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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end

compute G

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A

(0)G

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(b)=(

´

G

d

(0)C

S

); 1

_ _

; from Eqs. (6) and (7)

SOC = max(SOC

/

÷1=C

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E

aux

= 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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