrolyte. This forward reaction also depletes tional to the amount of electrode material in the bat-
the electrolyte, affecting its state of charge (SoC). When tery. This explains why a small cell has a lower capacity
the battery is recharged, the flow of electrons is re- than a large cell that is based on the same chemistry,
versed, as the external circuit does not have a load, but even though the open circuit voltage across the cell will
102 6. Components of PV Systems
Post straps
Vent plugs
Plate lugs
Positive plate
Container Envelope
separators
Negative plate
be the same for both the cells. Thus, the voltage of the Ah battery corresponds to a discharge current of 10 A
cell is more chemistry based, while the capacity is more over 1 hour. A C-rate of 2 for the same battery would
based on the quantity of the active materials used. correspond to a discharge current of 20 A over half an
hour. Similarly, a C-rate of 0.5 implies a discharge cur-
The capacity Cbat is measured in ampere-hours (Ah).
rent of 5 A over 2 hours. In general, it can be said that
Note that charge usually is measured in coulomb (C).
a C-rate of n corresponds to the battery getting fully
As the electric current is defined as the rate of flow of
discharged in 1/n hours, irrespective of the battery ca-
electric charge, Ah is another unit of charge. Since 1 C
pacity.
= 1 As, 1 Ah = 3600 C. For batteries, Ah is the more
convenient unit, because in the field of electricity the
amount of energy usually is measured in watt-hours Battery efficiency
(Wh). The energy capacity of a battery is simply given
by multiplying the rated battery voltage measured in For designing PV systems it is very importent to know
Volt by the battery capacity measured in Amp-hours, the efficiency of the storage system. For storage systems,
Ebat = Cbat V, (6.24) usually the round-trip efficiency is used, which is given
as the ratio of total storage system input to the total
which results in the battery energy capacity in Watt- storage system output,
hours.
Eout
ηbat = 100%. (6.25)
Ein
C-rate For example, if 10 kWh is pumped into the storage sys-
tem during charging, but only 8 kWh can be retrieved
A brand new battery with 10 Ah capacity theoretically during discharging, the round trip efficiency of the stor-
can deliver 1 A current for 10 hours at room temperat- age system is 80%. The round-trip efficiency of batteries
ure. Of course, in practice this is seldom the case due to can be broken down into two efficiencies: first, the vol-
several factors. Therefore, the C-rate is used, which is a taic efficiency, which is the ratio of the average dischar-
measure of the rate of discharge of the battery relative ging voltage to the average charging voltage,
to its capacity. It is defined as the multiple of the cur-
rent over the discharge current that the battery can sus- Vdischarge
ηV = 100%. (6.26)
tain over one hour. For example, a C-rate of 1 for a 10 Vcharge
104 6. Components of PV Systems
This efficiency covers the fact that the charging voltage Thus, a 10 Ah rated battery that has been drained by
is always a little above the rated voltage in order to 2 Ah is said to have a SoC of 80%. Also the Depth of
drive the reverse chemical (charging) reaction in the Discharge (DoD) is an important parameter. It is defined
battery. as the percentage of the battery capacity that has been
discharged,
Secondly, we have the coulombic efficiency (or Faraday
C V − Ebat
efficiency), which is defined as the ratio of the total DoD = bat . (6.30)
charge got out of the battery to the total charge put into Cbat V
the battery over a full charge cycle, For example, a 10 Ah battery that has been drained
Qdischarge by 2 Ah has a DoD of 20%. The SoC and the DoD are
ηC = 100%. (6.27) complimentary to each other.
Qcharge
The battery efficiency then is defined as the product of
these two efficiencies, Cycle lifetime
Vdischarge Qdischarge
ηbat = ηV · η C = 100%. (6.28)
Vcharge Qcharge The cycle lifetime is a very important parameter. It is
defined as the number of charging and discharging
When comparing different storage devices, usually
cycles after that the battery capacity drops below 80%
this round-trip efficiency is considered. It includes all
of the nominal value. Usually, the cycle lifetime is spe-
the effects of the different chemical and electrical non-
cified by the battery manufacturer as an absolute num-
idealities occurring in the battery.
ber. However, stating the battery lifetime as a single
number is a oversimplification because the different
State of charge and depth of discharge battery parameters discussed so far are not only related
to each other but are also dependent on the temperat-
At another important battery parameter is the State of ure.
Charge (SoC), which is defined as the percentage of the Figure 6.30 (a) shows the cycle lifetime as a function of
battery capacity available for discharge, the DoD for different temperatures. Clearly, colder op-
Ebat erating temperatures mean longer cycle lifetimes. Fur-
SoC = . (6.29) thermore, the cycle lifetime depends strongly on the
Cbat V
6.4. Batteries 105
(a) (b) use too. As seen from Fig. 6.30 (b), the lower the tem-
perature, the lower the battery capacity. This is because,
the chemicals in the battery are more active at higher
temperatures, and the increased chemical activity leads
to increased battery capacity. It is even possible to reach
an above rated battery capacity at high temperatures.
However, such high temperatures are severely detri-
mental to the battery health.
Ageing
DoD. The smaller the DoD, the higher the cycle life-
time. Thus, that the battery will last longer if the av-
erage DoD can be reduced during the lifetime of the The major cause for ageing of the battery is sulphation.
battery. Also, battery overheating should be strictly con- If the battery is insufficiently recharged after being dis-
trolled. Overheating can because of overcharging and charged, sulphate crystals start to grow, which cannot
subsequent over-voltage of the lead acid battery. To pre- be completely transformed back into lead or lead oxide.
vent this, charge controllers are used that we address in Thus the battery slowly loses its active material mass
the next section. and hence its discharge capacity. Corrosion of the lead
grid at the electrode is another common ageing mech-
anism. Corrosion leads to increased grid resistance due
Temperature effects to high positive potentials. Further, the electrolyte can
dry out. At high charging voltages, gassing can occur,
While the battery lifetime is increased at lower tem- which results in the loss of water. Thus, demineralised
peratures, one more effect must to be considered. The water should be used to refill the battery from time to
temperature affects the battery capacity during regular time.
106 6. Components of PV Systems
during severe winter days at low irradiance, the load If no blocking diodes are used, it is even possible that
exceeds the power generated by the PV array, such that the battery can “load” the PV array, when the PV ar-
the battery is heavily discharged. Over-discharging the ray is operating at a very low voltage. This means that
battery has a detrimental effect on the cycle lifetime, as the battery will impose a forward bias on the PV mod-
discussed above. The charge controller prevents the bat- ules and make them consume the battery power, which
tery from being over-discharged by disconnecting the leads to heating up the solar cells. Traditionally, block-
battery from the load. ing diodes are used at the PV panel or string level to
prevent this back discharge of the battery through the PV
For optimal performance, the battery voltage has to be array. However, this function is also easily integrated in
within specified limits. The charge controller can help the charge controller.
in maintaining an allowed voltage range in order to
ensure a healthy operation. Further, the PV array will We distinguish between series and shunt controllers, as
have its Vmpp at different levels, based on the temper- illustrated in 6.32. In a series controller, overcharging is
ature and irradiance conditions. Hence, the charge con- prevented by disconnecting the PV array until a partic-
troller needs to perform appropriate voltage regulation ular voltage drop is detected, at which point the array
to ensure the battery operates in the specified voltage is connected to the battery again. On the other hand, in
range, while the PV array is operating at the MPP. a parallel or shunt controller, overcharging is prevented
Modern charge controllers often have an MPP tracker by short-circuiting the PV array. This means that the PV
integrated. modules work under short circuit mode, and that no
current flows into the battery. These topologies also en-
As we have seen above, certain C-rates are used as bat- sure over-discharge protection using power switches for
tery specifications. The higher the charge/discharge the load connection, which are appropriately controlled
rates, the lower the coulombic efficiency of the battery. by the algorithms implemented into the charge con-
The optimal charge rates, as specified by the manufac- troller algorithm. As charge controllers are a necessary
turer, can be reached by manipulating the current flow- component of of stand-alone PV systems, they should
ing into the battery. A charge controller that contains have a very high efficiency.
a proper current regulation is also able to control the
C-rates. Finally, the charge controller can impose the As we have seen above, temperature plays a crucial
limits on the maximal currents flowing into and from role in the functioning of the battery. Not only does
the battery. temperature affect the lifespan of the battery, but it also
108 6. Components of PV Systems
Ibat Ibat
Charge Charge
controller Vbat Load controller Vbat Load
PV core PV core
Array Array
Figure 6.32: Basic wiring scheme of (a) a series and (b) a shunt charge controller.
6.5. Cables 109
changes its electrical parameters significantly. Thus, • black is used for interconnecting the −-contacts.
modern charge controllers have a temperature sensor
For AC wiring, different colour conventions are used
included, which allows the charge controller to adjust
around the world.
the electrical parameters of the battery, like the operat-
ing voltage, to the temperature. The charge controller • For example, in the European Union,
thus keeps the operating range of the battery within blue is used for neutral,
the optimal range of voltages. The charge controller is green-yellow is used for the protective earth and
usually kept in close proximity to the battery, such that brown (or another color) is used for the phase.
the operating temperature of the battery is close to that
• In the United States and Canada,
of the charge controller. However, when the battery is
silver is used for neutral,
heavily loaded, the battery might heat up, leading to
green-yellow, green or a bare conductor is used
differences between the temperature expected by the
for the protective earth
charge controller and the actual temperature of the bat-
and black (or another color) is used for the phase.
tery. Therefore, high end charge controllers also take
temperature effects due to high currents into account. • In India and Pakistan,
black is used for neutral,
green is used for the protective earth and
6.5 Cables blue, red, or yellow is used for the phase.
Therefore it is very important to check the standards
The overall performance of PV systems also is strongly of the country in that the PV system is going to be in-
dependent on the correct choice of the cables. We there- stalled.
fore will discuss how to choose suitable cables. But we The cables have to be chosen such that resistive losses are
start our discussion with color conventions. minimal. For estimating these losses, we look at a very
PV systems usually contain DC and AC parts. For cor- simple system that is illustrated in Fig. ?? The system
rectly installing a PV system, it is important to know consists of a power source and a load with resistance
the color conventions. For DC cables, R L . Also the cables have a resistance Rcable , which also
is sketched. The power loss at the cables is given as
• red is used for connecting the +-contacts of the
different system components with each other while Pcable = I · ∆Vcable , (6.31)
110 6. Components of PV Systems
` 1 `
Rcable = ρ = , (6.35)
A σA
where ∆Vcable is the voltage drop across the cable, For electrical cables it is convenient to have ` in metres
which is given as and A in mm2 . When using this convention, we find
Rcable the following units for ρ and σ:
∆Vcable = V . (6.32)
R L + Rcable
Using m2 106 mm2 mm2
V = I ( R L + Rcable ) (6.33) 1Ω·m = 1Ω = 1Ω = 106 Ω ,
m m m
we find S m m m
Pcable = I 2 Rcable . (6.34) 1 = 1S 2 = 1Ω 6 2
= 10−6 Ω .
m m 10 mm mm2
Hence, as the current doubles, four times as much heat
will be dissipated at the cables. It now is obvious why
modern modules have connected all cells in series.
Let us now calculate the resistance of a cable with The most widely used metals used for electrical cables
length ` and cross section A. It is clear, that if ` is are copper and aluminium. Their resistances and con-
6.5. Cables 111
ductivities are
mm2
ρCu = 1.68 · 10−8 Ω · m = 1.68 · 10−2 Ω ,
m
S m
σCu = 5.96 · 107 = 59.6 S ,
m mm2
mm2
ρAl = 2.82 · 10−8 Ω · m = 2.82 · 10−2 Ω ,
m
S m
σAl = 3.55 · 107 = 35.5 S .
m mm2
Standard Test Conditions (STC) of Photovoltaic (PV) In this section, an accurate thermal model for estimat-
modules are generally not representative of the real ing the PV module working temperature as a function
working conditions of a solar module. High level of of meteorological parameters will be developed. The
incident irradiation, for example, may cause the tem- model is based on a detailed energy balance between
perature of a module to rise many degrees above the the module itself and the surrounding environment.
STC temperature of 25°C therefore lowering the module Both the installed configuration of the array together
performances. In a climate such as the one of the Neth- with external parameters such as direct incident solar
erlands, real operating conditions for PV systems gen- irradiance on the panels, wind speed and cloud cover
erally correspond to relatively low levels of irradiance will be taken into account. From the calculated equilib-
combined with a cold and windy weather. In order to rium temperature, the overall efficiency of the PV array
effectively estimate real time energy production from will be calculated by separately assessing the temperat-
113
114 7. PV System Design
Table 7.1: Derivation of the INOCT from the NOCT for vari- complex and has been approached by developing a de-
ous mounting configurations [25]. tailed thermal energy balance between the module and
the surroundings. Here, the module is assumed to be a
Rack Mount INOCT = NOCT − 3◦ C single uniform mass at temperature TM . The three types
Direct Mount INOCT = NOCT + 18◦ C of heat transfer between the module itself with the sur-
Standoff INOCT = NOCT + X roundings are conduction, convection and radiation. The
where X is given by contributions considered in the model are:
W (inch) X (°C) • Heat received from the Sun in the form of insola-
1 11 tion ϕG M , where ϕ is the absorptivity of the mod-
3 2 ule.
6 − 1 • Convective heat exchange with surrounding air
from the front and rear side of the module
mental conditions such as wind speed and mounting hc ( TM − Ta ),
configuration of the array into account [24].
where hc denotes the overall convective heat trans-
In order to take into mounting configuration of the
fer coefficient of the module.
module into account, the Installed Nominal Operating
Conditions Temperature (INOCT) has been defined [25]. • Radiative heat exchange between the upper mod-
This value is described as the cell temperature of an in- ule surface and the sky
stalled array at NOCT conditions. Its value can there-
fore be obtained from the NOCT and the mounting 4
etop σ TM 4
− Tsky ,
configuration. The evaluation of how the INOCT var-
ies with the module mounting configuration has been where etop = 0.84 is the emissivity of the module
experimentally determined by measuring the NOCT at front glass and σ is the Stefan-Boltzmann constant
various mounting heights. The results can be found in as defined in Eq. (3.19). and between the rear sur-
literature and are here summarised in the Table 7.1. face and the ground
Evaluating the influence of external meteorological
4 4
parameters on the the module temperature is more eback σ TM − Tgr. ,
116 7. PV System Design
qrad, sky Tsky Before providing a solution for this differential equa-
tion, it is appropriate to remark the fact that we are
qconvection considering the entire module as a uniform piece at a
temperature TM . However, this is not entirely realistic
qsun since modules are made of various layers of different
Ta qrad, ground
materials surrounding the actual solar cells. It is the
a+θ purpose of this section to evaluate the temperature of
qconvection
the inner cell, which is the place where absorption of
TM solar radiation effectively place. This temperature will
a Tgr. θ be to some extent higher than the surface module tem-
perature TM due to the heat produced in the cell due
to light absorption. The approximation of considering
Figure 7.2: Representation of heat exchange between a tilted
the temperature uniform throughout the module layers
module surface and the surroundings.
however is justified because of the relatively low thick-
ness of the active cell together with the low heat capa-
where the emissivitiy of the back is assumed to be city of the cell material compared to the other layers
eback = 0.89. [26]. This results into a very low thermal resistance of
the cell to heat flow and therefore justifies the uniform
• Conductive heat transfer between the module and temperature approximation.
the mounting structure. We neglect this contribu-
tion due to the small area of the contact points [26]. In addition to this, a steady state approach will here be
considered, meaning that the module temperature will
By separately considering each of the above mentioned not change over time for each of the10 minutes time
contributions, we write down the balance for the heat steps. In reality, the temperature follows an exponential
transfer, [25] decay lagging behind variations in irradiation level. It
dTM is defined as Time Constant of a module, which is ‘the
4 4
mc = ϕGM − hc ( TM − Ta ) − eback σ TM − Tgr. time it takes for the module to reach 63% of the total
dt change in temperature resulting from a step change
4 4
− etop σ TM − Tsky . in irradiance’ [26]. Time constants for PV modules are
(7.2) generally of the order of approximately 7 minutes. For
7.1. Effects of meteorological effects on the module efficiency 117
time steps greater that the Time Constant, as it is in our However, since hr, gr. and hr, sky are also function of the
case, the module can be approximated as being in a module temperature, the equation needs to be solved
steady state condition. In light of this assumption, the iteratively: an initial module temperature is assigned
term on the left hand side of Eq. (7.2) vanishes. and by hr, gr. and hr, sky are updated each iterations. A
nearly exact solution can be obtained after 5 iterations.
It is now possible to proceed with the solution of the
thermal energy balance equation. The formula can be Before solving iteratively Eq. (7.6) there are still many
linearised by noticing that unknown variables that need to be determined, which
we will do in the following sections.
a4 − b4 = a2 + b2 ( a + b)( a − b). (7.3)
Since
7.1.2 Calculating the convective transfer
2 2
TM + Tsky ( TM + Tsky )
coefficients
changes less than 5% for a 10°C variation in TM , we can
consider this term to be constant when T M varies [25].
Convection is a form of energy transfer from one place
Therefore the energy balance can be simplified becom-
to another caused by the movement of a fluid. Convect-
ing linear with respect to T M . By defining
ive heat transfer can be either free or forced depending
hr, sky = etop σ TM 2 2
+ Tsky ( TM + Tsky ), (7.4a) on the cause of the fluid motion.
2 2
In free convection, heat transfer is caused by temper-
hr, gr. = eback σ TM + Tgr. ( TM + Tgr. ), (7.4b)
ature differences which affect the density of the fluid
we can rewrite Eq. (7.2) and find itself. Air starts circulating due to difference in buoy-
ancy between hot (less dense) fluid and cold (denser)
ϕG M −hc ( TM − Ta ) − hr, sky ( TM − Tsky ) fluid. A circular motion is therefore initiated with rising
(7.5)
−hr, gr. ( TM − Tgr. ) = 0. hot fluid and sinking cold fluid. Free convection only
takes place in a gravitational field [27].
By rearranging the terms, the formula can be explicitly
Forced convection, on the other hand, is caused by a
expressed as a function of T M ,
fluid flow due to external forces which therefore en-
ϕG + hc Ta + hr, sky Tsky + hr, gr. Tgr. hance the convective heat exchange. The heat transfer
TM = (7.6)
hc + hr, sky + hr, gr. depends very much on whether the induced flow over
118 7. PV System Design
a solid surface is laminar or turbulent. In the case of bulent flow. We obtain for the laminar and turbulent
turbulent flow, an increased heat transfer is expected convective heat transfer coefficients
with respect to the laminar situation. This fact is due
to an increased heat transport across the main direction
of the flow. On the contrary, in laminar flow regime, 0.86 Re−0.5
only conduction is responsible for transport in the cross hlam.
forced = ρcair w, (7.8a)
Pr0.67
direction. For this reason, forced convection is always
0.028 Re−0.2
studied separately in the laminar and turbulent regime hturb.
forced = ρcair w. (7.8b)
[27]. The overall convection transfer is made of the two Pr0.4
relative contribution for free and forced components. Re is the Reynolds number that expresses the ratio of of
Mixed convective coefficient can be obtained by taking the inertial forces to viscous forces,
the cube root of the cubes of the forced and convective
coefficients according to the equation [23] wDh
Re = , (7.9)
ν
h3mixed = h3forced + h3free . (7.7)
where w is the wind speed at the height of the PV array,
Dh is the hydraulic diameter of the module, which is used
Since the convective heat transfer coefficients will be as relevant length scale, and ν is the kinematic viscosity
different on the top and rear surface of the module, de- of air. Pr is the Prandtl number which is the ratio of the
termining the total heat transfer coefficient has to be momentum diffusivity to the thermal diffusivity. It is
decoupled between the top hcT and rear hcB surfaces. The considered to be 0.71 for air. Finally, ρ and cair are the
overall heat transfer will eventually be determined by density and heat capacity of air, respectively. The hy-
the sum of the two components. draulic diameter of a rectangle of length L and width
W, and thus of the PV module, is given as
12 hlam. 0.5
forced ' w , (7.11a)
hforced (Wm-2 K-1 )
gβ( T − Ta ) Dh3
Gr = . (7.13)
ν2
overall increase in the heat transfer with increasing
wind speeds due to the increased force transfer com- Here, g is the acceleration due to gravity on Earth and β
ponent. There are two different regions in the graph is the volumetric thermal expansion coefficient of air,
which represent the laminar and turbulent regimes, which can be approximated to be β = 1/T.
respectively. The laminar flow extends till around 3 With the value of both free and forced coefficient we
m/s and is characterised by a lower convective heat can calculate the total mixed heat convective mass
exchange compared to the turbulent regime. transfer coefficient using Eq. (7.7).
q
hmixed = hcT = 3
h3forced + h3free . (7.14)
In a good approximation, hforced and w are proportional
120 7. PV System Design
Convective heat transfer coefficient on the rear sur- The back side convection is therefore given by
face of the module.
hcB = R · hcT . (7.18)
Convection on the back side of the module will be
lower than on the top because of the mounting struc- We therefore find the overal convective heat transfer
ture and the relative vicinity to the ground. For ex- coefficient to be
ample, a rack mount configuration, which is approx- hc = hcT + hcB . (7.19)
imately installed at 1 m height, will achieve a larger
heat exchange rate than a standoff mounted array that
is mounted 20 cm above the ground. We model the ef-
7.1.3 Other parameters
fect of the different mounting configurations by scaling
the convection coefficient obtained for the top of the Sky temperature evaluation
module. We determine the scaling factor by performing
an energy balance at the INOCT conditions [25], The sky temperature can be expressed as a function of
the measured ambient temperature, humidity, cloud
ϕG M − hcT ( TINOCT − Ta ) − hr, sky ( TINOCT − Tsky ) = cover and cloud elevation [25]. On a cloudy day, usu-
hcB ( TINOCT − Ta ) + hr, gr. ( TINOCT − Tgr. ) ally when the cloud cover is above 6 okta1 , the sky
(7.15) temperature will approach the ambient temperature,
We define R as the ratio of the actual to the ideal heat Tsky = Ta [26]. However, on a clear day the sky temper-
loss from the back side, ature can drop below Ta and can be estimated by
Substituting this into Eq. (7.15) at INOCT conditions Wind speed at module height evaluation
yields
Since the anemometer used for the evaluation of the
ϕG M − hcT ( TINOCT − Ta ) − etop σ ( TINOCT
4 4 )
− Tsky wind speed is at a higher height than the module array,
R= 4 4 )
.
hcT ( TINOCT − Ta ) + etop σ ( TINOCT − Tsky 1 Okta is a measure for the cloud cover, where 0 is clear sky and 8 is a com-
(7.17) pletely cloudy sky.
7.1. Effects of meteorological effects on the module efficiency 121
eration.
Current I
∂T
∂Isc
Isc ( TM , GSTC ) = Isc + (STC)( TM − TSTC ), (7.24)
∂T
∂Pmpp
Pmpp ( TM , GSTC ) = Pmpp + (STC)( TM − TSTC ),
∂T
(7.25)
Pmpp ( TM , GSTC )
η ( TM , GSTC ) = . (7.26)
GSTC Voltage V
If the efficiency temperature coefficient ∂η/∂T is not
given in the datasheet, it can be obtained by rearran- Figure 7.6: Effect of a temperature increase on the I-V solar
ging cell characteristic.
∂η
η ( TM , GSTC ) = η (STC) + (STC)( TM − 25◦ C). (7.27)
∂T
completely outweighed by the decrease in open circuit module with respect to STC. The evaluation of the ex-
voltage. The overall effect is of a general linear decrease tent of this reduction is however less straightforward
in the maximum achievable power and therefore a de- than for the case of the temperature since solar man-
crease in the system efficiency and fill factor. This effect ufacturer often do not explicitly provide a reduction
is due to an increase of the intrinsic carrier concetra- factor of the efficiency at every light intensity level.
tion at higher temperatures which in turns leads to an
By definition the efficiency is given by
increase of the reverse saturation current I0 , which rep-
resents a measure of the leakage of carriers across the Isc Voc FF
solar cell junctions, as we have seen in Chapter ?? . The η= (7.29)
GM
exponential dependence of I0 from the temperature is
the main cause of the linear reduction of Voc with the The maximum variation of the FF for light intensity
temperature, between 1 and 1000 W/m2 is about 2% for CdTe, 5%
kT Isc for a-Si:H, 22% for poly-cristalline silicon, and 23% for
Voc = ln . (7.28)
e I0 mono-crystalline silicon [29].
The short circuit current of a solar cell is directly pro-
On the other hand, the slight increase in the generated
portional to the incoming radiation,
current is due to a moderate increase in the photo-
generated current resulting from an increased number Isc ' λG M , (7.30)
of thermally generated carriers. The overall reduction of
power at high temperature shows that cold and sunny where λ is simply a constant of proportionality. By ex-
climates are the best environment where to place a solar pressing Voc as in Eq. (7.28), the efficiency can be writ-
system. ten as
kT
η ' FF λ (ln G M + ln λ − ln I0 ). (7.31)
e
By defining
7.1.6 Effect of light intensity on the solar
cell performance kT
a = FF λ , (7.32a)
e
Intuitively, performances of a solar cell decrease con- kT
b = FF λ (ln λ − ln I0 ), (7.32b)
siderably with decreasing light intensity incident on the e
7.1. Effects of meteorological effects on the module efficiency 125
the efficiency can be finally written as level of irradiance and temperature can be determined
as [30]
η (25◦ C, G M ) = a ln G M + b (7.33)
η ( TM , G M ) = η (25°C, G M ) [1 + κ ( TM − 25°C)] , (7.38)
The values of the coefficients a and b are device specific where
parameters and are rarely given by the manufacturer. ∂η 1
κ= . (7.39)
The overall trend of the efficiency is represented by a ∂T η (SCT)
straight line on a logarithmic scale [29]. Typical values for κ are −0.0025/◦ C for CdTe,
From this model the values of Isc , Voc and the efficiency −0.0030/◦ C for CIS, and −0.0035/◦ C for c-Si [30].
at a irradiance level G M can be determined from the All the other parameters such as Isc , Voc , and Pmpp can
STC as follows also be evaluated at every level of irradiance and mod-
ule temperature by simply adapting Eq. (7.38) to the
corresponding coefficients and parameters.
ln G M
Voc (25°C, G M ) = Voc (STC) , (7.34)
ln GSTC In Fig. 7.7 the difference between η (25°C, G M ) and
G η ( TM , G M ), which takes both irradiance G M and mod-
Isc (25°C, G M ) = Isc (STC) M , (7.35) ule temperature TM into account, is shown. The graph
GSTC
has been derived by keeping the wind speed constant
PMPP (25°C, G M ) = FF Voc (25°C, G M ) Isc (25°C, G M ),
at a value of 1 m/s and by determining TM iterating
(7.36)
the heat thermal model at each level of irradiance for
PMPP (25°C, G M ) a constant ambient temperature of 25°C. At low levels
η (25°C, G M ) = , (7.37)
A M GM of irradiance no effects of the module temperature are
observed. At higher level of incident light intensity the
where A M is the module area.
difference between the two curves becomes more pro-
nounced. since only ηmpp ( G M , TM ) takes into account
7.1.7 Overall module performance the marked reduction in efficiency resulting from the
rising TM .
By combining the two effects of temperature and light Figure 7.8 shows the overall efficiency at various light
intensity, the final efficiency of the module at every intensities in dependence of the wind speed. Once
126 7. PV System Design
16
14.0
14 13.8
12 13.6
Module efficiency (%)
PDC
YDC = · 100%. (7.41)
Pmax
128 7. PV System Design
Appendix
129
Derivations in Electrodynamics
A
A.1 Basics of Vector Calculus
∇ · D(r, t) = ρ F (r), (A.1a)
∂B(r, t)
∇ × E(r, t) = − , (A.1b)
∂t
∇ · B(r, t) = 0, (A.1c)
A.2 The Maxwell equations ∂D(r, t)
∇ × H(r, t) = + + J F ( r ), (A.1d)
∂t
131
132 A. Derivations in Electrodynamics
other via Now we take the fourth Maxwell equation, Eq. A.1d,
D = ee0 E, (A.2a) with j F = 0 and Eqs. (A.2),
∂2 E
B = µµ0 H, (A.2b) ∇ × (∇ × E) = −ee0 µµ0 . (A.4)
∂t2
where µ is the relative permeability of the medium in By using the relation
that the fields are observed and µ0 = 4π · 10−7 Vs/(Am)
is the permeability of vacuum. Eqs. (A.2) are only valid ∇ × (∇ × E) = ∇(∇ · E) − ∆E = −∆E, (A.5)
if the medium is isotropic, i.e. e and µ are independent
of the direction. We may assume all the materials im- we find
∂2 E
portant for solar cells to be nonmagnetic, i.e. µ ≡ 1. ∆E = ee0 µµ0 . (A.6)
∂t2
In Eq. (A.5) we used that we are in source free space,
i.e. ∇E = 0. Equation (A.6) is the wave equation for the
A.3 Derivation of the electromag- electric field. Note that the factor
netic wave equation 1
ee0 µµ0
We now derive de electromagnetic wave equations in
source-free space, ρ F ≡ 0 and j F ≡ 0. For the derivation has the unit of (m/s)2 , i.e. a speed to the square. In
of the electromagnetic wave equations we start with ap- easy terms, it is the squared propagation speed of the
plying the rotation operator ∇× to the second Maxwell wave.1 We now set
equation, Eq. (A.1b), 1
c20 := (A.7)
e0 µ 0
∂B
∇ × (∇ × E) = −∇ × . (A.3) 1 In reality, if the medium is absorbing, things are getting much more complex.
∂t
A.4. Properties of electromagnetic waves 133
In section A.3, we found that plane waves can be de- Thus, neither the electric nor the magnetic fields have
scribed by components in the propagation direction but only com-
ponents perpendicular to the propagation direction (the
E(x, t) = E0 · eikz z−iωt , (A.10a) x- and y-directions in our case).
H(x, t) = H0 · eikz z−iωt . (A.10b) Without loss of generality we now assume that the
electric field only has an x-component, E0 = ( Ex,0 , 0 0).
Substituting this electric field into the left hand side
In this section we study some general properties of
of the second Maxwell equation, Eq. (A.1b), yields
plane electromagnetic waves. Substituting Eq. (A.10a)
134 A. Derivations in Electrodynamics
−
∂ ∂
∂y Ez ∂z Ey
∂
∂x Ex
0
0
0
= ∂ Ex = Ex,0 ∂ eikz z−iωt = iEx,0 k z eikz z−iωt .
∇×E =
∂ ×E
∂x y =
∂z Ex − ∂x Ez
∂ ∂
∂z ∂z
(A.12)
Ez 0 0 0
∂x Ey − ∂y Ex
∂ ∂ ∂
∂x
where r
µ0
Z0 = cµ0 = = 376.7 Ω (A.20)
e0
is the impedance of free space.
The derivations in this section were done for plane
waves. However, it can be shown that the properties
of electromagnetic waves summarised in the itemisation
above are valid for electromagnetic waves in general.
136 A. Derivations in Electrodynamics
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heim, Germany, 2005). (2000).
137
138 A. Derivations in Electrodynamics
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