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Loss-of-load probability model for stand-alone photovoltaic
systems in Europe
J.H. Lucio

, R. Valde´s, L.R. Rodrı ´guez
Department of Physics, Polytechnic School of Engineering, University of Burgos, C/Villadiego s/n, 09001 Burgos, Spain
Received 7 November 2011; received in revised form 21 May 2012; accepted 22 May 2012
Available online 28 June 2012
Communicated by: Associate Editor Nicola Romeo
Abstract
Sizing of stand-alone photovoltaic systems (SAPVS) requires knowledge of their reliability. Because of primary influence of the solar
irradiance and meteorological conditions, simulations are the best way to compute an accurate reliability for a given location and fixed
sizing parameters. These studies have been developed for more than two decades, but have had a narrow geographical applicability.
In this paper, we perform a complete (in time and space) simulation of a standard SAPVS in Europe using 23 year radiation data
corresponding to almost 2300 geographical points. At each point, the tilt angle that maximizes the energy reaching the PV array in
December is estimated and the relation among sizing parameters and reliability is computed for wide ranges of values. Finally, multilayer
perceptrons are trained for both computations, allowing (after their training) simple and fast estimations of the sizing parameters of this
type of plants for any location in Europe.
The procedure presented in this paper, although focused particularly in Europe, can be easily extended to almost any other region in
the world.
Ó 2012 Elsevier Ltd. All rights reserved.
Keywords: Stand-alone PV systems (SAPVS); Loss-of-load probability (LLP); Multilayer perceptron (MLP)
1. Introduction
Autonomous or stand-alone photovoltaic systems (SAP-
VS) are installations with photovoltaic modules and batter-
ies designed to fit some load without any connection to the
electric grid. Even in Europe, those systems are interesting
for powering stations, plants or houses in rural areas,
where the grid is distant or is not very reliable. Additional
benefits of this kind of systems are independence and its
energy-saving and environmentally friendly character.
A SAPVS is characterized by two dimensionless param-
eters: C
S
, related to the capacity of the storage system, and
C
A
(b), the mean or minimum capacity of the PV panels
array, defined:
C
U
=
def
N
B
V
B
C
B
DOD; (1a)
C
S
=
def
C
U
L
; and (1b)
C
A
(b) =
def
gA
´
G
d
(b)
L
; (1c)
where N
B
is the number of batteries (supposed all equal),
V
B
the nominal voltage of one battery (in V), C
B
the nom-
inal or rated capacity of each battery (charge dimension, in
C = As), DOD the maximum allowable depth of discharge
of each battery (dimensionless), C
U
the maximum useful
capacity of the batteries (energy dimension, in J), L the
mean daily energy load (in J), g is the average whole energy
transmission efficiency of the PV system from the PV array
to the load, A the array area (in m
2
), and
´
G
d
(b) is a
0038-092X/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved.
http://dx.doi.org/10.1016/j.solener.2012.05.021

Corresponding author. Tel.: +34 947 259425.
E-mail address: [email protected] (J.H. Lucio).
www.elsevier.com/locate/solener
Available online at www.sciencedirect.com
Solar Energy 86 (2012) 2515–2535
Nomenclature
a; b parameters in Liu–Jordan correlation
A total area of PVA, m
2
AG auxiliary generator
ANN artificial neural network
b bias of a neuron
B beam irradiation, J=m
2
B
d
(0) daily beam irradiation on horizontal plane, J=m
2
B
d
(b) daily beam irradiation on tilted plane, J=m
2
B
e;d
(0) e.t. daily solar irradiation, horizontal plane,
J=m
2
B
l
e;h
average e.t. hourly solar irradiation, normal inci-
dence, J=m
2
B
h
(0) hourly beam irradiation on horizontal plane,
J=m
2
B
h
(b) hourly beam irradiation on tilted plane, J=m
2
BS batteries system
C
A
(0) capacity of the PVA (horizontal) relative to the
load
C
A
(b) capacity of the PVA relative to the load
C
B
nominal or rated capacity of each battery,
C = A s
C
S
capacity of the BS (C
U
) relative to the load (L)
C
U
maximum useful (energy) capacity of the BS, J
D diffuse irradiation, J=m
2
D
d
(0) daily diffuse irradiation on horizontal plane,
J=m
2
D
d
(b) daily diffuse irradiation on tilted plane, J=m
2
D
h
(0) hourly diffuse irradiation on horizontal plane,
J=m
2
D
h
(b) hourly diffuse irradiation on tilted plane, J=m
2
DOD maximum allowable depth of discharge of each
battery
E
aux
daily energy provided (if any) by AG to BS, J
E
aux;T
total energy provided by AG to BS, J
e.t. extraterrestrial (outside the atmosphere)
ET equation of time, h
f parameter in isoLLP model, Eq. (16)
f
1
; f
2
parameters in Eq. (5)
G global (total) irradiation, J=m
2
G
d
(0) daily global irradiation on horizontal plane,
J=m
2
G
d
(b) daily global irradiation on tilted plane, J=m
2
´
G
d
(0) representative daily insolation on the horizontal
plane, J=m
2
´
G
d
(b) representative daily insolation on the PVA
plane, J=m
2
G
d;Dec
(0) average of December-daily global irrad. hori-
zontal plane, J=m
2
G
d;m
(b) monthly average daily global irradiation on
tilted plane, J=m
2
G
h
(0) hourly global irradiation on horizontal plane,
J=m
2
G
h
(b) hourly global irradiation on tilted plane, J=m
2
h hour index in the day (h = 1; . . . ; 24)
H number of neurons in the hidden layer of the
MLP
j day index in simulation (j = 1; . . . ; N
days
)
k
1
parameter in Klucher’s model
K
Dd
daily fraction of diffuse to global irradiation
K
Td
daily clearness index
K
Td
average of daily clearness index
K
Td;y
yearly average of daily clearness index
L load (physical system)
L energy load (mathematical variable), J
LLP loss of load probability of plant
LLPt target value of LLP in that algorithm
M number of neurons in the output layer of the
MLP
MLP multilayer perceptron
n day index in the year (1st January is n = 1)
n unit vector normal to the PVA plane
N number of inputs in the MLP
N
B
number of batteries in BS
N
days
total number of days in the simulation
N
points
number of selected points in Europe (2280)
P number of patterns in MLP training
PV photovoltaic
PVA photovoltaic panels array
r
D
; r
G
parameters in Liu–Jordan correlation
R albedo irradiation, J=m
2
R
h
(b) hourly albedo irradiation on tilt plane, J=m
2
SAPVS stand-alone photovoltaic systems
SOC state of charge of BS
SOC
j
state of charge of BS at the end of day j (post-
load)
SOC
/
j
state of charge of BS at sunset of day j (pre-load)
u parameter in isoLLP model, Eq. (16)
u
1
; u
2
parameters in Eq. (5)
V
B
nominal voltage of each battery in BS, V
w
i
weight i in neuron
x
i
input i in neuron/MLP
y output in neuron
y
i
output i in MLP
a PVA azimuth angle (supposed null in this work),
rad
b; b
g
PVA tilt angle (0 6 b 6
p
2
), rad, °
b
opt
b value that maximises

j÷Dec
G
d;j
(b), rad
C day angle, rad
d declination, rad
D
÷
SOC
(positive) increment in SOC due to solar energy
income

0
squared ratio of mean-to-actual Sun–Earth dis-
tance
g global efficiency of PVA
h
s
Sun’s zenith relative to PVA plane, rad
h
zs
Sun’s zenith, rad
2516 J.H. Lucio et al. / Solar Energy 86 (2012) 2515–2535
representative daily insolation on the plane (with tilt b) of
the array (in J=m
2
).
The physical sense of C
S
and C
A
(b) is clear: C
S
repre-
sents the number of days the batteries at full capacity
and with no energy income could feed the load (assumed
constant), while C
A
(b) means the number of loads (each
one with value L) that are expected to be fed by the PV
array alone. We are concerned with fixed tilt plants (no
Sun tracking), so b is supposed to be constant for a certain
installation.
Regarding this last definition, there are basically two
approaches: one, with a monthly basis, in which
´
G
d
(b) is
taken as the monthly average of daily values (Klein and
Beckman, 1987), G
d;m
(b); and another approach, with an
annual basis, in which
´
G
d
(b) is usually taken as the worst
month mean, following Chapman (1989) and many other
authors. The reason for this election is that worst month
mean was found to be most correlated to irradiation on
PVplane. Additionally, worst month mean is a good indica-
tor of the reliability of the installation, as it gives a good
estimate of the systeminput in the worst conditions. We will
follow the second (annual) approach and take December as
the worst month, as it is typically the month with lowest
irradiation and therefore with lowest C
A
(b) for a given loca-
tion at Northern Hemisphere. Thus, C
A
(b) would mean the
minimum(monthly average) number of loads (each one with
value L) that could be fed by the PV array alone.
As the real irradiation for several tilt angles is seldom
known, the following parameter related to the irradiation
on horizontal surface, G(0), is defined:
C
A
(0) =
def
gA
´
G
d
(0)
L
= C
A
(b)
´
G
d
(0)
´
G
d
(b)
(2)
C
S
and C
A
(b), or C
A
(0), are called the sizing parameters of
the PV plant, taking into account that both are referred to
the daily load (L) and that C
A
(b) (and C
A
(0)) is obviously
dependent on the insolation.
Because of the irregularity of solar radiation, some days
the energy absorbed is too low and batteries may deplete.
For long periods, a (usually small) fraction of the energy
demanded by the load would become uncovered. The
loss-of-load probability (LLP) is defined as the probability
of the system (photovoltaic generator plus a battery set) fail-
ing to power the load, therefore it is complementary of the
reliability concept. For a given pair of the sizing parame-
ters, i.e. (C
S
; C
A
(0)), LLP can be estimated as the ratio
of the energy not served by the ‘original’ system (i.e., served
by the auxiliary generator) to the total demanded energy:
LLP =
total energy deficit
total energy demand
: (3)
In the following, for the sake of clarity, we will take LLP
estimation through Eq. (3) as the true LLP value.
Eq. (3) is usually focused on a long time period, typically
one year or, better, some decades. In many studies,
functions relating C
S
; C
A
(0) and LLP have been proposed.
Some authors (see, for instance, Bucciarelli, 1984; Gordon,
1986; Barra et al., 1984 or Bartoli et al. (1984)) tried a gen-
eral (analytic) probabilistic approach, but their methods
have several drawbacks, including the hypothesis of con-
sidering that the daily array output is a normally distrib-
uted variable. On the other hand, most of the work on
the (C
S
; C
A
(0), LLP) relationship is based on numerical
methods (Chapman, 1989; Egido and Lorenzo, 1992); these
authors perform simulations with real or artificial insola-
tion data in order to compute an LLP value for each given
pair (C
S
; C
A
(0)). A few decades of irradiation data are
enough in order to get accuracy for LLPJ0:01 (Klein
and Beckman, 1987). Other authors perform simulations
with synthetic (hourly or daily) data generated from
long-term averages (Cabral et al., 2010).
Egido and Lorenzo (1992) simulated a photovoltaic
plant for 42 locations in Spain and fitted simple but signif-
icant models:
C
A
(0) = fC
÷u
S
; (4)
for each LLP value (i.e., for constant LLP), and
f = f
1
÷f
2
log
10
LLP; (5a)
q reflectivity of the ground (supposed 0.2)
r; r
g
longitude (positive East), rad, °
r
K
Td
standard deviation of daily clearness index
r
st
; r
st;g
longitude of meridian of local standard time,
rad, °
/; /
g
latitude (positive North), rad, °
u() activation function
x solar hour angle (< 0 in the morning), rad
x
h
solar hour (0 h at noon), h
x
s
sunrise angle (< 0), rad
x
ss
sunrise angle relative to the PV plane (tilted an-
gle b), rad
x
ss;h
sunrise time relative to the PV plane (12 h at
noon), h
x
sp;h
sunset time relative to the PV plane (12 h at
noon), h
Sub and superscripts
d relative to a day
D relative to diffuse irradiation
Dec relative to December
e extraterrestrial
g expressed in degrees
G relative to global irradiation
h relative to an hour
m relative to a month
s relative to the Sun
l on plane normal to the rays
J.H. Lucio et al. / Solar Energy 86 (2012) 2515–2535 2517
u = exp(u
1
÷u
2
LLP); (5b)
for any LLP value. Thus the parameters f and u depend on
the location and on the LLP value, but the parameters
f
1
; f
2
; u
1
and u
2
depend only on the location, not on
LLP. Finally, these authors derived maps of f and u for
LLP = 0:1 and LLP = 0:01 for Spain.
Following a similar approach, Hontoria et al. (2005)
went one step ahead and tried to build a unique model
for isoLLP curves corresponding to the Iberian Peninsula.
They trained an MLP (multilayer perceptron, see Section
2.3) with one 3-variable (C
S
, LLP and K
Td;y
) input layer,
one 9-neuron hidden-layer and one 1-variable (C
A
(0)) out-
put layer. K
Td;y
is the yearly average clearness index,
defined as the yearly average of the quotients between the
daily total radiative energy arriving at a point and the daily
total irradiation arriving at top of the atmosphere, both at
an horizontal plane (see Duffie and Beckman, 1991). In
order to obtain the LLP data, they built a PV plant model
for as few as seven locations in Spain and few more than
10 years of data. Three other locations were reserved for
model validation. These authors found scarce coincidence
with (Egido and Lorenzo, 1992) results.
Many works include detailed estimations and optimisa-
tions for realistic scenarios: authors in Notton et al. (1998)
study the cost of a whole SAPVS; authors in Sidrach-
De-Cardona and Mora-Lo´ pez (1999) fit multivariate mod-
els to estimate optimal tilt angle and C
A
parameter for a
number of locations in USA, obtaining good correlations;
authors in Agha and Sbita (2000) study the optimisation
under different load patterns and different tilt angles, for
several locations in Libya; Celik (2007) also analyzes the
effect of different load profiles on the LLP; Posadillo and
Lo´pez Luque (2008) and Posadillo and Lo´ pez Luque
(2008) analyze the case of variable demand, variable tilt
angle and monthly LLP; Chel et al. (2009) and Chel and
Tiwari (2011) perform a whole analysis of SAPVS, includ-
ing costs, energy production, energy payback time, life-
times and CO
2
emissions; Cabral et al. (2010) perform a
stochastic approach for the sizing of SAPVS. There is a
growing number of works (Mellit et al., 2005, 2007a,b,
2008, 2009, 2010; Mellit and Benghanem, 2007; Mellit
and Kalogirou, 2008; Mellit, 2008; Benghanem and Mellit,
2010) that employ artificial neural networks to estimate the
sizing parameters of SAPVS. Recently, some authors have
adopted statistical approaches based on specific theories,
such as the extreme value theory (Fragaki and Markvart,
2008; Chen, 2009, 2012), getting reliable results.
In this paper, we will take a simple and thus more gen-
eral approach, assuming constant daily load, constant (the
optimal) tilt angle and null azimuth angle. Any particular
load profile or tilt angle can be fed to the algorithm to
get the consequent isoLLP curves. We present the results
of simulations of a plant with a PV array and a system
of batteries, similar to that simulated by Egido and Lore-
nzo (1992) and Hontoria et al. (2005), based on time series
of real hourly-averages radiation data on the horizontal
plane. However, unlike previous authors, we have focused
on an entire continent, particularly Europe, including its
seashores. For each geographical point, we have computed
firstly the optimal tilt angle (b
opt
) for the PV array in the
worst month and, finally, the LLP surface for the plane
(C
S
; C
A
(0)). We have trained two types of MLP’s: one
for the prediction of b
opt
(Section 6.1) and another one
for the prediction of LLP curves (Section 6.2).
2. Simulations: model, data and artificial neural networks
According to the method followed by Lorenzo, 1994, for
each geographical point we have performed a great number
of simulations in the plane (C
S
; C
A
(0)), in order to esti-
mate C
A
(0) for each C
S
value and each LLP target value.
Unlike cited authors, our goal has been almost whole Eur-
ope. Thus isoLLP lines are found for typical LLP values
(0.01, 0.05 and 0.1). Then, for each curve and through
Eq. (16), parameters f and u are estimated. Finally, Eq.
(5) are applied for LLP values 0.01 and 0.1 in order to esti-
mate the values of f
1
; f
2
; u
1
and u
2
.
2.1. Model
For the simulation, we have implemented the simple
model with a photovoltaic array (PVA), a battery system
(BS), a constant load (L) and an auxiliary generator
(AG) used by other authors (Egido and Lorenzo, 1992),
shown in Fig. 1. Both PVA and AG are connected to BS,
and this one supplies energy directly to L. In practice, a
power conditioner must be connected to PVA, AG and
BS, to adapt the different electrical powers, but we will
consider its presence only through the average whole
energy transmission efficiency g, Eq. (1c).
Everyday irradiation impinges on PVA, which charges
BS (if possible, up to its full capacity), and this one tries
to feed L. When the energy provided by BS is less than
L, AG starts up and supplies the difference. As it is usual
in the bibliography, we consider a constant daily load, L.
An important parameter characterizing BS is the so-called
state of charge, SOC: it is defined as the energy that BS can
Fig. 1. Simulated model.
2518 J.H. Lucio et al. / Solar Energy 86 (2012) 2515–2535
provide to the load relative to the maximum energy it could
provide at full charge.
The model used for simulation has a daily basis and lies
on two main assumptions: the consumption occurs at night
and BS has no losses. Each day BS increases its SOC due to
the solar energy income during daylight (D
÷
SOCj
, being SOC
restricted by its maximum: 1) and then reduces it due to
load feed at night (L=C or a fraction of it, being SOC
restricted by its minimum: 0). At sunset, if the available
energy in BS is greater than or equal to the load (i.e.
SOC PL=C), no extra energy will be needed from AG.
Otherwise, AG has to provide the energy left to BS and
then its charge vanishes (SOC becomes 0). This criterion
keeps AG supply to a minimum.
Let E
aux;j
be the energy provided by the AG on day j,
SOC
/
j
be the SOC at sunset of day j (before the feed of
L) and SOC
j
be the SOC at the end of day j (after the feed
of L). Then the following equations apply:
D
÷
SOC
j
=
gAG
d;j
(b)
C
=
C
A
(b)G
d;j
(b)
´
G
d
(b)C
S
=
C
A
(0)G
d;j
(b)
´
G
d
(0)C
S
; (6)
SOC
/
j
= min SOC
j÷1
÷D
÷
SOCj
; 1
_ _
; (7)
SOC
j
= max SOC
/
j
÷
L
C
; 0
_ _
; (8)
E
aux;j
= max L ÷CSOC
/
j
; 0
_ _
: (9)
After applying Eqs. (9), (7)–(9) for all the periodof sim-
ulation (with a total of N
days
days, i.e. j = 1; . . . ; N
days
), fi-
nally LLP is computed simply by summing all the energy
provided by the AG and dividing the result by all the en-
ergy demanded by the load:
LLP =

N
days
j=1
E
aux; j
N
days
L
: (10)
This model differs from those in Egido and Lorenzo (1992),
Lorenzo (1994) or Lorenzo and Narvarte (2000), in two
points:
(a) we have taken into account the possible saturation of
BS by day, before sunset, and thus before the load
feed, Eq. (7);
(b) we have supposed that AG supplies only the energy
left to L, no more to charge BS, Eq. (9).
−20 −10 0 10 20 30 40 50 60 70
30
40
50
60
70
Fig. 2. Map of Europe with the points used for simulation.
4000 5000 6000 7000 8000 9000 10000 11000 12000 13000 14000
0
2
4
x 10
7
Daily irradiation (J/m
2
) at Madrid, from WRDC
4000 5000 6000 7000 8000 9000 10000 11000 12000 13000 14000
0
2
4
x 10
7
Daily irradiation (J/m
2
) at Madrid, from NASA
time (days since 1−1−1970)
Fig. 3. Irradiation data for Madrid from WRDC and from NASA.
J.H. Lucio et al. / Solar Energy 86 (2012) 2515–2535 2519
Several authors have performed simulations with more
sophisticated models (including one hour step, tempera-
ture influence on PVA and BS, nonlinearities in some
components, variable load and more), but implementing
a more exact model improves the representation of a par-
ticular plant, but no more generality or predictability are
achieved (see Lorenzo and Narvarte, 2000). Furthermore,
Ambrosone et al. (1985) found that a daily-based simula-
tion considering constant g yielded similar LLP than an
hourly-based simulation considering the influence of tem-
perature, radiation and the SOC on g, especially for
C
S
P2.
In order to run the simulation, values of daily solar irra-
diation on the plane of the PVA, G
d
(b), must feed the
model. These values are seldom available and are often
estimated from values on horizontal plane, G
d
(0), by well
known correlations (see, for instance, Collares-Pereira
and Rabl, 1979; Liu and Jordan, 1960; Klucher, 1979;
Hay and Davies, 1980; Duffie and Beckman, 1991;
Lorenzo, 1994 or Muneer, 2004). Let n be the unit vector
normal to the PVA plane, pointing outward. Firstly, astro-
nomical and PVA angular position quantities (solar decli-
nation (d), Earth-Sun distance correction (
0
), solar hour
angle (x), Sun’s zenith (h
zs
), sunrise angle (x
s
) and the an-
gle between Sun and n (h
s
)) are computed starting from the
day number in the year (n), geographical coordinates and
PVA orientation angle (b, tilt or angle between n and the
vertical). PVA azimuth (a), i.e. the angle between the pro-
jection of n on horizontal plane and the vector pointing to-
wards Equator, is supposed null.
Extraterrestrial daily irradiation on horizontal plane is
easily computed and then daily global and daily diffuse irra-
diations on horizontal plane at ground are estimated
through clearness index and diffuse fraction (for instance,
by the Collares–Pereira–Rabl correlation (Collares-Pereira
and Rabl, 1979)). Hourly irradiation values on horizontal
plane can be estimated from daily ones through Liu–Jordan
correlations (Liu and Jordan, 1960). Later, hourly irradia-
tion values on tilted plane for any b can be estimated from
previous values through different correlations for the three
components: direct, diffuse and albedo (diffuse component
can be estimated by means of an anisotropic model like Klu-
cher’s (1979) or Hay and Davies’ (1980); albedo component
is computed supposing a reflectivity of the ground q = 0:2).
Simply adding these components, one gets hourly global
irradiation on tilted surface. Finally, daily global irradia-
tion is estimated summing hourly values from sunrise to
sunset. All these equations are summarized in Appendix B.
2.2. Data
Usually, as it was mentioned above, irradiations are only
known at notilt angle (i.e. horizontal plane, b = 0). The sim-
ulations start with daily values of irradiation served by
NASA (2011) for a regular grid of equally spaced points
1
·
latitude ×1
·
longitude. The grid covers almost all Eur-
ope (all continental Europe plus Great Britain, Ireland,
and the Mediterranean islands), to the East up to Ural
Mountains and to the North up to latitude 63:5
·
, as can
be seen in Fig. 2 (Iceland and Northern lands have been ex-
cluded because they have too many days with little or no
radiation at all and the annual LLP concept for plants in
these areas would have no sense). Finally, the area selected
for the simulation comprises a total of 2280 points, includ-
ing part of the Mediterranean Sea and part of the Atlantic
Ocean.
Data are daily and cover the period from July 1, 1983 to
June 30, 2006 (23 years). For each point, data are served as
Table 1
Several stations for which irradiation data from NASA (2011) and WRDC (2011) are compared. The name of each location (and country, in first column)
and its spacial coordinates (latitude, longitude and elevation, columns 2–4) are shown. The fifth column shows the correlation coefficient between both
sources of data, corresponding to graphics in Fig. 4.
Location: number and name Lat. (°) Long. (°) Elev. (m) Corr. coeff.
1. Madrid (Spain) 40.45 ÷3.72 664 0.9597
2. Oviedo (Spain) 43.35 ÷5.87 335 0.8953
3. Uccle (Belgium) 50.80 4.35 100 0.9726
4. Sofia (Bulgaria) 42.65 23.38 586 0.9059
5. Zagreb (Croatia) 45.82 16.03 123 0.9509
6. Hradec Kralove (Czechia) 50.25 15.85 241 0.9640
7. Copenhagen (Denmark) 55.67 12.30 28 0.9734
8. Jokioinen (Finland) 60.82 23.50 104 0.9699
9. Bordeaux (France) 44.83 ÷0.70 49 0.9736
10. Macon (France) 46.30 4.80 221 0.9674
11. Bremen (Germany) 53.05 8.80 3 0.9644
12. Hohenpeissenberg (Germany) 47.80 11.02 977 0.9536
13. Aberdeen (United Kingdom) 57.20 ÷2.22 65 0.9333
14. Athens (Greece) 37.97 23.72 107 0.9628
15. Budapest (Hungary) 47.43 19.18 138 0.9610
16. Kilkenny (Ireland) 52.67 ÷7.27 63 0.9619
17. Bologna (Italy) 44.53 11.30 36 0.9306
18. De Bilt (Netherlands) 52.10 5.18 2 0.9508
19. Lisboa (Portugal) 38.72 ÷9.15 77 0.9406
20. Samara (Russia) 53.25 50.45 44 0.9591
2520 J.H. Lucio et al. / Solar Energy 86 (2012) 2515–2535
an ASCII file, variables arranged in columns. The variables
are: year, month, day of month and global daily irradiation
on horizontal surface at ground level (symbol G
d
(0)).
The NASA’s solar energy data “is generated using the
Pinker/Laszlo shortwave algorithm. Cloud data is taken
from the International Satellite Cloud Climatology Project
DX dataset (ISCCP). ISCCP DX data is on an equal area
grid with an effective 30 km×30 km pixel size. The output
data is generated on a nested grid containing 44 016
regions. The nested grid has a resolution of 1
·
latitude
globally, and longitudinal resolution ranging from 1
·
in
the tropics and subtropics to 120
·
at the poles”.
Finally, these data are interpolated to get the final
1
·
×1
·
resolution. The great advantages of NASA’s data
are: they are global (available for any point on Earth),
continuous in time, reliable (within their accuracy), fast
to get and free (public access).
Daily clearness index data (for the estimation of K
Td
and
r
K
Td
) can be downloaded from NASA web page (NASA,
2011) too, or calculated through Eqs. (A.11) and (A.12).
2.2.1. Validation of irradiation data
Radiation satellite data from NASA (2011) used in this
work have been compared to data measured at ground sta-
tions (stored at World Radiation Data Center (WRDC,
2011), which manages solar radiation data collected at over
1000 measurement sites throughout the world), showing in
general good accordance. In order to compare both sources
of data, for each selected station in WRDC, the closest four
points of NASAgrid were chosen, and a linear interpolation
0 0.5 1 1.5 2 2.5 3 3.5
x 10
7
0
0.5
1
1.5
2
2.5
3
3.5
x 10
7
Comparison of irradiation values at Madrid
daily irradiation (J/m
2
), from WRDC
d
a
i
l
y

i
r
r
a
d
i
a
t
i
o
n

(
J
/
m
2
)
,

f
r
o
m

N
A
S
A
Comparison of irradiation values at Oviedo (Spain) Comparison of irradiation values at Uccle (Belgium) Comparison of irradiation values at Sofia (Bulgaria)
Comparison of irradiation values at Zagreb (Croatia) Comparison of irradiation values at Hradec Kralove (Czechia) Comparison of irradiation values at Copenhagen (Denmark) Comparison of irradiation values at Jokioinen (Finland)
Comparison of irradiation values at Bordeaux (France) Comparison of irradiation values at Macon (France) Comparison of irradiation values at Bremen (Germany) Comparison of irradiation values at Hohenpeissenberg (Germany)
0 0.5 1 1.5 2 2.5 3 3.5
x 10
7
0
0.5
1
1.5
2
2.5
3
3.5
x 10
7
daily irradiation (J/m
2
), from WRDC
d
a
i
l
y

i
r
r
a
d
i
a
t
i
o
n

(
J
/
m
2
)
,

f
r
o
m

N
A
S
A
0 0.5 1 1.5 2 2.5 3 3.5
x 10
7
0
0.5
1
1.5
2
2.5
3
3.5
x 10
7
daily irradiation (J/m
2
), from WRDC
d
a
i
l
y

i
r
r
a
d
i
a
t
i
o
n

(
J
/
m
2
)
,

f
r
o
m

N
A
S
A
0 0.5 1 1.5 2 2.5 3 3.5
x 10
7
0
0.5
1
1.5
2
2.5
3
3.5
x 10
7
daily irradiation (J/m
2
), from WRDC
d
a
i
l
y

i
r
r
a
d
i
a
t
i
o
n

(
J
/
m
2
)
,

f
r
o
m

N
A
S
A
0 0.5 1 1.5 2 2.5 3 3.5
x 10
7
0
0.5
1
1.5
2
2.5
3
3.5
x 10
7
daily irradiation (J/m
2
), from WRDC
d
a
i
l
y

i
r
r
a
d
i
a
t
i
o
n

(
J
/
m
2
)
,

f
r
o
m

N
A
S
A
0 0.5 1 1.5 2 2.5 3 3.5
x 10
7
0
0.5
1
1.5
2
2.5
3
3.5
x 10
7
daily irradiation (J/m
2
), from WRDC
d
a
i
l
y

i
r
r
a
d
i
a
t
i
o
n

(
J
/
m
2
)
,

f
r
o
m

N
A
S
A
0 0.5 1 1.5 2 2.5 3 3.5
x 10
7
0
0.5
1
1.5
2
2.5
3
3.5
x 10
7
daily irradiation (J/m
2
), from WRDC
d
a
i
l
y

i
r
r
a
d
i
a
t
i
o
n

(
J
/
m
2
)
,

f
r
o
m

N
A
S
A
0 0.5 1 1.5 2 2.5 3 3.5
x 10
7
0
0.5
1
1.5
2
2.5
3
3.5
x 10
7
daily irradiation (J/m
2
), from WRDC
d
a
i
l
y

i
r
r
a
d
i
a
t
i
o
n

(
J
/
m
2
)
,

f
r
o
m

N
A
S
A
0 0.5 1 1.5 2 2.5 3 3.5
x 10
7
0
0.5
1
1.5
2
2.5
3
3.5
x 10
7
daily irradiation (J/m
2
), from WRDC
d
a
i
l
y

i
r
r
a
d
i
a
t
i
o
n

(
J
/
m
2
)
,

f
r
o
m

N
A
S
A
0 0.5 1 1.5 2 2.5 3 3.5
x 10
7
0
0.5
1
1.5
2
2.5
3
3.5
x 10
7
daily irradiation (J/m
2
), from WRDC
d
a
i
l
y

i
r
r
a
d
i
a
t
i
o
n

(
J
/
m
2
)
,

f
r
o
m

N
A
S
A
0 0.5 1 1.5 2 2.5 3 3.5
x 10
7
0
0.5
1
1.5
2
2.5
3
3.5
x 10
7
daily irradiation (J/m
2
), from WRDC
d
a
i
l
y

i
r
r
a
d
i
a
t
i
o
n

(
J
/
m
2
)
,

f
r
o
m

N
A
S
A
0 0.5 1 1.5 2 2.5 3 3.5
x 10
7
0
0.5
1
1.5
2
2.5
3
3.5
x 10
7
daily irradiation (J/m
2
), from WRDC
d
a
i
l
y

i
r
r
a
d
i
a
t
i
o
n

(
J
/
m
2
)
,

f
r
o
m

N
A
S
A
Fig. 4. Comparison of irradiation data from WRDC and from NASA for 20 locations in Europe, detailed in Table 1.
J.H. Lucio et al. / Solar Energy 86 (2012) 2515–2535 2521
was done to estimate the radiation values at the geographi-
cal location of the station (in Fig. 3 an example of compar-
ison of irradiation data from both sources is shown).
In Table 1 we present the name and geographical char-
acteristics of 20 stations throughout Europe, for which we
compare daily irradiation data from NASA (2011) to the
same from WRDC (2011), obviously for the same days.
In Fig. 4 both sets of irradiation data are represented for
each location, and the resulting correlation coefficient is
shown in last column of Table 1.
2.3. Artificial neural network: MLP
Since seven decades ago, scientists have emulated some-
how biological neural networks through simplified models.
Many artificial neural networks (ANN from now on) have
been proposed and used for different purposes, being func-
tion approximation and pattern recognition the main ones
(Haykin, 2009). ANN’s are basically parallel architectures
of simple processors, the neurons. The two strengths of
ANN’s are parallelism and nonlinearity (some of them also
include a third feature: feedback).
Each neuron in the model is fed by a number of inputs
(emulating the synaptic connections through dendrites)
and yields one output (emulating synaptic connections
through the axon), see Fig. 5a. Typically, the artificial
0 0.5 1 1.5 2 2.5 3 3.5
x 10
7
0
0.5
1
1.5
2
2.5
3
3.5
x 10
7
Comparison of irrad. values at Aberdeen (United Kingdom)
daily irradiation (J/m
2
), from WRDC
d
a
i
l
y

i
r
r
a
d
i
a
t
i
o
n

(
J
/
m
2
)
,

f
r
o
m

N
A
S
A
Comparison of irradiation values at Athens (Greece) Comparison of irradiation values at Budapest (Hungary) Comparison of irradiation values at Kilkenny (Ireland)
Comparison of irradiation values at Bologna (Italy) Comparison of irradiation values at De Bilt (Netherlands) Comparison of irradiation values at Lisboa (Portugal) Comparison of irradiation values at Samara (Russia)
0 0.5 1 1.5 2 2.5 3 3.5
x 10
7
0
0.5
1
1.5
2
2.5
3
3.5
daily irradiation (J/m
2
), from WRDC
d
a
i
l
y

i
r
r
a
d
i
a
t
i
o
n

(
J
/
m
2
)
,

f
r
o
m

N
A
S
A
0 0.5 1 1.5 2 2.5 3 3.5
x 10
7
0
0.5
1
1.5
2
2.5
3
3.5
daily irradiation (J/m
2
), from WRDC
d
a
i
l
y

i
r
r
a
d
i
a
t
i
o
n

(
J
/
m
2
)
,

f
r
o
m

N
A
S
A
0 0.5 1 1.5 2 2.5 3 3.5
x 10
7
0
0.5
1
1.5
2
2.5
3
3.5
daily irradiation (J/m
2
), from WRDC
d
a
i
l
y

i
r
r
a
d
i
a
t
i
o
n

(
J
/
m
2
)
,

f
r
o
m

N
A
S
A
0 0.5 1 1.5 2 2.5 3 3.5
x 10
7
0
0.5
1
1.5
2
2.5
3
3.5
daily irradiation (J/m
2
), from WRDC
d
a
i
l
y

i
r
r
a
d
i
a
t
i
o
n

(
J
/
m
2
)
,

f
r
o
m

N
A
S
A
0 0.5 1 1.5 2 2.5 3 3.5
x 10
7
0
0.5
1
1.5
2
2.5
3
3.5
daily irradiation (J/m
2
), from WRDC
d
a
i
l
y

i
r
r
a
d
i
a
t
i
o
n

(
J
/
m
2
)
,

f
r
o
m

N
A
S
A
0 0.5 1 1.5 2 2.5 3 3.5
x 10
7
0
0.5
1
1.5
2
2.5
3
3.5
daily irradiation (J/m
2
), from WRDC
d
a
i
l
y

i
r
r
a
d
i
a
t
i
o
n

(
J
/
m
2
)
,

f
r
o
m

N
A
S
A
0 0.5 1 1.5 2 2.5 3 3.5
x 10
7
0
0.5
1
1.5
2
2.5
3
3.5
daily irradiation (J/m
2
), from WRDC
d
a
i
l
y

i
r
r
a
d
i
a
t
i
o
n

(
J
/
m
2
)
,

f
r
o
m

N
A
S
A
x 10
7
x 10
7
x 10
7
x 10
7
x 10
7
x 10
7
x 10
7
Fig 4. (continued)
(a) (b)
Fig. 5. Neuron model. (a) External scheme. (b) Internal scheme showing
weights, bias and activation function.
Fig. 6. MLP model.
2522 J.H. Lucio et al. / Solar Energy 86 (2012) 2515–2535
neuron multiplies each input (x
i
) by a certain weight (w
i
)
and sums all the products, along with another value (b),
called bias. The result is transformed by a function (u(),
called activation function), as shown in Fig. 5b. u() can
be any function, but special performance is obtained when
it is nonlinear. Typical elections for u(s) are sigmoid
functions, as tanh(as) or the logistic function: 1=(1÷
exp(÷as)), where a is a free constant parameter. The neu-
ron output is thus given by the following expression:
y = u b ÷

n
i
w
i
x
i
_ _
: (11)
0
1
2
3
4
5
6
7
8
9
10
Fig. 7. Map of Europe with
´
G
d
(0) isolines (expressed in 10
6
J=m
2
). Values range from 0 to 10 in increments of 0.25. (For interpretation of the references to
color in this figure legend, the reader is referred to the web version of this article.)
10
15
20
25
Fig. 8. Map of Europe with b
opt
÷/ isolines (in degrees). Values range from 10 to 25 in increments of 1. (For interpretation of the references to color in
this figure legend, the reader is referred to the web version of this article.)
J.H. Lucio et al. / Solar Energy 86 (2012) 2515–2535 2523
ANN’s can operate in two modes: training (or learning)
and remembering (or simulation). In the former, the
weights and bias are adjusted in order to approximate
the outputs to certain given targets (supervised learning)
or to separate the input patterns into groups with common
features. Once trained, an ANN can apply its ‘knowledge’
to produce new outputs: this is called remembering. A
good training should produce good approximations for
any future input and not only for the training set: this is
known as generalization.
A widely used type of ANN is the multilayer feedfor-
ward network, being the most popular the multilayer per-
ceptron (called MLP). This ANN has several layers of
neurons: each layer output is the input of the following
one. The first and last layers are called respectively input
and output layers. There can be other layers inside: they
are called hidden layers. An MLP with at least one hidden
layer is capable of separating patterns nonlinearly. It has
been proved that an MLP with only one hidden layer can
virtually approximate any function (Funahashi, 1989)
(such an ANN is said to be a universal function approxi-
mator). Fig. 6 shows an MLP with one hidden layer. The
activation functions applied in MLP’s usually are sigmoid
in the hidden layers and linear in the output layer. We have
used this configuration.
MLP’s can be trained by several methods, being gradi-
ent descent and second-order the main groups. The former
(being BP, back propagation algorithm, the most widely
used) are simple and secure but slow. The latter (like
Newton, quasi-Newton or conjugate gradient) are more
complex (as usually the Hessian matrix or approximations
must be computed or estimated at each iteration) but fas-
ter. One of the most preferred methods is the Levenberg–
Marquardt algorithm (a mix between the Gauss–Newton
algorithm and the method of gradient descent), as it is
one of the fastest. Any method tries to reach the minimum
of the sum of squares of differences between outputs and
targets, but any of them can stop at a local minimum.
Usually, the set of patterns is divided into three groups
(the third one being optional): training, validation and test
(this method is known as cross-validation). Only the
training set is used in the optimization procedure to adjust
weights and bias of the neurons. The validation set is used
50 55 60 65 70 75 80 85 90
50
55
60
65
70
75
80
85
90
β
opt
(
º
) estimated from PVGIS
β
o
p
t

(

º

)

e
s
t
i
m
a
t
e
d

f
r
o
m

N
A
S
A

a
n
d

t
h
i
s

p
a
p
e
r
Fig. 9. Values of b
opt
for some points in Europe from NASA (this paper)
vs. the same from PVGIS.
Fig. 10. Map of Europe with f
0:1
isolines. Values range from 0.08 to 0.60 in increments of 0.02. (For interpretation of the references to color in this figure
legend, the reader is referred to the web version of this article.)
2524 J.H. Lucio et al. / Solar Energy 86 (2012) 2515–2535
to check the goodness of the fit and particularly to stop the
training procedure when the error in this set rises. The test
set can be useful to compare the results from the different
sets. Typically, the set of patterns feeds the training
algorithm several times and the individual patterns are ran-
domly permuted. Each resulting set of patterns (inputs and
targets), as well as the corresponding iteration in the train-
ing algorithm, is called an epoch. The ANN training proce-
dure usually stops at the first minimum of the error in the
validation group of patterns. This is called early-stopping
method.
About the number of neurons in the hidden layer,
bounds on approximation errors have been estimated (see
Haykin, 2009), giving that the size of the hidden layer is
optimized when H is
H ’ C
f
P
N log P
_ _1
2
; (12)
Fig. 11. Map of Europe with u
0:1
isolines. Values range from 0.00 to (0.27 in increments of 0.01. (For interpretation of the references to color in this figure
legend, the reader is referred to the web version of this article.)
Fig. 12. Map of Europe with f
0:01
isolines. Values range from 0.3 to 4.5 in increments of 0.1. (For interpretation of the references to color in this figure
legend, the reader is referred to the web version of this article.)
J.H. Lucio et al. / Solar Energy 86 (2012) 2515–2535 2525
where C
f
is the first absolute moment of the target function
of the approximation f (C
f
quantifies the smoothness of f ),
P is the number of samples or patterns and N is the number
of inputs of the MLP. Previous expression, among other
ideas, indicates that supposing C
f
, N and error bound con-
stant, H may be incremented only if P can be increased. In
our case, P will be fixed, so H should not be increased
arbitrarily. Other authors propose simpler estimations of
H, like (Kalogirou, 2006):
H ’
N ÷M
2
÷
ffiffiffi
P
_
; (13)
where M is the number of outputs of the MLP. As a gen-
eral rule, H should be great enough for a better approxi-
mation (generalization, is the ability to estimate outputs
different from the training patterns), but should be
bounded so as to avoid learning of the errors
(memorization).
About the inputs, they should be chosen among signifi-
cant variables influencing the outputs, but increasing the
number of inputs has to be done with caution, as a greater
dimensionality usually reduces the smoothness of the func-
tion to be approximated and a denser sampling (and so
more patterns) should be needed.
Fig. 13. Map of Europe with u
0:01
isolines. Values range from 0.00 to 0.90 in increments of 0.05. (For interpretation of the references to color in this figure
legend, the reader is referred to the web version of this article.)
−4
−3
−2
−1
0
1
2
3
4
Fig. 14. Map of Europe with [MLP1
output
÷(b
opt
÷/)[ isolines. Their values range from ÷3:5
·
to 4:3
·
in increments of 0:5
·
. (For interpretation of the
references to color in this figure legend, the reader is referred to the web version of this article.)
2526 J.H. Lucio et al. / Solar Energy 86 (2012) 2515–2535
3. Estimation of
´
G
d
(0)
Prior to any other computation, it is interesting here to
estimate the average irradiation on horizontal plane at
December,
´
G
d
(0), for all the points chosen in Europe
(Fig. 2), in order to relate it to the other parameters
appearing in Eq. (2).
The estimation is straightforward, we have just to com-
pute the mean of the daily irradiation values corresponding
to December for all the years. The values are represented as
isolines, in 10
6
J=m
2
units, in Fig. 7.
4. Estimation of b
opt
For continuous-operation PV plants (i.e., PV plants
running all days in the year, for several years), the LLP
concept is directly and mainly influenced by worst
time-periods (obviously, the plant will fail most the epochs
with least radiation), by convention the worst month. As
mentioned above, December is usually the worst month
in Northern Hemisphere, and we will take this as true for
all the points. If we fix the rest of parameters in the plant
(basically g and A in PVA, C
S
in BS and L), LLP would
be minimized by maximizing the irradiation at worst
month. Therefore, in this type of plants (with no Sun track-
ing) b should be chosen so that

Dic
G
d
(b), or equivalently
´
G
d
(b), is maximum. We have called this value b
opt
. Follow-
ing the algorithm shown in B, together with the equations
appearing in Appendix B (G
d
(b) estimation from G
d
(0)),
we have estimated b
opt
for all the points chosen in Europe,
with available data (commented in Section 2.2).
For simplicity, let’s call here y(b) =

Dic
G
d
(b). Particu-
larly, as the shape of function y(b) near the maximum is
very similar to quadratic, the method of maximisation
was rather simple:
1. first, we took the latitude / as the initial value for b
0
and chose a certain angle shift (Db, actually 5
·
) and
a maximum admitted error (db, actually 0:02
·
);
2. then, we computed y(b) for the three values:
b
1
= b
0
÷Db; b
2
= b
0
and b
3
= b
0
÷Db, obtain-
ing the corresponding ordinates y
1
; y
2
and y
3
;
3. we fitted a parabola (y = ab
2
÷bb ÷c) to the three
points;
4. then, we found the maximum of the parabola,
simply: b
max
= ÷
b
2a
= b
0
÷
(y
1
÷y
3
)
2(y
1
÷2y
2
÷y
3
)
Db;
5. if [b
max
÷b
0
[ < db, exit: the maximum has been
reached;
6. next, we move the central point of searching to the
previous maximum: b
0
= b
max
;
7. we decreased the increment in b (by a factor 0.8):
Db = 0:8Db;
8. the loop goes back to step 2 until the condition 5 is
met.
The previous algorithm never failed, and never needed
more than seven iterations to converge. Note that step 3
is only conceptual (nothing needs to be done at that step)
if the formula in step 4 is to be applied.
The result of b
opt
estimation, in the form of (b
opt
÷/)
isolines (after computing cubic interpolation in a 4 ×4 den-
ser grid), is shown in Fig. 8. Isoline values range from 10
·
to
25
·
in increments of 1
·
. It should be noted that this map
reflects the particular weather conditions (especially cloud-
iness) of each location. One can appreciate three main areas:
the Mediterranean area, in which (b
opt
÷/) is about 22
·
;
the continental area, in which (b
opt
÷/) is about 18
·
; and
the British Islands area, in which (b
opt
÷/) is about 13
·
.
In the map it is also apparent that, in general, b
opt
grows
slower than latitude. This could be caused mainly by the
greater weight of diffuse vs. direct component in the global
radiation at higher latitudes.
In the extreme Northern areas, the estimated values of
b
opt
should be considered with caution, as the correlation
equations used to estimate diffuse radiation (Eqs. (A.13),
(A.14), (A.24), (A.26), (A.30) and (A.31)) usually are
obtained for medium latitudes and therefore will have little
validity for high latitudes. An additional problem (in the
−4 −2 0 2 4
0
50
100
150
200
250
300
350
(a)
−0.6 −0.4 −0.2 0 0.2 0.4 0.6
0
500
1000
1500
2000
2500
3000
3500
4000
−0.6 −0.4 −0.2 0 0.2 0.4 0.6
0
2000
4000
6000
8000
10000
12000
(b)
(c)
Fig. 15. Histograms of MLP1 absolute errors (a), MLP2 relative errors (b), and MLP3 relative errors (c). The symmetry of the distributions is high in
MLP1 and MLP3 errors and low in MLP2 errors. In MLP1 and MLP3 cases, most of the errors are close to zero. In MLP2 case, errors are rather big. It is
clear the improvement from MLP2 to MLP3.
J.H. Lucio et al. / Solar Energy 86 (2012) 2515–2535 2527
sense of being a source of error) in b
opt
estimation for high
latitudes is the greater influence of the albedo component
due to persistent snow in winter. For that reason, the value
of ground reflectivity is expected to be much greater at
those latitudes, and much higher in winter than in summer
days. On the other hand, at those high latitudes (/J60
·
),
continuous-operation SAPVS have (at least economically)
little sense, as the radiation is extremely poor in winter
and, in order to yield a reasonable LLP value, C
S
should
be excessively high.
It should be noted that a little shift in b near the abscissa
in the maximum, b
opt
, yields very little change in the esti-
mated total irradiation (particularly, computations in sev-
eral points give changes lower than 1% in

G
d
(b) per
10
·
change in b). As

G
d
(b) is very similar to a parabola
near the maximum, we can derive that light deviations
from estimated value of b
opt
will cause negligible variations
in the dimensioning parameters C
A
(0) and LLP.
4.1. Validation of b
opt
Finally, in order to validate our estimation, we have ran-
domly chosen 100 points inside the map of Europe (Fig. 2)
and have compared the results to data taken from PVGIS
database (Photovoltaic Geographical Information System)
(PVGIS, 2012; S
ˇ
u´ ri et al., 2007, 2008). From this database
and for each point, we have downloaded irradiation data
for 13 tilt angles, from 25
·
to 85
·
. b
opt
(the value of b at
maximum irradiation in December) has been estimated fit-
ting a parabola to the data and taking the abscissa of its
maximum. Four points (with coordinates 44
·
N3
·
E;
48
·
N7
·
E; 55
·
N5
·
W and 62
·
N 7
·
E) have shown a strange
behaviour in PVGIS data: total irradiation corresponding
to December for those points decreased monotonically as
the tilt angle (b) increased. The same database gives for
those bizarre points a value of 0 for b
opt
at December.
The only explanation we can think of, is that those loca-
tions are extremely cloudy in December, and the diffuse
component is predominant. But surely the strange behav-
iour is simply due to error in data. Therefore, we have
excluded those four points. For the rest, we have estimated
b
opt
from PVGIS data, and compared those values to our
estimations. The comparison is shown in Fig. 9. The max-
imum deviation is 11
·
but, on average, the results are very
similar (the correlation coefficient between both sets is
0.927).
Apart from the previous study, the values obtained for
b
opt
(around 62
·
for the Iberian Peninsula, and general dif-
ferences b
g;opt
÷/
g
of 15–20
·
) are very similar to those
found in the literature (Egido and Lorenzo, 1992; Fragaki
and Markvart, 2008).
5. Computation of LLP and derived parameters
For each geographical point, after finding the optimal
tilt angle (b
opt
), estimated in previous section, we compute
three isoLLP curves (corresponding to LLP values 0.01,
0.05 and 0.1) by algorithm in C. Finally, also for each
place, we fit expression in Eq. (16) for LLP = 0.1 and
LLP = 0.01, to find corresponding values of f and u, thus
called f
0:1
; u
0:1
; f
0:01
and u
0:01
. In 96.7% of the
(2 ×2280 = 4560) fitted curves, the correlation coefficient
is greater than or equal to 0.95. These parameters are
shown in Figs. 10–13 as contour lines or isolines. In all
the Figures, isolines are shown as alternating thick-red
lines (with labels of each isoline value) and thick-green lines
(with no labels). A fine-red line, continuation of the thick-
red isoline, is drawn below each label for greater clarity.
(For a better inspection of dense line zones, we recommend
viewing the PDF document and applying first rotation and
then zoom where needed, as all the graphics in this paper
are vectorial and thus fully scalable.)
If the desired LLP for a PV system is 0.01 or 0.1, from
the maps it is straightforward (excepting uncertain values
due to figure limitations, especially in borders) to find the
parameters f and u and thus to know the relation
C
S
~ C
A
(0) through Eq. (16) for the selected location.
−10 0 10 20 30 40 50 60
40
50
60
zone 1
zone 4
zone 2
zone 3
Fig. 16. Map of Europe with zones 1–4.
2528 J.H. Lucio et al. / Solar Energy 86 (2012) 2515–2535
For other values of LLP, the four parameters
(f
0:1
; u
0:1
; f
0:01
and u
0:01
) should be taken from Figs. 10–
13, and then relations in Eq. (5) for LLP values 0.1 and
0.01 could be used to find f
1
; f
2
; u
1
and u
2
:
f
2
= f
0:1
÷f
0:01
; (14a)
f
1
= f
0:1
÷f
2
; (14b)
u
2
=
100
9
ln
u
0:1
u
0:01
; (14c)
u
1
= ln u
0:1
÷0:1u
2
; (14d)
where ln() means natural logarithm. Finally, again Eq. (5)
would give the parameters f and u for the desired LLP.
These last steps (when LLP is different from 0.1 and
0.01) can be shorten by eliminating the parameters
f
1
; f
2
; u
1
and u
2
: simply one should read the four values
(f
0:1
, u
0:1
; f
0:01
and u
0:01
) from the maps, and then he or
she should apply the following expressions to find f and u
directly as a function of LLP:
f = 2f
0:1
÷f
0:01
÷(f
0:1
÷f
0:01
)log
10
LLP; (15a)
u = exp
1
9
ln
u
10
0:01
u
0:1
÷
100
9
ln
u
0:1
u
0:01
_ _
LLP
_ _
: (15b)
6. MLP training and simulations
6.1. MLP for (b
opt
÷/): MLP1
In order to get (b
opt
÷/) value for any point inside Eur-
ope, any type of interpolation could be implemented. But
artificial intelligence through ANN is suitable for a vast
(virtually any) type of relation among input and output
variables. Particularly, we have chosen a feedforward layer:
an MLP (see Section 2.3) with one hidden layer. We will
call it MLP1. MLP1 has latitude and longitude as inputs
and (b
opt
÷/) as output.
Starting at H value given by expression in Eq. (13):
H = 50, we tried several numbers of neurons in the hidden
layer and found that, in order to achieve sufficient preci-
sion, a higher number of neurons was needed. Finally we
trained an MLP with 70 neurons in the hidden layer, as this
number gave the first minimum in mean squared error.
With a modern computer, in a few seconds MLP1 was
trained and a mean squared error 0.0016 was reached at
epoch number 24. The pattern set was divided into two
groups: training (90% of the values) and validation (10%
of the values) sets. The Levenberg–Marquardt algorithm
was applied for the training, as it is usually very fast,
−0.2 0 0.2
0
1000
2000
3000
4000
−0.2 0 0.2
0
200
400
600
800
1000
−0.2 0 0.2
0
50
100
150
−0.2 0 0.2
0
50
100
150
200
−0.2 0 0.2
0
50
100
150
(a) (b) (c) (d) (e)
Fig. 17. Histograms of MLP3 relative errors, corresponding to whole Europe (a) and zones 1 (b), 2 (c), 3 (d) and 4 (e). The errors are clearly smaller in the
small areas (MLP3.1 to MLP3.4) than in the MLP applied to whole Europe (MLP3).
Table 2
MLP3.1 through MLP3.4 characteristics and some statistics of their
relative errors, compared to MLP3 (first numerical column). Fraction n
means the fraction of relative errors between ÷n and n. The improvement
by reducing the area estimated is great.
Concept MLP3 MLP3.1 MLP3.2 MLP3.3 MLP3.4
Number of points 2280 265 44 74 44
´
H (by Eq. (13)) 222 78 33 42 33
H used 300 100 30 100 50
Relative errors
Mean 0.003 0.001 0.000 ÷0:001 0.002
Standard deviation 0.064 0.026 0.029 0.038 0.027
Fraction 0.20 (%) 98.8 100.0 100.0 99.9 100.0
Fraction 0.10 (%) 89.7 99.5 99.7 98.5 100.0
Fraction 0.05 (%) 65.5 94.9 92.0 84.4 92.7
Table 3
Characteristics of SAPVS at Villafrı ´a (Burgos, Spain).
Concept Value Unit
Location
Latitude /
g
= 42:36 °
Longitude r
g
= ÷3:63 °
Elevation h = 870 m
Daily load
Load L = 2:5 kW h
Photovoltaic modules
Area A = 13:2 m
2
Nominal power P
PV
= 170 W
Mean estimated efficiency g
PV
= 0:13
System of batteries
Nominal capacity C
B
= 650 A h
Nominal voltage V
B
= 2 V
Depth of discharge DOD = 0.75
Mean estimated efficiency g
B
= 0:85
Power conditioner
Mean estimated efficiency g
PC
= 0:95
J.H. Lucio et al. / Solar Energy 86 (2012) 2515–2535 2529
especially compared with the famous back-propagation
algorithm.
Once trained MLP1, a simulation with any pair of input
values is extremely fast. Differences between MLP1 output
(called MLP1
output
) and its targets (b
opt
÷/) are shown as
isolines in Fig. 14. Although these values range between
÷3:5
·
and 4:3
·
(highest errors occur in some coastal zones),
in general the fit is very good.
The distribution of the errors in MLP1 can be appreci-
ated in Fig. 15a. It is clear that, on average, the errors are
small, distributed about 0 and that their distribution is very
symmetrical (their mean is ÷0:011
·
and their standard devi-
ation is 0:711
·
). 98.5% of the errors are between ÷2
·
and 2
·
,
and 87.0% between ÷1
·
and 1
·
. These absolute errors
should be compared to MLP1 output target: (b
opt
÷/) in
degrees, ranging from 10:4
·
to 25:8
·
: therefore the errors
are generally small, as the change in long range global irra-
diation due to a 1
·
change in b is usually negligible.
6.2. MLP’s for isoLLP lines: MLP2 and MLP3
Two MLP’s have been built to predict the isoreliability
curves in Europe. Each one has only one output: the
C
A
(0) value. They differ in the variables selected as inputs.
These MLP’s, once trained, could be used to generate vir-
tually the isoLLP curve for any LLP value for any location
in Europe.
6.2.1. MLP2
Inspired by Hontoria et al.’s work (Hontoria et al.,
2005), we tried to estimate LLP curves through daily clear-
ness index, K
Td
. Thus we created one MLP (called MLP2)
with four inputs: C
S
, LLP, K
Td
and r
K
Td
(being last two
ones K
Td
mean and standard deviation) and with one out-
put: the corresponding C
A
(0) value. Hontoria et al. only
took the first three variables, but we added r
K
Td
in order
to distinguish between locations with similar mean daily
clearness index but different regularity (high or low fluctu-
ations). In fact, we got a significant improvement adding
the fourth variable (mean squared absolute error falls from
0.014 to 0.009). To estimate r
K
Td
, we simply computed the
standard deviation of the daily values of clearness index
K
Td
(see paragraph 2.2).
Seven values (between 2 and 9) of C
S
, three of LLP and
all the geographical points (2280) were used to form the
patterns: thus finally there were 47,880 different patterns.
The number of neurons in the hidden layer, H, should
be 222, according to Eq. (13), but the error bounds would
then be high, following Eq. (12), and also the training time
would become excessive.
Finally MLP2, with four inputs and one output, has
been implemented with 200 neurons in the hidden layer,
as increasing H did not improve the results significantly.
The same proportions for training and validation as in
MLP1 (90% and 10%, respectively) were used in MLP2.
Also the Levenberg–Marquardt algorithm was employed
for the training. In a few tens epochs the training con-
verged, what supposed about ten minutes in our computer
but, as it is typical in artificial neural networks, a very short
computing time will be needed later to simulate any input
after the MLP has been trained.
Relative errors distribution of MLP2 is shown in Fig. 15
b. In the histogram three features can be appreciated: rela-
tive errors are in general very big, distributed about 0 and
their distribution is a bit asymmetrical (their mean is 0.090
and their standard deviation is 0.368). 62.4% of the errors
are between ÷0:2 and 0.2, and only 40.3% are between
÷0:1 and 0.1. These errors are not small, and then MLP2
output should be considered only an approximate estima-
tion. The limited success of this MLP can be understood
by the lack of determination of isoLLP curves through only
clearness index. Following we present another MLP for a
better estimation of LLP curves, substituting inputs K
Td
and r
K
Td
by the geographical coordinates /
g
and r
g
.
6.2.2. MLP3
In order to model the isoLLP curves, we created another
MLP (called MLP3) with four inputs: C
S
, LLP, /
g
and r
g
(being last two ones latitude and longitude) and the same
output as in MLP2: C
A
(0).
As inMLP2, seven values (between 2 and 9) of C
S
, three of
LLP and all the geographical points (2280) were used to
formthe patterns (47,880). AlsoinMLP3 the number of neu-
rons in the hidden layer, H, should be 222, but we finally
chose H = 300. The same proportions for training and vali-
dation as in MLP1 (90%and 10%, respectively) were used in
MLP3. Levenberg–Marquardt algorithm was employed
again for the training. MLP3 needed about 500 epochs to
converge, what supposedabout three hours in our computer.
Relative errors distribution of MLP3 is shown in Fig. 15
c. In the histogram three features can be appreciated: errors
are in general very small, distributed about 0 and their distri-
bution is very symmetrical (MLP3 relative errors mean is
0.003 and their standard deviation is 0.064). 98.8% of the
errors were between ÷0:2 and 0.2; 89.7% were between
÷0:1 and 0.1; and 65.5%were between ÷0:05 and0.05. These
errors are relatively small, and thus MLP3 output could be
used as a first estimation. In order to get higher accuracy
and precision, smaller areas should be estimated through
an MLP like MLP3. This is to be done in next paragraph.
6.2.3. MLP3 in 4 zones
It is difficult to find an MLP capable of modeling pre-
cisely isoLLP curves for such a vast territory as Europe.
The attempts have given poor results (MLP3 and worse
MLP2). These large-region MLP’s can yield tentative or
first-estimation results. As any other interpolation method,
the MLP will benefit from the fact that, fixed the sampling
density, a smaller sampling grid generally will be approxi-
mated by a smoother and thus a simpler function. There-
fore, in order to estimate precise and accurate isoLLP
curves, it is better taking into account smaller regions with
more uniformsizing parameters or smaller regions including
the locations of interest.
2530 J.H. Lucio et al. / Solar Energy 86 (2012) 2515–2535
For that purpose, we have chosen four regions, called
zones 1–4, shown in Fig. 16.
v Zone 1 (Northeast Europe, mainly West Russia
and part of Belarus and Ukraine): although very
large, presents small variations in estimated param-
eters f and u (see Figs. 10–13).
v Zone 2 (East Adriatic Sea and nearby countries):
presents special variations in f and u parameters.
v Zone 3 (Alps region: Northern Italy and part of
Switzerland and France): presents special varia-
tions in f and u parameters.
v Zone 4 (interior of Iberian Peninsula): seems a
rather constant region in f and u parameters.
We have trained an MLP for each zone, with the same
inputs and outputs as MLP3, thus we will call these MLP’s:
MLP3:1 MLP3:4. InFig. 17we showrelative errors of these
MLP’s compared to MLP3. In Table 2 several characteristics
of the four MLP’s and some statistics of their relative errors
are shown. As expected, MLP3 applied to zones 1–4 give
much smaller errors than MLP3 applied to whole Europe.
MLP3 can give a first estimate of the LLP curves for any
place inside Europe, but it is clear that MLP’s trained with
smaller areas will give better results.
6.3. Representative case study
In order to apply the results obtained for b
opt
, isoLLP
curves, and the MLP’s, here we expose the study and imple-
mentation of a SAPVS at Villafrı ´a (Burgos, Spain). In Table
3 we show some data on the location and the installation.
Next we present the necessary steps to estimate a suit-
able sizing for our experimental SAPVS:
1. First, some unit conversions:
L = 2:5 kW h
3:6 ×10
6
J
kW h
= 9 ×10
6
J; and
C
B
= 650 A h ’ 2:34 ×10
6
A s:
2. Then, multiplying all the efficiencies shown in Table 3,
we obtain an estimation of the average total efficiency
of the system:
g ’ g
PV
g
B
g
PC
= 0:13 ×0:85 ×0:95 = 0:105: (16)
3. We estimate
´
G
d
(0) in Eq. (2) as G
d;Dec
(0) (average of
December-daily global irradiation on horizontal plane)
for the location of the plant (for instance, simply taking
the value fromthe mapinFig. 7):
´
G
d
(0) ’5:15 10
6
J=m
2
.
4. With previous results, we can find a value for C
A
(0) in
Eq. (2):
C
A
(0) =
gA
´
G
d
(0)
L
’ 0:793:
5. Then, by applying the trained MLP1 (Section 6.1) in
order to estimate b
opt
for the plant, with /
g
and r
g
as
inputs, we get the value: b
g; opt
= 62:31
·
.
6. We choose an LLP value for the SAPVS, for instance:
LLP = 0.01.
7. By means of MLP3.4 (4th zone in Fig. 16) and a simple
iterative process, we get the value for C
S
: 4:73.
8. Finally, we get the number of batteries using Eqs. (1a)
and (1b):
N
B
= round
C
S
L
C
B
V
B
DOD
_ _
’ round
4:73 ×9 ×10
6
2:34 ×10
6
×2 ×0:75
_ _
= 12:
7. Conclusions
In this paper, we present the results of a vast work mod-
eling stand-alone PV plants for almost any location in Eur-
ope, using daily irradiation data from NASA’s satellites,
covering 23 years.
First, estimations of the average daily irradiation at hori-
zontal plane in December (
´
G
d
(0)) and the optimal tilt angle
(b
opt
) of the PV array for the same month at any place have
been performed. This tilt angle also approximately minimizes
the loss-of-load probability (LLP) for a given installation.
Maps of Europe with both parameters are shown, so that
finding their values is easy for any point in the area. Previous
works (Egido and Lorenzo, 1992; Lorenzo, 1994; Lorenzo
and Narvarte, 2000) employed common tilt angles for vast
regions, but each zone can have special radiation features.
Then, we have estimated several LLP curves for every
point, through simulations with the irradiation data, apply-
ing well-known irradiation models. Once the target LLP is
fixed, these curves allow anyone to choose the best sizing
parameters for agivengeographical point, accordingtosome
criterion, typically minimum cost. Potential functions have
been fixed for LLP 0.01 and 0.1, and their parameters are
shown in corresponding maps of Europe. This way, virtually
any LLP curve for any point in Europe can be estimated.
Finally, artificial intelligence techniques have been
employed to approximate b
opt
and LLP curves. Three mul-
tilayer perceptrons have been trained, so that they can eas-
ily be used to estimate optimum tilt angle and sizing
parameters for any place inside the selected area. Previous
works (see, for instance, Hontoria et al., 2005) only took
into account average values of clearness index and trained
an MLP with as few as 7 geographical points for a climat-
ically so complex region as Spain. We have trained the
MLP’s with 2280 point data and have taken into account
also the clearness index variation.
Two different MLP’s (MLP2 and MLP3) have been used
to predict LLP curves for Europe. Clearly, MLP3 is more
precise, and could be used to estimate isoreliability curves
inside Europe, as geographical coordinates are part of its
inputs. Nevertheless, in order to get more accurate and
precise results, an MLP covering a smaller area, for instance
an area including only the locations of interest, should be
trained. In order to show this approach, we have trained
MLP’s for four different zones in Europe, giving much
smaller relative errors than MLP3 (for the whole area).
This work, as far as we know, is the first one in estimat-
ing the optimal tilt angle and LLP curves for a whole
J.H. Lucio et al. / Solar Energy 86 (2012) 2515–2535 2531
continent. The given maps allow easy estimations of
´
G
d
(0); b
opt
and the LLP curves parameters. Additionally,
the trained MLP’s allow fast and easy calculations of those
parameters and could be added to a computer application
to size a stand-alone PV plant without previous knowledge
of the radiation history of the selected point.
Finally, the procedure presented in this paper, although
focused in Europe, could be easily extended to any other
region in the world.
We hope that this vast work can be employed by many
PV researchers and designers across Europe.
Acknowledgements
The authors thank Atmospheric Science Data Center
(ASDC) at NASA Langley Research Center (NASA,
2011), World Radiation Data Centre (WRDC) (WRDC,
2011) and Photovoltaic Geographical Information System
(PVGIS) (PVGIS, 2012) for their kind distribution of
radiation data.
Appendix A. G
d
(b) estimation from G
d
(0)
In this appendix we reproduce the equations applied to
get G
d
(b) from G
d
(0), following Duffie and Beckman
(1991), Lorenzo (1994), and Muneer (2004). Other correla-
tions will give somewhat different results.
PVA azimuth is taken a = 0, usual in practice and in the
literature, because that value (for fixed b) approximately
optimizes total irradiation at the PVA plane.
Angles are expressed in rad, except explicitly indicated
(if expressed in degrees, their symbol has a ‘g’ subscript).
Times are expressed in hours, although otherwise
indicated.
The input parameters for the general model (not the
algorithms) are the following:
Name Description Units
n Day index in the year (1st January is n = 1)
h Hour index in the day (h = 1; . . . ; 24)
/
g
Latitude (positive North) °
r
g
Longitude (positive East) °
b
g
PVA tilt angle °
G
d
(0) Daily global irradiation on horizontal
plane
J=m
2
A.1. Global equations
The following equations are applied only once, at the
beginning of the algorithm:
/ = /
g
p
180
·
; (A:1)
r = r
g
p
180
·
; (A:2)
b = b
g
p
180
·
; (A:3)
B
l
e;h
= 1367 W=m
2
×3600 s = 4:9212 ×10
6
J=m
2
; (A:4)
q = 0:2: (A:5)
A.2. Equations for each day
The following equations are applied once per day
(j = 1; . . . ; N
days
), in the same order:
C =
2p
365
(n ÷1); (A:6)
d = 0:006918 ÷0:399912 cos(C) ÷0:070257 sin(C)
÷0:006758 cos(2C) ÷0:000907 sin(2C)
÷0:002697 cos(3C) ÷0:00148 sin(3C); (A:7)

0
= 1:00011 ÷0:034221 cos(C) ÷0:00128 sin(C)
÷0:000719 cos(2C) ÷0:000077 sin(2C); (A:8)
ET = 3:8197 0:000075 ÷0:001868 cos(C) ÷0:032077 sin(C) [
÷0:014615 cos(2C) ÷0:04089 sin(2C)[; (A:9)
x
s
= ÷arccos(÷tan d tan /); (÷p < x
s
< 0); (A:10)
B
e;d
(0) = ÷
24
p
B
l
e;h

0
x
s
sin d sin / ÷cos d cos /sin x
s
( );
(A:11)
K
Td
=
G
d
(0)
B
e;d
(0)
; (A:12)
K
Dd
=
0:99 if K
Td
60:17;
1:188 ÷2:272K
Td
÷9:473K
2
Td
;
÷21:856K
3
Td
÷14:648K
4
Td
_
if 0:17 < K
Td
60:75;
÷0:54K
Td
÷0:632 if 0:75 < K
Td
< 0:80;
0:2 if K
Td
P0:80;
_
¸
¸
¸
¸
¸
¸
_
¸
¸
¸
¸
¸
¸
_
(A:13)
D
d
(0) = K
Dd
G
d
(0); (A:14)
a = 0:409 ÷0:5016 sin x
s
÷
p
3
_ _
; (A:15)
b = 0:6609 ÷0:4767 sin x
s
÷
p
3
_ _
; (A:16)
x
ss
= max[x
s
; ÷arccos(÷tan d tan(/ ÷b))[; (A:17)
x
ss;h
= round
d
12 1 ÷
x
ss
p
_ _ _ _
;
(round downward to nearest integer) (A:18)
x
sp;h
= round
u
12 1 ÷
x
ss
p
_ _ _ _
;
(round upward to nearest integer): (A:19)
A.3. Equations for each hour
The following equations are applied once per hour
(h = x
ss;h
; . . . ; x
sp;h
), in the same order:
x
h
= h ÷12 ÷ET ÷
r
g
÷r
st;g
15
; (A:20)
2532 J.H. Lucio et al. / Solar Energy 86 (2012) 2515–2535
x =
p
12
x
h
; (A:21)
cos h
zs
= sin d sin / ÷cos d cos /cos x; (A:22)
cos h
s
= sin d sin(/ ÷b) ÷cos d cos(/ ÷b) cos x; (A:23)
r
D
= max 0;
p
24
cos x ÷cos x
s
x
s
cos x
s
÷sin x
s
_ _
; (A:24)
r
G
= max[0; r
D
(a ÷b cos x)[; (A:25)
D
h
(0) = r
D
D
d
(0); (A:26)
G
h
(0) = r
G
G
d
(0); (A:27)
B
h
(0) = G
h
(0) ÷D
h
(0); (A:28)
B
h
(b) =
B
h
(0)
cos h
zs
max(0; cos h
s
); (A:29)
k
1
= 1 ÷
D
h
(0)
G
h
(0)
_ _
2
; (A:30)
D
h
(b) =
D
h
(0)
2
(1
÷cos b) 1 ÷k
1
cos
2
h
s
sin
3
h
zs
_ _
1 ÷k
1
sin
3
b
2
_ _
;
(A:31)
R
h
(b) =
q
2
G
h
(0)(1 ÷cos b); (A:32)
G
h
(b) = B
h
(b) ÷D
h
(b) ÷R
h
(b): (A:33)
A.4. Equation at the end of each day
The following equations are applied once per day, just
after previous group:
G
d
(b) =

x
sp;h
h=x
ss;h
G
h
(b): (A:34)
Appendix B. Algorithm to estimate b
opt
The target is to find the b value maximizing the total
energy reaching the PVA the worst month (December).
Let us call it b
opt
. It is computed for all the points
selected in Europe (Fig. 2). Next we present a pseudocode
version.
In this algorithm, obviously any optimization procedure
can be used to find b
opt
. Fast convergence is guaranteed, as
the function G
Dec
(b) is almost quadratic in the neighbor-
hood of the maximum and b = / is near it.
J.H. Lucio et al. / Solar Energy 86 (2012) 2515–2535 2533
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D
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G
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h
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h
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h
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h
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end
compute G
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/
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A
(0)G
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´
G
d
(0)C
S
); 1
_ _
; from Eqs. (6) and (7)
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/
÷1=C
S
; 0); from Eq. (8)
E
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= max(L ÷CSOC
/
; 0); from Eq. (9)
E
aux;T
÷E
aux;T
÷E
aux
; partial sum of energy from AG
end
LLP = E
aux;T
=(N
days
L); compute actual LLP
C
A
(0) ÷C
A
(0) ÷DC
A
(0); change C
A
(0) seeking LLP ’ LLPt
if [LLP ÷LLPt[=LLPt < 0:01 found an acceptable LLP
break exit the while loop
end
end
end
end
end
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