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932
ENERGY COLLECTED AND DELIVERED BY PV MODULES
20.5.3.4 Daily irradiation
The most accurate way of calculating
G
dm
(β, α)
from
G
dm
(
0
)
is, first, to calculate the
hourly horizontal irradiation components
G
hm
(0),
D
hm
(0) and
B
hm
(0); second, to transpose
them to the inclined surface
G
hm
(β, α)
,
D
hm
(β, α)
and
B
hm
(β, α)
; and, finally, to integrate
during the day.
Such a procedure, summarised in Figure 20.16, allows us to account for the
anisotropic properties of diffuse radiation, and leads to good results whatever the
orientation of the inclined surface. However, it is laborious to apply and a computer
must be used. It is interesting to mention that, for the case of surfaces tilted to the equator
(
α
=
0), frequently encountered in photovoltaic applications, if the diffuse radiation is
taken to be isotropic, the following expression may be applied
G
d
(β,
0
)
=
B
d
(
0
)
×
RB
+
D
d
(
0
)
1
+
cos
β
2
+
ρG
d
(
0
)
1
−
cos
β
2
(
20
.
39
)
where the factor
RB
represents the ratio between the daily direct irradiations on an inclined
surface and on an horizontal surface, and may be approximated by setting it equal to the
corresponding ratio between daily extraterrestrial irradiations on similar surfaces. Hence,
RB
is given as follows:
RB
=
ω
SS
π
180
[sign
(φ)
] sin
δ
sin
(
|
φ
| −
β)
+
cos
δ
cos
(
|
φ
| −
β)
sin
ω
SS
ω
S
π
180
sin
δ
sin
φ
+
cos
δ
cos
φ
sin
ω
S
(
20
.
40
)
where
ω
SS
is the sunrise angle on the inclined surface, which is given by
ω
SS
=
max[
ω
S
,
−
arccos
(
−
[sign
(φ)
] tan
δ
tan
(
abs
(φ)
−
β))
]
(
20
.
41
)
It is interesting to observe that for the equinox days,
δ
=
0
⇒
ω
S
=
ω
SS
and
equation (20.40) becomes
RB
=
cos[
abs
(φ)
−
β
]
/
cos
φ
.
Example
: Estimate the average daily irradiation in January at Changchun–China
(
φ
=
43
.
8
◦
) over a fixed surface facing south and tilted at an angle
β
=
50
◦
with respect
to the horizontal, knowing that the mean value of the global horizontal irradiation is
G
dm
(
0
)
=
1861 Wh/m
2
and the ground reflection
ρ
=
0
.
2. The solution is as follows:
January
⇒
d
n
=
17;
δ
= −
20
.
92
◦
φ
=
43
.
8
◦
⇒
ω
S
= −
68
.
50 and
B
0d
(
0
)
=
3586 Wh
/
m
2
⇒
K
Tm
=
0
.
519
⇒
F
Dm
=
0
.
414
D
dm
(
0
)
=
770 Wh/m
2
;
B
dm
(
0
)
=
1091 Wh/m
2
arccos
(
−
tan
δ
tan
(φ
−
β))
= −
92
.
38
◦
⇒
ω
SS
= −
68
.
5
◦
⇒
RB
=
2
.
741
DIURNAL VARIATIONS OF THE AMBIENT TEMPERATURE
933
G
dm (0)
D
dm (0)
G
hm (0)
D
hm (0)
B
hm (0)
=
G
hm (0)
−
D
hm (0)
D
hm (
b
,
a
);
b
hm (
b
,
a
);
R
hm (
b
,
a
)
G
dm
(
b
,
a
)
= ∑
G
hm
(
b
,
a
)
w
s
−
w
s
Equation
(20.18)
(20.23
−
20.26)
(20.28, 20.29)
(20.32
−
20.38)
Figure 20.16
Diagram explaining the calculation of the daily irradiation on an inclined surface
G
dm
(β, α)
from the corresponding horizontal value
G
dm
(0)
D
dm
(
50
)
=
633 Wh
/
m
2
;
B
dm
(
50
)
=
2990 Wh
/
m
2
;
R
dm
(
50
)
=
66 Wh
/
m
2
G
dm
(
50
)
=
3689 Wh
/
m
2
It is worth mentioning that a more detailed calculation, following the procedure outlined in
Figure 20.16, would lead to
G
dm
(
50
)
=
3956 Wh/m
2
. That means the error associated to
equation (20.40) is below 8%. This difference is mainly due to the different consideration
of the diffuse radiance distribution.
20.6 DIURNAL VARIATIONS OF THE AMBIENT
TEMPERATURE
The behaviour of the photovoltaic modules depends, to some extent, on the ambient
temperature. Just as it is for solar radiation, sometimes it is necessary to determine how
this parameter varies throughout the day. The data available as a starting point for this
calculation are, in general, the maximum and minimum temperature of the day,
T
aM
and
T
am
, respectively.
A model that is simple but, nevertheless, gives a good fit to the experimental values
is obtained from the fact that the temperature evolves in a similar manner to the global
radiation but with a delay of about 2 h. This fact allows the following three principles to
be deduced
•
T
am
occurs at sunrise (
ω
=
ω
S
).
•
T
aM
occurs two hours after midday (
ω
=
30
◦
).
•
Between these two times, the ambient temperature develops according to two semi
cycles of a cosine function: one from dawn to midday, and the other between midday
and sunrise of the following day.
934
ENERGY COLLECTED AND DELIVERED BY PV MODULES
The following set of equations based on the above, permits the ambient temperature
throughout a day
j
to be calculated as follows:
For
−
180
< ω
≤
ω
S
T
a
(j, ω)
=
T
aM
(j
−
1
)
−
T
aM
(j
−
1
)
−
T
am
(j )
2
[1
+
cos
(aω
+
b)
]
(
20
.
42
)
with
a
=
−
180
ω
S
+
330
and
b
= −
aω
S
For
ω < ω
S
≤
30
T
a
(j, ω)
=
T
am
(j )
+
T
aM
(j )
−
T
am
(j )
2
[1
+
cos
(aω
+
b)
]
(
20
.
43
)
with
a
=
180
ω
S
−
30
and
b
= −
30
a
For 30
< ω
≤
180
T
a
=
T
aM
(j )
−
T
aM
(j )
−
T
am
(j
+
1
)
2
[1
+
cos
(aω
+
b)
]
(
20
.
44
)
with
a
=
180
ω
S
+
330
and
b
= −
(
30
a
+
180
)
To apply these equations, it is necessary to know the maximum temperature on the
previous day,
T
aM
(j
−
1
)
, and the minimum temperature of the following day,
T
am
(j
+
1
)
.
If these data are unavailable, then it can be assumed, without introducing too much error,
that they equal those for the day in question.
20.7 EFFECTS OF THE ANGLE OF INCIDENCE
AND OF THE DIRT
The reflectance and transmittance of optical materials depends on the angle of incidence.
Glass covers of solar collectors are not an exception and therefore the optical input of
photovoltaic modules is affected by their orientation with respect to the sun, due to the
angular variation of the glass reflection. Theoretical models, based on the well-known
Fresnel formulae, have been developed for clean surfaces. The most popular formula-
tion is from ASHRAE [32]. For a given incidence angle,
θ
S
, it can be described by the
simple expression
FT
B
(θ
S
)
=
1
−
b
0
1
cos
θ
S
−
1
(
20
.
45
)
where
FT
B
(θ
S
)
is the relative transmittance, normalised by the total transmittance for
normal incidence, and
b
0
is an adjustable parameter that can be empirically determined
for each type of photovoltaic module. If this value is unknown, a general value
b
0
=
0
.
07
may be used. The effect of the angle of incidence on the successfully collected solar
radiation can be calculated by applying equation (20.45) to the direct and circumsolar
irradiances, and by considering an approximated value,
FT
=
0
.
9, for the isotropic diffuse
and reflected radiation terms. Figure 20.17 shows a plot of
FT
B
(θ
S
)
versus
θ
S
. It presents
EFFECTS OF THE ANGLE OF INCIDENCE AND OF THE DIRT
935
0
0.2
0.4
0.6
0.8
1
0
20
40
60
q
S
80
100
FT
B
Clean
Dirty
Figure 20.17
The relative transmittance
FT
B
is plotted against the angle of incidence
θ
S
, for a
clean surface and also for a dust-covered surface
a pronounced knee close to 60
◦
. In practical terms, that means the effects of the angle
of incidence are negligible for all the
θ
S
values well below this figure. For example,
FT
B
(
40
◦
)
=
0
.
98.
The ASHRAE model is simple to use but has noticeable disadvantages. It cannot
be used for
θ
S
>
80
◦
, and, still worse, it cannot take into consideration the effects of dust.
Dust is always present in real situations, and not only reduces the transmittance at normal
incidence but also influences the shape of
FT
B
(θ
S
)
. Figure 20.17 shows that the relative
transmittance decreases because of dust at angles from about 40 to 80
◦
. Real
FT
B
(θ
S
)
are
best described [33] by
FT
B
(θ
S
)
=
1
−
exp
−
cos
θ
S
a
r
−
exp
−
1
a
r
1
−
exp
−
1
a
r
(
20
.
46
)
where
a
r
is an adjust parameter mainly associated with the degree of dirtiness, as shown
in Table 20.4. Note that the degree of dirtiness is characterised by the corresponding
relative normal transmittance,
T
dirt
(
0
)/T
clean
(
0
)
. Equation (20.46) applies for direct and
circumsolar radiation components. The angular losses for isotropic diffuse and albedo
radiation components are, respectively, approximated by [33]
FT
D
(β)
=
1
−
exp
−
1
a
r
c
1
sin
β
+
π
−
β
·
π
180
−
sin
β
1
+
cos
β
+
c
2
sin
β
+
π
−
β
π
180
−
sin
β
1
+
cos
β
2
(20.47)
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