Solar Energy Materials and Solar Cells 55 (1998) 199—214
A simple model for sizing stand alone
photovoltaic systems
M. Sidrach-de-Cardona!,*, Ll. Mora López"
! Departamento Fı& sica Aplicada II, ETSI Informa& tica, Universidad de Ma& laga, 29071 Ma& laga, Spain
" Departamento Lenguajes y C. Computacio& n, ETSI Informa& tica, Universidad de Ma& laga, 29071 Ma& laga, Spain
Received 5 November 1997
Abstract
We consider a general model for sizing a stand-alone photovoltaic system, using as energy
input data the information available in any radiation atlas. The parameters of the model are
estimated by multivariate linear regression. The results obtained from a numerical sizing
method were used as initial input data to fit the model. The expression proposed allows us to
determine the photovoltaic array size, with a coefficient of determination ranging from 0.94 to
0.98. System parameters and mean monthly values for daily global radiation on the solar
modules surface are taken as independent variables in the model. It is also shown that the
proposed model can be used with the same accuracy for other locations not considered in the
estimation of the model. ( 1998 Elsevier Science B.V. All rights reserved.
Keywords: Photovoltaic stand-alone systems; Photovoltaic sizing; Multivariate linear method
1. Introduction
The operation of a stand-alone photovoltaic (PV) system depends, among other
factors, on the energy input on the panel surface and on the energy demand or load.
Incident solar radiation has a random component which makes it impossible to
accurately know how much energy the system will receive during a given period. So,
before building a stand-alone photovoltaic system it is necessary to calculate the size
of the generator and the storage system for the expected load via a sizing method.
* Corresponding author. E-mail: msidrach@ctima.uma.es
0927-0248/98/$ see front matter ( 1998 Elsevier Science B.V. All rights reserved.
PII S 0 9 2 7 - 0 2 4 8 ( 9 8 ) 0 0 0 9 3 - 2
200 M. Sidrach-de-Cardona, Ll. M. Lo& pez/Solar Energy Materials and Solar Cells 55 (1998) 199—215
There are several such methods and the use of one or the other will depend on the
initial data available. Generally speaking, sizing methods can be classified as follows:
1.1. Intuitive methods
A simplified calculation of the size of the system is carried out without establishing
any relationship between the different subsystems nor taking into account the random
nature of solar radiation. One of the approaches most frequently used is the “worst
month” method; the energy balance per month is calculated and the parameter for
energy input is taken to be the month with the worst conditions for the system. The
rationale behind this method is that if it works this month it will work for the rest of
the year. This usually results in oversizing the installation, and prevents any kind of
energy or economic optimisation, Sidrach [1].
1.2. Numerical methods
A system simulation is used in this case. For each time period considered, usually
a day or an hour, the energy balance of the system and the battery load state is
calculated. These methods offer the advantage of being more accurate, and the
concept of energy reliability can be applied in a quantitative manner. System reliability is defined as the load percentage satisfied by the photovoltaic system for long
periods of time. These methods allow us to optimise the energy and economic cost of
the system. One of the more commonly used ones is the loss of load probability
method: in this case, system reliability is described with a parameter that represents
the mean load percentage, for large periods of time, that is not supplied by the
photovoltaic system. This parameter is appropriately called loss of load probability
(LOLP) and has an inverted relationship to system reliability. The disadvantage of
such methods is the need to have available hourly or daily exposure radiation series
for long enough periods of time. Klein and Beckman [2] claim that with data
currently available we can only calculate accurate sizes with LOLP values '10~2.
1.3. Analytical methods
By the use of equations these methods try to describe system size as a function of
reliability. For example, Barra et al. [3] and Bartoli et al. [4] proposed analytical
expressions to achieve this. The main disadvantage of these methods is that either they
are not accurate enough or they require the determination of coefficients for the
expressions for each location. Their strongest advantage is that the calculation of the
different subsystem sizes is very simple.
Egido and Lorenzo [5] proposed a method consisting in creating reliability maps
for each LOLP value considered — isoreliability lines. The main disadvantage is the
difficulty in applying it to locations for which we do not have a daily radiation series.
In this case, as these authors state in their work, it would be necessary to recalculate
all coefficients in the model if we are not to suffer a significant loss of accuracy.
M. Sidrach-de-Cardona, Ll. M. Lo& pez/Solar Energy Materials and Solar Cells 55 (1998) 199—214 201
In this work we propose a new model for sizing stand-alone photovoltaic systems.
This model is obtained by analysing the results of calculating system size with
numerical methods. The size of the generator in relation to the storage system size and
the loss of load probability is obtained with a numerical method. Daily radiation
series from different Spanish locations are used. The data obtained with this method is
then processed via a regression linear analysis. As explanatory or independent
variables we use both climatic and system parameters. Six of all the parameters
analysed were found to be significant: three related to the system’s energy input (as
a mean monthly value); and three related to the system’s configuration (array surface
tilt, battery capacity and LOLP.) Using these parameters a general expression is
proposed which enables us to calculate the generator size. The suitability of the model
for locations not used in the analysis has also been tested.
2. A numerical sizing method
The PV array size is calculated using the LOLP method mentioned earlier. This
method is based on the ideas proposed by Gordon [6], and Klein and Beckman [2].
LOLP is defined as the dimensionless energy deficit, for a PV system, carried out over
a sufficiently long period of time which allows us to fully characterise the statistical
nature of the solar radiation. To determine the array capacity and the battery sizes for
a specified LOLP, the long-term photovoltaic behaviour has been simulated and
calculated day by day.
The first step has been to calculate the solar radiation incident on the tilted surface.
With this purpose in mind, and from the values of daily global radiation on a horizontal surface, direct and diffuse radiation values were obtained according to the model
developed by Collares-Pereira and Rabl [7]. Total irradiance on the array plane is
calculated using an isotropic model for the ground reflected diffuse radiation and
Hay’s anisotropic model for the diffuse radiation [8].
In order to simulate the behaviour of a photovoltaic system a daily energy balance
is carried out each day. The generator capacity value obtained, C , is divided by the
A
daily load under consideration. This balance is carried out assuming that the load is
constant every day of the year. Thus, the array capacity C is defined as the generator
A
size divided by the daily load and its uints are W /Wh. The storage capacity, C , is
1
B
defined as the rate between the available energy in the battery and the daily load, and
therefore it represents the number of days of autonomy.
The analysis has been carried out for 2—9 days of autonomy since most stand-alone
photovoltaic installations are sized for a period of time within this range.
In order not to lose generality in the results we did not consider any factor
representing losses in the system, such as battery efficiency, losses due to wiring or
mismatch losses, etc. These factors have to be taken into account once the C and
A
C sizes have been obtained in order to calculate the actual size of the installation.
B
In order to calculate the value of C and thus obtain a value for LOLP we use an
A
iterative procedure. The process is repeated until we obtain an LOLP value the same
as the reliability to which the system is sized, with a 2% error. In our case we have
202 M. Sidrach-de-Cardona, Ll. M. Lo& pez/Solar Energy Materials and Solar Cells 55 (1998) 199—215
used three LOLP values, 0.01, 0.05, and 0.1. We found that the results obtained are
independent of the initial load status of the battery.
As a starting point to carry out these simulations, we have used a series of daily
radiation data received on a horizontal surface at 31 Spanish meteorological stations.
These data correspond to locations distributed throughout Spanish territory and
cover Melilla, a town in North Africa with a latitude of 35.28°; Santander, situated in
the north of Spain at a latitude of 43.47°; Vigo, in the northwest; and Palma de
Mallorca in the east. So, the climatic conditions are very different from each other.
Table 1 shows the number of available observations, specifying the years when data
was collected and the latitude of the place where they where recorded (North latitude).
As some months in each year are not available, the total number of months for which
observations are available in each location appears in the table. This table also
includes annual average values of daily global radiation, G .
$,:
Table 1
Data set used in the analysis
Location
Latitude
(°N)
Years of data
Months
G
$:
(MJ m~2)
Alicante
Albacete
Almeria
Arenosillo
Badajoz
Bilbao
Burgos
Ceuta
Cofrentes
Cordoba
C. Real
Granada
León
Logron8 o
Lubia
Lugo
Madrid
Málaga
Mallorca
Melilla
Menorca
Oviedo
Salamanca
Santander
Sevilla
Tarragona
Toledo
Valencia
Vigo
Zaragoza
38.36
38.93
36.85
37.10
38.88
43.30
42.37
35.91
39.20
37.85
38.98
37.18
42.58
42.65
41.60
43.25
40.45
36.66
39.55
35.28
39.88
43.35
40.95
43.47
37.41
40.95
39.55
39.48
42.23
41.67
1976—1982
1976—1983
1976—1984
1976—1984
1976—1982
1976—1984
1976—1982
1976—1980
1974—1983
1976—1984
1976—1983
1976—1984
1976—1984
1975—1980
1978—1984
1976—1984
1973—1984
1975—1982
1973—1984
1976—1984
1976—1984
1975—1984
1976—1984
1975—1982
1975—1982
1975—1984
1979—1984
1975—1980
1977—1984
1971—1983
83
68
106
102
84
92
78
59
119
72
65
104
103
69
67
106
108
83
92
105
83
120
108
108
71
106
68
72
95
74
17.4
17.7
17.9
18.5
16.7
10.9
14.4
18.5
15.4
16.6
16.3
17.3
14.2
15.0
15.5
14.0
16.7
17.0
15.3
17.7
15.7
11.0
15.3
11.4
17.6
15.6
15.6
16.2
13.2
16.0
M. Sidrach-de-Cardona, Ll. M. Lo& pez/Solar Energy Materials and Solar Cells 55 (1998) 199—214 203
Fig. 1 shows the loss of load probability curve for some of the locations studied
with a LOLP value of 0.01 and an array surface tilt, b, of 60°. There are differences in
the array capacity C for the different days of autonomy considered depending on
A
whether the locations are southern (Melilla) or northern (Santander). This confirms
that the locations belong to different climatic areas.
Fig. 2 shows the results obtained for the previously mentioned locations with
a LOLP value of 0.05 and a surface tilt of 60°. In this case, although the differences
between the curves are less marked, capacity values for the array are different
according to location.
Fig. 1. LOLP curves for differents locations. LOLP"0.01, b"60° (L) Santander, (.) Oviedo, (v)
Madrid, (j) Mallorca, (h) Melilla.
Fig. 2. LOLP curves for differents locations. LOLP"0.05, b"60° (L) Santander, (.) Oviedo, (v)
Madrid, (j) Mallorca, (h) Melilla.
204 M. Sidrach-de-Cardona, Ll. M. Lo& pez/Solar Energy Materials and Solar Cells 55 (1998) 199—215
For each location we have studied how the system size (C , C ) changes according
A B
to the value of LOLP and different values of b. Figs. 3—5 show these results for some
of the sites analysed, for two LOLP values (0.01 and 0.05) and two b values (60° and
30°).
All these results show how the array capacity significantly changes when the given
LOLP value is modified. For a certain LOLP the tilt of the array from the horizontal
is less important. The size of the generator C is less when the tilt angle is 60°. This is
A
due to the fact that when considering the annual daily average load to be constant,
a 60° tilt is optimal for the latitude of the locations studied. It is worth pointing out
Fig. 3. LOLP curves for Málaga, Lat. 36,66°. (L) LOLP"0.01, b"30° (v) LOLP"0.01, b"60° (h)
LOLP"0.05, b"30° (j) LOLP"0.05, b"60°.
Fig. 4. LOLP curves for Oviedo, Lat. 43,35°. (L) LOLP"0.01, b"30° (v) LOLP"0.01, b"60° (h)
LOLP"0.05, b"30° (j) LOLP"0.05, b"60°.
M. Sidrach-de-Cardona, Ll. M. Lo& pez/Solar Energy Materials and Solar Cells 55 (1998) 199—214 205
Fig. 5. LOLP curves for Madrid, Lat. 40,45°. (L) LOLP"0.01, b"30° (v) LOLP"0.01, b"60° (h)
LOLP"0.05, b"30° (j) LOLP"0.05, b"60°.
that variations on the system size, with these parameters, are much larger in those
locations with greater climatic variability, (e.g. like those in the north of Spain, such as
Oviedo, Fig. 4).
3. Analysis of the results obtained with the numerical method
Using the values obtained with the loss of load probability method, we have used
a regression linear analysis to determine which variables enable us to estimate the
capacity of the array C with the minimum error possible. C values are the
A
A
dependent variable of the model. As independent variables we have used two kinds of
variables: the first type refers to the photovoltaic system; and the other to energy
availability or radiation values for each location.
The variables related to the photovoltaic system are:
f Battery capacity (number of days of autonomy), C . An autonomy of 2—9 days has
B
been considered.
f Array surface tilt, b.
The variables related to energy availability are:
f Yearly average of daily global radiation, on tilted surface.
f Minimum, mean and maximum value of monthly average of daily global radiation,
on tilted surface.
f Minimum, mean and maximum value of monthly average of daily atmosferic
clearness index
f Radiation variability. This parameter relates the yearly mean value to the minimum
value of monthly average of daily global radiation.
206 M. Sidrach-de-Cardona, Ll. M. Lo& pez/Solar Energy Materials and Solar Cells 55 (1998) 199—215
Radiation values were calculated for the photovoltaic array surface tilt. All variables related to energy availability were calculated in two different ways: on the one
hand, using the entire data series; and on the other hand, using only the mean monthly
values published in the radiation atlas from the Spanish National Institute of Meteorology (INM) [9]. The main reason for using the latter calculations was to
determine whether having only twelve mean monthly values would yield similar
results to those obtained with estimated mean values from the series. Our results show
that it is possible to use these 12 mean monthly values to characterise the radiation,
since the coefficients of determination of the regression models do not change
significantly when values calculated from the series are used.
Of all the values analysed the ones we give below have proven to be significant, i.e.
they can be used for determining the C values:
A
f The mean annual value of daily global radiation on the tilted surface G
cal$,y,b
culated from the 12 monthly mean values on a horizontal surface:
12
+G
$,m,b
m/1
G
"
,
$,y,b
12
(1)
where G
is the mean monthly value of daily global radiation on the tilted
$,m,b
surface. These values are calculated from mean monthly values of daily global
radiation on the horizontal surface. Direct and diffuse radiation values were
obtained using Page’s Method [10]. Total irradiance on the array plane is calculated using an isotropic model for the ground reflected diffuse radiation and
Hay’s anisotropic model for the diffuse radiation [8].
f The minimum value, G
, of the 12 mean monthly values for daily global
$,.*/,b
radiation on a tilted surface, G
:
$,m,b
G
"minMG
, m"1,2,12N.
(2)
$,.*/,b
$,.,b
f The minimum value, K
, for the 12 mean monthly values of the daily clearness
$,.*/
index, calculated according to the expression:
K
"minMK , m"1,2,12N.
(3)
$,.*/
$,m
The K
values are calculated using the mean monthly value of daily global
$,m
radiation G , and the extraterrestrial radiation value corresponding to the 15th
$,m
day of each month:
G
K " $,m .
(4)
$,m G
0,15
f The variability of monthly mean daily radiation, VG
defined by the following
$,y,b
expression:
G
!G
$,.*/,b.
»G
" $,y,b
$,y,b
G
$,y,b
f The number of days of autonomy, C .
B
(5)
M. Sidrach-de-Cardona, Ll. M. Lo& pez/Solar Energy Materials and Solar Cells 55 (1998) 199—214 207
From the analysis of the array sizes obtained in the previous section, it is clear that
there is a relationship between array sizes and radiation values; variability in size does
not seem to be uniform for the different levels of radiation. For this reason, dummy
variables were used to group the different observations according to the different
levels of radiation. Dummy variables were also used for the variable number of days of
autonomy, C ,. With these dummy variables it was possible to make groups from the
B
observations according to the number of days of autonomy and the mean annual
values of the daily exposure series, G
(in MJm~2). These dummy variables have
$,y,b
been defined, for each observation, as follows:
G
1 if G
3I and C "j#1,
$,y,b i
B
F "
i,j
0 otherwise.
1)i)3, 1)j)8,
(6)
Variable I represents the group to which the observation belongs. These groups are
i
defined as follows:
I "[10, 14),
I "[14, 18),
I "[18, 22].
1
2
3
Dummy variables allow us to take into account non linearities in the way that the
value G
affects the dependent variable C for the different days of autonomy. On
$,y,b
A
the other hand, it is assumed that all non-dummy independent variables have the
same effect on the dependent variable irrespective of the group it belongs to.
4. The proposed model
The multivariate linear regression model proposed is as follows:
3 8
C "c G
#c G
#c »G
#c K
#+ + a F ,
(7)
A
1 $,y,b
2 $,.*/,b
3 $,y,b
4 $,.*/
i,j i,j
i/1 j/1
where c (1)k)4) and a (1)i)3, 1)j)8) are unknown parameters.
k
ij
Eq. (7) has been estimated by ordinary least squares. The estimated values of
coefficients c and a for the LOLP values considered are shown in Tables 2—5. The
k
ij
coefficient of determination, R2, obtained in each fit is also shown in the tables. Note
that it is possible to assert that the selected variables are sufficient to determine C (the
A
generator size) with a coefficient of determination that ranges from 0.94, for LOLP
values of 0.01, to 0.98 for LOLP values of 0.1.
Table 2
Estimates of coefficients c of Eq. (7) for the three values of LOLP analysed
*
LOLP
c
1
c
2
c
3
c
4
0.01
0.05
0.10
!0.0538
!0.0454
!0.0326
0.0497
0.0386
0.0251
1.1090
0.8809
0.5458
!0.7598
!0.0232
0.0660
208 M. Sidrach-de-Cardona, Ll. M. Lo& pez/Solar Energy Materials and Solar Cells 55 (1998) 199—215
Table 3
Estimates of coefficients a
*,+
of Eq. (7) for LOLP"0.01. Coef. of determination (R2)"0.940
G
CC
$,y,b B
2
3
4
5
6
7
8
9
I (10—14 MJm~2)
1
I (14—18 MJm~2)
2
I (18—22 MJm~2)
3
0.9532
0.7929
0.7745
0.8477
0.7327
0.7416
0.7777
0.7024
0.7250
0.7360
0.6836
0.7144
0.7075
0.6713
0.7060
0.6859
0.6602
0.6988
0.6655
0.6508
0.6924
0.6502
0.6426
0.6866
Table 4
Estimates of coefficients a
i,j
of Eq. (7) for LOLP"0.05. Coef. of determination (R2)"0.962
G
CC
$,y,b B
2
3
4
5
6
7
8
9
I (10—14 MJm~2)
1
I (14—18 MJm~2)
2
I (18—22 MJm~2)
3
0.4055
0.3328
0.3282
0.3717
0.3191
0.3211
0.3535
0.3110
0.3168
0.3418
0.3052
0.3137
0.3332
0.3007
0.3112
0.3262
0.2969
0.3090
0.3198
0.2941
0.3069
0.3130
0.2911
0.3051
Table 5
Estimates of coefficients a
i,j
of Eq. (7) for LOLP"0.1. Coef. of determination (R2)"0.975
G CC
$,:,b B
2
3
4
5
6
7
8
9
I (10—14 MJm~2)
1
I (14—18 MJm~2)
2
I (18—22 MJm~2)
3
0.3275
0.2815
0.2793
0.3128
0.2758
0.2768
0.3055
0.2731
0.2752
0.2996
0.2709
0.2737
0.2952
0.2693
0.2728
0.2921
0.2675
0.2717
0.2891
0.2663
0.2708
0.2852
0.2651
0.2699
It is worth pointing out that only mean monthly parameters are used to calculate
the value of C . In other words, it is not necessary to have available the daily radiation
A
series values.
5. Results from the model
In this section we will compare the results obtained using the LOLP method in
a stand-alone photovoltaic system with those obtained with our model.
5.1. For locations used in the regression
In order to compare the results obtained, for each C and LOLP, it has been
B
calculated the mean error, e , on estimating C , defined according to the following
.
A
expression:
n
+ [(C !C* )/C ]
A,i A,i
A,i
e (%)"i/1
100,
.
n
(8)
M. Sidrach-de-Cardona, Ll. M. Lo& pez/Solar Energy Materials and Solar Cells 55 (1998) 199—214 209
where C is the array capacity calculated using the LOLP method, for location i,
A,i
C* is the array capacity calculated using the proposed method, for location i, n is the
A,i
total amount of locations, and [A] respresent the absolute value of A.
In Table 6 the results obtained are shown for various LOLP and C . As expected,
B
the mean error is bigger for LOLP values of 0.01 since the coefficient of determination
is 0.94 in this case.
When sizing is done in a less strict manner as regards system reliability
(LOLP"0.1) the coefficient of determination, and therefore the error obtained
between both methods, is smaller.
Fig. 6 shows the results obtained for Santander using both methods, for three
LOLP values and with b"60°. This location belongs to group I . The sizing is fairly
1
similar with both methods. In addition, when the LOLP value increases, the discrepancy between both methods goes down.
Figs. 7 and 8 show the results obtained for Salamanca (group I ) and Málaga
2
(group I ), respectively. Generally speaking, the values are very similar to those
3
presented in these figures. In the case of Malaga (Fig. 8) when the battery capacity is
low a bigger discrepancy is found between the calculated values for the two methods.
Table 6
Mean error (in %) between values of C calculated by the numerical method and values of C calculated by
A
A
the proposed method. Different LOLP and C are considered
B
LOLPCC
B
2
3
4
5
6
7
8
9
0.01
0.05
0.10
9.7
5.1
3.0
7.9
4.0
2.5
6.6
3.6
2.4
5.9
3.6
2.5
6.1
3.6
2.7
6.3
3.9
2.8
6.4
4.3
3.0
6.7
4.6
3.2
Fig. 6. Results obtained for Santander using both methods, with b"60°. LOLP"0.01: (v) numerical
method (L) proposed method; LOLP"0.05: (j) numerical method (h) proposed method; LOLP"0.1:
(.) numerical method, (+) proposed method.
210 M. Sidrach-de-Cardona, Ll. M. Lo& pez/Solar Energy Materials and Solar Cells 55 (1998) 199—215
Fig. 7. Results obtained for Salamanca using both methods, with b"60°. LOLP"0.01: (v) numerical
method (L) proposed method; LOLP"0.05: (j) numerical method (h) proposed method; LOLP"0.1:
(.) numerical method, (+) proposed method.
Fig. 8. Results obtained for Málaga using both methods, with b"60°. LOLP"0.01: (v) numerical
method (L) proposed method; LOLP"0.05: (j) numerical method (h) proposed method; LOLP"0.1:
(.) numerical method, (+) proposed method.
5.2. For locations not used in the regression
The advantage of having a model such as the one proposed here is that this enables
sizing a photovoltaic system in locations for which the daily radiation series is
not available. In order to analyse the results obtained in such cases, when we fitted
the regression model we did not use all of the series available. For example,
data corresponding to Santiago de Compostela (latitude: 42.88°) and Lanzarote
M. Sidrach-de-Cardona, Ll. M. Lo& pez/Solar Energy Materials and Solar Cells 55 (1998) 199—214 211
(latitude: 28.95°) have only been used to determine the validity of the model. Santiago
has a very similar latitude to some of the locations used to fit the model. On the other
hand, Lanzarote, situated in the Canary Islands, has a very different latitude from
those used in the fitting, although the levels of incident radiation on a tilted surface are
very close to the groups considered.
Fig. 9 shows the results obtained for Santiago de Compostela. There is a good
match between the values calculated with both models for loss of load probabilities of
Fig. 9. Results obtained for Santiago de Compostela using both methods, with b"60°. LOLP"0.01: (v)
numerical method (L) proposed method; LOLP"0.05: (j) numerical method (h) proposed method;
LOLP"0.1: (.) numerical method, (+) proposed method.
Fig. 10. Results obtained for Lanzarote using both methods, with b"30°, LOLP"0.01: (v) numerical
method (L) proposed method; LOLP"0.1: (.) numerical method, (+) proposed method.
212 M. Sidrach-de-Cardona, Ll. M. Lo& pez/Solar Energy Materials and Solar Cells 55 (1998) 199—215
0.05 and 0.1. When the LOLP value is 0.01 the differences are slightly bigger,
especially if the battery capacity is low.
Fig. 10 shows the results for Lanzarote. In this case the angle is taken to be 30° since
this is optimal for this latitude. For higher LOLP values (0.1) both methods yield
similar values. When LOLP is 0.01 greater discrepancies are found, although errors
are very similar to those obtained for other locations included in the study.
6. Conclusions
This paper proposes a simple model for calculating the size of stand-alone photovoltaic installations. This model is obtained using regression linear analysis. As the
dependent variable of the model we have used the result obtained from calculating the
generator size using one of the numerical methods of sizing stand-alone photovoltaic
systems, i.e. the LOLP. The independent variables in the model are on the one hand,
system characteristics such as battery capacity (as days of autonomy). On the other
hand, the energy the system receives is taken into account by the model via the
following variables: the mean annual value of daily global radiation on the tilted
surface; the mean minimal monthly value of daily radiation on the tilted surface; a
variable which relates both of these; and the mean monthly minimum value of the
clearness index. All these variables are calculated by using just twelve mean monthly
values of daily radiation, i.e., it is not necessary to have available a daily radiation series.
In order to fit a single model for each LOLP considered, dummy variables were
used to register the influence of battery capacity as well as the different levels of
radiation observed. These dummy variables allow us to detect non-linearities in the
way annual mean values of daily global radiation influence the independent variable,
i.e. the generator size, for the different number of days of autonomy. On the other
hand, all non-dummy independent variables are assumed to affect the dependent
variable uniformly.
The results obtained allow us to conclude that, using the chosen variables, it is
possible to determine the size of the generator of a stand-alone photovoltaic installation with a coefficient of determination ranging from 0.94 to 0.98.
With the proposed model the best results are obtained for loss of load probability
values of 0.05 and 0.1. In fact, the coefficient of determination 0.94 is obtained for
a LOLP of 0.01. This is so because, in order to size with very small LOLP values (high
reliability), it is necessary to have a daily radiation series available for a sufficiently
large number of years, so that all solar radiation statistical data can be taken into
account.
It has also been confirmed that the proposed model can be used with the same
accuracy for locations not used in the fit of the multivariate linear regression model.
This indicates that the model is valid for any location with similar mean radiation
levels.
In this paper mean annual values of daily radiation on tilted surfaces for all the
locations used range between 10 and 22 MJm~2. The validity of the model for
locations with mean radiation levels different from these remains to be tested.
M. Sidrach-de-Cardona, Ll. M. Lo& pez/Solar Energy Materials and Solar Cells 55 (1998) 199—214 213
Appendix A. An example using the proposed model
For a given location, the calculation of the size of a photovoltaic generator for
a given LOLP value is carried out using just the 12 mean monthly radiation values for
this location as the system’s input energy. For example, for Málaga, according to the
Atlas of the Spanish Institute of Metereology [9] these values, in MJm~2, are:
Jan
Feb Mar Apr Mai Jun
Jul
Aug Sep
Oct Nov Dec
8.5
11.8 17.0 19.0 24.0 25.9 25.8 22.7 18.5 13.4 10.0 7.5
The values for the PV system to be used are:
f LOLP: 0.05.
f Days of autonomy: 6.
f Array surface tilt: 60°.
(1) The mean monthly values of daily exposure on the tilted surface are calculated.
The models used are the one proposed by Page [10] to obtain beam and diffuse mean
monthly values and the model proposed by Hay [8] to calculate the diffuse component on tilted surface. The daily exposure series values of global radiation on the tilted
surface so obtained are:
f
f
f
f
Yearly average value: G
"17.5 MJm~2.
$,y,b
The minimum monthly average value: G
"14.6 MJm~2.
$,.*/,b
Coefficient VG
"0.166.
$,y,b
The minimum monthly clearness index: K
"0.48.
$,.*/
The coefficients c are selected from Table 2. The coefficient a are selected from
*
i,j
Table 4 (LOLP"0.05). As the value of G
,e[14, 18] (MJm~2) and C "6 the
$,y,b
B
coefficient a is selected from row 2 and column 6 of Table 4.
2,6
(2) Using the equation proposed it can be obtained:
C "!0.0454G
#0.0386G
#0.8809»G
A
$,y,b
$,.*/,b
$,:,b
!0.0232K
#0.3007.
$,.*/
In this case it is obtained
C "0.205.
A
(3) As regards the load L in watts-hour (Wh), and the general coefficients for losses in
the system g , the size of the generator in watts-peak (¼ ) can be expressed as follows:
-044
P
¼ "C ¸g
P
A -044
and the battery capacity, in amperes hour, can be calculated using the expression:
C ¸
B
C "
A) g D »
#~$ .!9
is the maximum
where, g — is the efficiency of charge—discharge of the battery, D
.!9
#$
depth of discharge of the battery and » is the nominal voltage of battery.
214 M. Sidrach-de-Cardona, Ll. M. Lo& pez/Solar Energy Materials and Solar Cells 55 (1998) 199—215
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