Probabilistic Approach to Assess the Performance of GridConnected PV Systems

 SourceAvailable from: Ciprian NemesBuletinul AGIR. 01/2012; 3.
 SourceAvailable from: Giuseppe Marco Tina[Show abstract] [Hide abstract]
ABSTRACT: EU Governments are working towards increasing the amount of renewable energy sources in order to reduce the environmental impact of energy generation (Directive 2001/77/CE). Further, recent technical developments (e.g. fuelcells, micro CHP using novel turbines or Stirling engines and photovoltaic (PV) devices integrated into the fabric of buildings) and the liberalisation of the electricity market encourage the use of renewable resources in terms of distributed generation (DG) and CHP units. On the other hand, the presence of large amount of DG can have a negative impact on distribution networks, which are designed and operated as passive networks supplied by planned centralized generation. In LV distribution networks the effect of gridconnected PV DG units is worth to be investigated as such systems have a natural inclination to be easily integrated into high density urban LV distribution networks. If a large capacity of PVDG is to be connected in the future, then it will be necessary to account for consequences on important technical aspects, some of which related to power quality supplied to customers (voltage variations, protection, harmonic distortion, etc.) [1]. In order to study the effects of PVDG on distribution operation, the present paper deals with the development of PV units models suitable for large scale systems studies, for both steady state and transient analysis. The first section presents a general description of gridconnected PV systems to introduce the system modelling. The second section discusses the basic PLL concept and design criteria for gridconnected PV applications. In the third section modelling of the current reference generator of singlephase inverters based on the PLL technique is dealt with. The models have been developed by means of Simulink<sup>®</sup> (Dynamic System Simulation for MATLAB<sup>®</sup>). Finally, as an example of the application of the models developed, the fourth section presents a study which shows the effect of gridconnected PV units on feeders voltage profile in a LV distribution network.Electricity Distribution, 2005. CIRED 2005. 18th International Conference and Exhibition on; 07/2005
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Abstract—This paper deals with the problem of assessing the
performance of gridconnected PV systems. An analytical
probabilistic approach, based on the convolution technique, is
used for the evaluation of the net energy injected into the grid by
a PV system supplying loads. Then, a power duration curve for
the random variable “Powerdeliveredtothegrid” has been
derived.
Index Terms—Photovoltaic Systems, GridConnected Systems,
PV Modelling, Probabilistic Methods.
I. INTRODUCTION
HE recent technical developments and the current changes
in the electricity industry are expected to favour the
spreading of Distributed Generation (DG) units. In this
context, due to the problems related to air pollution and to
climate changes, the renewable DG is gaining more and more
importance.
Integration of DG units with the electrical utility grid is
critical especially for wind and photovoltaic systems because
of the variable nature of wind speed and solar irradiance.
Research and development in wind and solar energy systems
continue in order to improve their performance, establish
techniques for accurate prediction of their output and reliably
integrate them with other conventional generating sources. In
this respect probabilistic approaches continue to be developed
to assess their performance as standalone or gridconnected
systems.
In particular, as for the treatment of solar irradiance and
electricity demand, two approaches have been adopted. One is
a deterministic approach where the system performance is
evaluated on the basis of original data of irradiance and
electricity demand obtained through measurement [1], [2]. The
other is a probabilistic approach which is based on probability
distribution of solar irradiance and electricity demand assumed
from their original data [3], [4], [5]. Although theoretical work
for the latter has been done a detailed numerical study has not
been conducted using data obtained through real measurement.
Actually, in the model proposed in [4] and [5] it is assumed
that a maximum of solar irradiance is estimated as the
deterministic reference value for each sampling time on each
representative day and that the probability distribution of solar
The Authors are with the Dipartimento Elettrico, Elettronico e
Sistemistico, Università degli Studi di Catania, viale A. Doria, 6, 95125
Catania, Italy (email: gtina@dees.unict.it).
irradiance for each sampling time is subject to a β distribution.
This is based on the fact that the solar irradiance significantly
depends on the amount of cloud and that the probability
distribution of the amount of cloud can be expressed with
accuracy by a β distribution [6], but this model refers only to
solar radiation incident on an horizontal surface.
In the present paper the objective to evaluate the long term
performance of PV systems with fixed sloped array surface is
posed. To do this, a probability distribution function of solar
irradiance on a surface sloped at the generic angle β to the
horizontal is obtained. Then, this function is used to develop
an analytical probabilistic approach, based on the convolution
technique, to evaluate the power duration curve for the random
variable “power injected into the grid” and some important
global parameters, such as the energy injected into grid by the
PV system and the energy drawn from the grid to supply the
load.
II. SYSTEM DESCRIPTION AND MODELS
Utility gridconnected PV systems consist, essentially, of a
PV array and a power conditioner connected to the utility grid
and to local electrical loads (Fig. 1).
In order to have maximum exploitation of solar energy,
during overproduction hours, any PV generation exceeding the
load will be injected into the utility network; otherwise, when
the PV system will be unable to meet load demand, power will
be drawn from the network. Clearly, the power S(t) injected
into the grid is given by:
S(t) = P(t)  L(t) (1)
where P(t) is the power generated by the PV system and
L(t) is the power drawn by the load. The grid is assumed to be
an infinite bus.
The proposed PV system model requires a time
discretization of the quantities of interest. The choice of the
time step is mainly influenced by the smoothness of the load
profile. In the present work the time step is set to one hour.
However, this choice does not affect the generality of the
results since the PV system model allows to adopt time steps
shorter than one hour if necessary. The performance of the
system under study can be assessed by employing suitable
models for the power output of the PV system and for the load.
These models will be presented in the two following
subsections.
Probabilistic Approach to Assess the LongTerm
Performance of GridConnected PV Systems
S. Conti, Member, IEEE, T. Crimi, S. Raiti, G. Tina, Member, IEEE, and U. Vagliasindi
T
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PV
ARRAY
POWER
CONDITIONER
UTILITY
GRID
LOAD
L(t)
S(t)
P(t)
Fig. 1. Block diagram of a utility gridconnected PV system
A. PV system model
The amount of solar radiation that reaches the ground,
besides on the daily and yearly apparent motion of the sun,
depends on the geographical location (latitude and altitude)
and on the climatic conditions (cloud cover). Many studies
have proved that cloudiness is the main factor affecting the
difference between the values of solar radiation measured
outside the atmosphere and on earthly surface. To account for
the difference between these two values a daily clearness
index, Kt, has been defined as the ratio of a particular day’s
total solar radiation, Ht [MJ/m2], to the extraterrestrial total
solar radiation, Ho [MJ/m2], for that day, both referred to an
horizontal surface [7]:
H
K =
(2)
o
t
t
H
The higher the solar radiation amount that reaches the
ground, the higher the clearness index.
Since to maximize the received solar energy the PV arrays
have an appropriate inclination β to the horizontal plane, to
evaluate the received energy it is necessary to refer to the three
components of the total daily solar radiation: diffuse, Hd,
beam, Hb and ground reflected, Hr, radiation. These
components can be calculated from the daily clearness index,
Kt, and the daily diffuse fraction, K, given by:
H
K =
(3)
t
d
H
In literature, there are several different correlations of K
with Kt, obtained from experimental data gathered in different
sites in many years. In general it possible to approximate this
correlation with a linear function. To show this, as an example,
in Fig. 2 three correlations of K with Kt are shown [8]. In
particular the correlation based on the data relative to the city
of Palermo (Italy) can be represented by the following
expression:
qKpK
−=
(4)
where p = 1.0303 and q = 1.1515.
From the knowledge of the daily quantities and under the
assumption that (4) holds for the corresponding hourly
quantities, k and kt:
qkpk
−=
(4’)
it is possible to determine Iβ [kW/m2] that is the irradiance
on a surface with inclination β to the horizontal plane, by the
t
t
following expression:
TI
=
β
2
,,
,
'
tjmtjm
jm
kTk
⋅−⋅
(5)
where:
m is an integer indicating the month of the year;
j is an integer indicating the hour of the day;
Tm,j = Tm,j (β, φ, δ, ρ, ω, ωS, p, n, GSC );
T’m,j = T’m,j (β, φ, δ, ω, ωS, q, n, GSC );
where:
φ is the latitude, that is the angular location north or south
of the equator, north positive (90° ≤ φ ≤ 90°);
δ, is the declination, that is the angular position of the sun
at solar noon with respect to the plane of the equator, north
positive (23.45° ≤ δ ≤ 23.45°);
ρ is the reflectance of the ground;
ω is the hour angle, that is the angular displacement of the
sun east or west of the local meridian due to rotation of the
earth on its axis at 15° per hour, morning negative,
afternoon positive;
ωS is the sunset hour angle, in degrees;
n is the day of the year;
GSC is the solar constant (1353 W/m2);
Since the PV system is usually equipped with a maximum
power point tracker (MPPT) and the relationship between the
maximum power per unit area of array surface available from
the PV system (P’
the PV system is given by:
'
jm
jmcjmcjm
[
tut
kk
, 0
∈
(6)
where:
Ac is the array surface area;
ηm,j is the efficiency of the PV system;
ktu is the upper bound for kt .
From (6), if the probability density function for the random
variable kt is known, it is possible to obtain the probability
density function,
)(
,
,
jm
jm
P
pf
fundamental theorem for function of a random variable [9].
m,j ) and Iβm,j is linear, the power output of
(
,
,
,
mjmc
)
2
,,,,
'
tjmtj
kTkTAIAPAP
−===ηη
β
]
, for Pm,j by applying the
Fig. 2. Three correlations of K with Kt: data for the cities of Palermo and
Genoa (Italy) and proposed by Liu and Jordan [8].
In this work the expression used for the probability density
Page 3
3
function of kt is the one proposed by Hollands e Huget [10] as
follows:
(
=
k
kkf
tu
tkt
0
(7)
where:
(
⋅−−
tujm
ke
,
1 λ
)
[
, 0
]
∈
⋅
⋅
−
⋅
otherwise
kk
tkjm
e
kk
C
tut
t tu
jm
t
,
,
,
)(
λ
)
Γ
⋅
⋅
=
tujm
tujm
jm
k
k
C
,
2
,
,
λ
λ
(8)
tu
jmjm
jm
jm
k
ee
Γ−
−
−
. 1
−Γ
=
,,
,
,
0426 . 5
1062
3118
519. 172
λ
(9)
()
t
tu
tu
−
jm
kk
k
=Γ,
(10)
t k is the hourly average clearness index, obtained from the
monthly average daily clearness index
Depending on to the sign of the parameters Tm,j e T’m,j, the
probability density function
f
t
K [7].
)(
,
,
jm
jm
P
p
has four different
expressions. The sole two expressions which have a physical
meaning, given by (11’) and (11’’), are the ones in which the
function
)(
,
tjm
kgP
=
increases in the interval [0, ktu]:
[
, 0
]
∈
−+
⋅
⋅
−−
−+−
=
otherwise
kPp
AT
j
p
e
AT
p
TAk
AT
p
kC
pf
tujm
cjm
'
m
jm
jmjm
jm
2
cjmjm
jm
jmjmjmc tu
cjmjm
jm
jmjmtujm
jmjPm
0
)(
4
'
4
'
'
4
2
1
)(
,
,,
,
2
,,
,
,,
,
2
,,,
,,
,
2
,,,
,,
η
αα
λ
η
αη
η
αα
if Tm,j>0 and T’m,j<0 (11’)
[
, 0
]
∈
−−
⋅
⋅
−
−−−
=
otherwise
kPp
AT
j
p
e
AT
p
TAk
AT
p
kC
pf
tujm
cjm
'
m
jm
jmjm
jm
2
cjmjm
jm
'
jmjmjmctu
cjmjm
jm
'
jmjm tujm
jmjPm
0
)(
4
4
'
4
2
1
)(
,
,,
,
2
,,
,
,,
,
2
,,,
,,
,
2
,,,
,,
η
αα
λ
η
αη
η
αα
if Tm,j > 0 and T’m,j ≥ 0 (11”)
where:
jm
jm
'
jm
T
T
,
,
,
=
α
Typical plots of the probability density function and the
probability distribution function of Pm,j are shown in Figg. 3
and 4, respectively.
Fig. 3. Plot of the probability density function of Pm,j
1
Fig. 4. Plot of the probability distribution function Pm,j
B. Load model
The procedure assumes a random load, lm,j, that is
uniformly distributed [10] between a maximum value, Lmax m,j,
and a minimum value, Lmin m,j, that depend on the hour, j, of the
day and the month, m, of the year. The probability density
function of the random variable lm,j is expressed by the
following:
−
=
LL
lf
jmjm
jmjmL
0
≤≤
otherwise
LlL
jmjmjm
1
)(
,max,,min
, min,max
,,
(12)
By choosing suitable values for Lminm,j and Lmaxm,j, a wide
variety of loads can be included in the study. In case of
constant load (Lmaxm,j = Lminm,j) the function
becomes:
)()(
, max,,,
jmjmjmjLm
Lllf
−=δ
(13)
)(
,,
jmjLm
lf
FPm,j(pm,j)
pm,j
Pmax m,j
FPm,j(pm,j)
Pmax m,j
pm,j
Page 4
4
III. PROBABILITY DENSITY FUNCTION AND PROBABILITY
DISTRIBUTION FUNCTION FOR POWER INJECTED INTO THE GRID
A. Probability density function
From the probability density functions of the power
generated by the PV system, Pm,j, and the power drawn by the
load, Lm,j, assuming that the two functions are statistically
independent, the probability density function of the random
variable Sm,j is found by convolving the probability density
functions of Pm,j and Lm,j:
)()(
,,,,
jmjmPjmjmS
pfsf
∗=
In evaluating of the convolution integral of (14) three cases
are of interest:
1)
m,j m,j m,j
LLP
minmaxmax
−<
=
)(
,
,
jmS
sf
jm
)(
,,
jmjmL
lf
−
(14)
∫
0
+
⋅
−
=
jm
f
Ls
jmjmjmP
jmjm
dpp
LL
,max
,,,
,min ,max
)(
1
if
jmjmjmjm
LPsL
, max,max ,,max
−≤≤−
∫
0
⋅
−
=
jm
f
P
jmjmjmP
jmjm
dpp
LL
,max
,,,
,min,max
)(
1
if
jmjmjmjm
LsLP
,min,, max, max
−≤≤−
∫
+
−
⋅
−
=
jm
f
P
jmLs
jmjmjmP
jmjm
dp p
LL
,max
,min
L
min
,,,
,min, max
)(
1
if
jmjmjmjm
LPs
,min
(15’)
,max,,
−≤≤
2)
m,jm,jm,j
(
LLP
minmax
=
max
f
−≥
)
,
,
jmS
s
jm
∫
0
+
⋅
−
=
jm
f
Ls
jmjmjmP
jmjm
dp p
LL
,max
,,,
, min,max
)(
1
if
jmjmjm
LsL
,min,,max
−≤≤−
∫
+
+
−
⋅
−
=
jm
f
Ls
jmL
L
s
jmjmjmP
jmjm
dpp
LL
,max
,min
,,,
, min,max
)(
1
if
jmjmjmjm
LPs
,max, max,,min
−≤≤
∫
+
−
⋅
−
=
jm
f
P
jmLs
jmjmjmP
jmjm
dpp
LL
,max
,min
L
max
,,,
,min
P
,max
)(
1
if
jmjmjmjmjm
LPs
,min
(15’’)
,max,,,max
−≤≤
3)
m,jm,j
LL
minmax
=
=⋅=
−+
∫
jmjmjm
L
L
jmjmjPmjmjSm
dlff
Lllss
jm
jm
,,max,,,,,,
)()()(
,max
,min
δ
)(
,max,,
jmjmjPm
Ls
f
+
=
(15’’’)
where
Pmaxm,j is the value of Pm,j for kt equal to ktu.
B. Probability distribution function
The probability density function fx(x) and the probability
distribution function Fx(x) are related through:
x
dfxF
ςς)()(
(16)
∫
∞−
=
xx
Then, the function
)(
,,
jmjmS
sF
can be obtained through
(16) from
)(
,,
jmjmS
sf
calculated in the previous section.
Three different cases have to be highlighted to determine
the expression of the probability distribution function of
S
jm
,as well:
P
max
F
jm
F
=
1)
m,jm,jm,j
(
LL
minmax
=
)
−<
,
,
jmS
s
)(
,1
,
jmS
s
jm
if
L
jmjm
)
jmjm
LPs
, max,
max
s
,,max
−<≤
+
−
−
()(
,,2,max
L
if
,max
P
,1
j
≤
mjSm
L
jmjmjsm
FF
=
j
s
mjmjm
(
j
+
m
LsP
,
(
min ,,
−
max,max
−<−
)))(
,,3, min,2,max,max
P
,1
jmjsmjmjsmjmjmj sm
FLFLF
+−=
if
jmjmjmjm
LPsL
,min
(17’)
,max ,,min
−<≤−
where:
∫
−
=
s
jmL
jmjmjmS
,
jmjmS
ds sfsF
max
,,,,,1
)()(
∫
−
=
s
jmLjmP
jmjmjmSjmjmS
dssfsF
,max, max
,,,,,2
)()(
∫
−
=
s
jmL
−
jmjmjmS
,
jmjmS
dssfsF
min
L
,,,,,3
)()(
2)
m,jm,j
m,j
LP
minmax
=
)
max
F
≥
(
,,
jmjsm
F
s
)(
,
−
,1
jmjsm
s
=
if
jmjmjm
)
LsL
,min
)
,,
+
max
−<≤
(
<
(
,,2, min
−
,1
jm
P
jsm
s
jmjsm
sF
≤
LF
−
if
=
jm
+
jmjm
(
≤
jm
F
LL
,
F
max
)
,
j
−
,max,,min
)
+
−
−
)((
,,
(17’’)
3max,
<
max
P
s
,2,min,1
jmjsmmjmjsmjmjsm
sLLF
−
P
−
P
=
if
jmjmjmjmjm
LL
,min,max,,max,max
where:
∫
−
=
s
jmL
jmjmjmS
,
jmjmS
dssfsF
max
,,,,,1
)()(
Page 5
5
∫
−
=
s
jmL
jmjmjmS
,
jmjmS
ds sfsF
min
,,,,,2
)()(
∫
−
=
s
jmLjmP
jmjmjmS
max
jmjmS
dssfsF
,, max
,,,,,3
)()(
3)
m,jm,j
LL
min max
=
∫
max
−
=
s
L
jmjmjSmjmj Sm
dssfsF
,,,,,
)()(
(17’’’)
The expressions of
)(
,,1
jmjmS
sF
,
)(
,,2
jmjmS
sF
and
)(
,,3
jmjmS
In the case of utility gridconnected PV systems the amount
of energy injected into the grid and the amount of energy
drawn from the grid, during the study period, are of primary
interest because detailed economic assessments, which include
the time value of energy, can be undertaken. These amounts of
energy can be calculated by means of the power duration curve
of the random variable S(t). The power duration curve is a plot
of S(t) versus the time S(t) equalled or exceeded the particular
value. In Fig. 5 a typical power duration curve for S is
reported. The area E1 represents the energy injected by the PV
system into the grid, while E2 represents the energy drawn by
the utility grid to supply the load during the study period.
These areas can be evaluated for an hour time period using the
expressions of FSm,j(sm,j) presented in the Appendix. To be
more specific:
(
∫
−=
mjSmjm
sFE
0
sF
are reported in the Appendix.
)
−
jmL
1
jmP
jmj
ds
,min ,max
,,,,1
)(
(18)
∫
−
=
0
,max
,,,,2
)(
jmL
jmjmjSmjm
ds sFE
(19)
Fig. 5. Typical duration curve for S
In Figg. 6 and 7 the power duration curves of S are reported
for different PV array surfaces (Ac) with reference to the same
hour of the day (10 a.m.) and for two different months (May
and December, respectively).
Fig. 6. Power duration curves of S in May at 10 a.m. Lmax=754.8W,
Lmin=581.2W.
Fig. 7. Power duration curves of S in December at 10 a.m. Lmax=914.2W,
Lmin=703.8W
Page 6
6
IV. CONCLUSIONS
A closed form solution probabilistic approach, based on the
convolution technique, has been developed and presented for
the performance assessment of utilityinteractive PV systems
supplying a local load. The solar irradiance has been modelled
using a modified Gamma distributed random variable called
“clearness index”.
This model allows to take into account the inclination of the
PV array that is a very important parameter to evaluate the
long term energy assessment of PV systems. Closedform
expressions have been derived to obtain the power duration
curve for the power injected into the grid. The amount of
energy injected or drawn from the grid during the study period
can then be calculated from this duration curve.
The analysis is based on the statistical independence of
solar irradiance and load. Further, hourly load demand is
assumed to be uniformly distributed between a maximum and
a minimum value which depend on the hour of day and the
month of the year under consideration.
It should be pointed out that the employed probabilistic
method does not provide any information on the sequence in
which events happen, i.e. solar irradiance and load; it only
gives an overall view over the period under study.
However, this fact does not affect the expressions of the
energy injected by the PV system into the grid and the energy
drawn by the utility grid to supply the load. Actually, the
developed expressions are applicable at any penetration level.
They can be used to study the influence of different
parameters, e.g. surface area of PV array, efficiency, ratio
between PV rating and maximum load, and parameters
characterizing the solar irradiance, all of which have a bearing
on the operation of the system.
The results of such evaluations can ultimately lead to the
determination of energy and capacity credits for grid
connected PV systems.
V. APPENDIX
Case A:
P
max
1. T’
F
jmjmjm
LL
,min,max ,
−<
m,j < 0
)(' ')(')(
,,1,,1,,1
jmjmSjmjmSjmjmS
sFsFs
+=
where:
()
⋅
+
−
⋅=
jmcjm
jm
jm
jm
2
tu jm
jmc
2
jm
jmj sm
1
'
TA
Ls
e
k
TAD
sF
,,
,max
2
,
,
,
,,
,,
'
4
η
2
'
)(
α
λ
λ
η
(
α
)
()
+
+
−−−⋅
jmcjm
jm
jmtujmjmjm
TA
Ls
k
,,
,max
2
,,,,
'
4
η
62
αλλ
()
⋅
+
−
⋅−
jmcjm
jm
jm
jm
2
tujm
TA
Ls
e
k
D
,,
,max2
,
,
3
,
'
4
η
2
)
α
λ
λ
((
λ
)
[]
{}
342'4
,,,
2
,
2
,,,,max
2
,
++−−+⋅
tujmjmjmjmjmjmcjmjmjm
kTALs
λαλαηλ
and
(
L
)
()(
2
+
)+
)
3
+⋅
−
+
−=
jm
jm
2
jmtu
λ
jm
[
tujmjm
jmj sm
1
Ls
Lk
kC
sF
,max
,
)
min
+
,max
(
j
,
2
,
,,
,,
1
)(' '
λ
λ
(
λ
]
()
jmjmtujm
tujmtu
L
j
−
mm
L
jm
4
jm
λ
jmjmc
k
kkCTA
,min,max,
,,,,,,
'
+
−
λαη
()
⋅
−
⋅
+−
=
jmcjm
jm
jm
jm
2
tujm
jmjm
2
jmj sm
TA
P
e
k
sPLD
sF
,,
, max2
,
,
,
,max,
λ
max
,,2
'
4
)(
η
α
λ
(
α
) +
) 1
+
−−−−−⋅
22
'
4
,,,
,,
,max
A
2
,,
tujmjmjm
jmcjm
jm
jmjm
k
T
(
k
P
λλ
η
αλ
)
()
(
λ
,
,min
L
,max
L
2
,
,max
−
,max
L
,
⋅
+−−
λ
+
tujm
jmjmtu jm
jmjmjm
k
sPC
)()()(
,,3,,3,,3
' ''
jmjmSjmjmSjmjmS
sss
FFF
+=
where:
()
⋅
+
−
⋅
−
=
jmcjm
jm
jm
jm
2
tujm
jmc
k
jm
jmjSm
TA
Ls
e
TAD
sF
,,
,min
2
,
,
2
,
,,
,,3
'
4
η
2
'
)('
α
λ
λ
η
(
α
)
()
+
+
−−−⋅
jmcjm
jm
jmtujmjmjm
TA
Ls
k
,,
(
,min
2
,,,,
'
4
η
62
αλλ
)
⋅
+
−
⋅+
jmcjm
jm
jm
jm
2
tu
(
jm
TA
Ls
e
k
D
,,
,min
2
,
,
3
,
'
4
η
2
α
λ
λ
()(
λ
)
)
[]
342'4
,,,
2
,
2
,,,,min
2
,
++−−+⋅
tujmjmjmjmjmjmcjmjmjm
kTALs
λαλαηλ
and
()
⋅
−
⋅
+
=
jmcjm
jm
jm
jm
2
tujm
jm
jmjSm
TA
P
e
k
Ls
2
D
sF
,,
,max
2
,
,
,
,min
,,3
'
4
)(' '
η
α
λ
λ
(
α
)
−−−−−⋅
22
'
4
,,,
,,
,max
A
2
,,
tujmjmjm
jmcjm
jm
jmjm
k
T
P
λλ
η
αλ
2. T’
F
m,j ≥ 0
)(' ')(')(
,,1,,1,,1
jmjmSjmjmSjmjmS
sFsFs
+=
Page 7
7
where:
()
⋅
+
−
−
⋅
−
=
jmcjm
jm
jm
jm
tujm
jmc
k
jm
jmj sm
1
'
TA
Ls
e
TAD
sF
,,
,max
2
,
,
2
,
,,
,,
'
4
η
2
2
'
)(
α
λ
λ
η
(
α
)
()
+
+
−−−⋅
jmcjm
jm
jm tujmjmjm
TA
Ls
k
,,
,max
2
,,,,
'
4
η
62
αλλ
()
⋅
+
−
−
⋅−
jmcjm
jm
jm
jm
tu
(
jm
TA
Ls
e
k
D
,,
, max
2
,
,
3
,
'
4
η
2
2
α
λ
λ
()(
λ
)
)
[]
342'4
,,,
2
,
2
,,,,max
2
,
++−−+⋅
tujmjmjmjmjmjmcjmjmjm
kTALs
λαλαηλ
and
(
L
)
()(
)++⋅
−
+
−=
jm
jmjmtu jm
tujmjm
jmjsm
1
Ls
Lk
kC
sF
,max
,min,max
2
,
,,
,,
1
)(' '
λ
λ
(
λ
)(
λ
)
[]
()
jmjmtu jm
tujmtu
L
j
−
mjm
L
jm
4
jm
λ
jmjmc
k
kkCTA
,min,max,
,,,,,,,
322'
+++
−
λαη
() sPL
TA
P
e
k
D
sF
jmjm
jmcjm
jm
jm
jm
tujm
jmjsm
+−⋅
−
−
⋅=
,max,max
,,
,max
2
,
,
,
,,2
'
4
2
2
)(
η
α
λ
λ
(
α
) +
) 1
+
−−−−⋅
22
'
4
,,,
,,
,max
A
2
,,
tujmjmjm
jmcjm
jm
jmjm
k
T
P
λλ
η
αλ
()(
) (
⋅
,, max
P
,max
,min,max
2
,
,
+−⋅
−
−
tujmjmjm
jmjm tujm
jm
ksL
LLk
C
λ
λ
)(' ')(')(
,,3,,3,,3
jmjmSjmjmSjmjmS
sFsFsF
+=
where:
()
⋅
+
−
−
⋅=
jmcjm
jm
jm
jm
tujm
jmj
2
mc
jmjsm
TA
Ls
e
k
T DA
sF
,,
,min
2
,
,
,
,,
,,3
'
4
η
2
2
'
)('
α
λ
λ
η
(
α
)
()
+
+
−−−⋅
jmcjm
jm
jmtujmjmjm
TA
Ls
k
,,
,min
2
,,,,
'
4
η
62
αλλ
()
⋅
+
−
−
⋅+
jm
'
cjm
jm
jm
jm
tujm
TA
Ls
e
k
D
,,
,min
2
,
,
3
,
4
η
2
2
α
λ
λ
()(
λ
)
()
[]+++−−+⋅
342'4
,,,
2
,
2
,,,,min
L
2
,
tujmjmjmjmjmjmcjmjmjm
kTAs
λαλαηλ
(
2
)
+
−⋅
−
−
⋅
+
+
jmcjm
jm
jmjm
jmcjm
jm
jm
jm
tujm
jm
TA
P
TA
P
e
k
LsD
,,
,max
2
,,
,,
,max
2
,
,
,
,min
'
4
'
4
2
η
αλ
η
α
λ
λ
()
(
α
)
[]
22
'
4
2
2
,,,
,,
,max
A
2
,
,
,
,min
k
−−
−
−
⋅
+
λ
−
tujmjmjm
jmcjm
jm
jm
jm
tujm
jm
k
T
P
e
LsD
λλ
η
α
λ
and
()
(
λ
)(
λ
)
[
α
]
322
'
)( ' '
,,,,
, min
L
, max
L
4
,
,,,
,,3
+++⋅
−
=
tujmtujmjmjm
jmjmtujm
jmjmcjm
jmjsm
kk
k
TAC
sF
λ
λ
η
Case B:
P
jmjmjm
, min, max, max
LL
−≥
1. T’m,j < 0
(
,1
jmS
F
where:
)()()
,,1,,1,
' ''
jmjmSjmjmSjm
sss
FF
+=
()
⋅
+
−
⋅=
jmcjm
jm
jm
jm
2
tujm
jmc
2
jm
jmjsm
1
'
TA
Ls
e
k
TAD
sF
,,
, max
2
,
,
,
,,
,,
'
4
η
2
'
)(
α
λ
λ
η
(
α
)
()
+
+
−−−⋅
jmcjm
jm
jmtujmjmjm
TA
Ls
k
,,
(
,max
2
,,,,
'
4
η
62
αλλ
)
⋅
+
−
⋅−
jmcjm
jm
jm
jm
2
tu
(
jm
TA
Ls
e
k
D
,,
,max
2
,
,
3
,
'
4
η
2
α
λ
λ
()(
λ
)
)
[]
342'4
,,,
2
,
2
,,,,max
2
,
++−−+⋅
tujmjmjmjmjmjmcjmjmjm
kTALs
λαλαηλ
and
(
L
)
( )(
2
)++⋅
−
+
−=
jm
jmj
λ
mtujm
tujmjm
jmjsm
1
Ls
Lk
[
kC
sF
,max
,
+
min
k
,max
λ
2
,
,,
,,
1
)(' '
λ
λ
()(
λ
)
]
()
jmjmtu jm
tu jmtujmjmjm
4
jmjmjmc
LLk
kC TA
,min ,max,
,, ,,,, ,
3 2' 2
−
++
−
λ
αη
)() ( )() (
, ,2,, 2, ,2 ,, 2
' ' ' ' ''
jm jmSjmjmSjmjmSjmjmS
ssss
FFFF
where:
++=
()
⋅
−
T
−
⋅
−
=
jmc
A
L
'
L
e
k
TAD
sF
jm
jmjm
jm
jm
2
tujm
jmcjm
jmjsm
,
4
2
'
)('
,
,min,max
η
2
,
,
2
,
,,
,,2
α
λ
λ
η
(
α
)
()
+
−
T
−−−⋅
jmcjm
jmjm
jmtujmjmjm
A
L
'
L
k
,,
,min,max
η
(
L
4
2
,,,,
4
62
αλλ
)
⋅
−
T
−
⋅+
jmc
A
L
'
e
k
)
D
jm
jmjm
jm
jm
2
tujm
,
2
,
,min,max
η
2
,
,
3
,
α
λ
λ
((
λ
)
()
[]
342'4
,,,
2
,
2
,,,,min
L
,max
L
2
,
++−−−⋅
tujmjmjmjmjmjmcjmjmjmjm
kTA
λαλαηλ
Page 8
8
()
⋅
+
−
⋅=
jmcjm
jm
jm
jm
2
tujm
jmc
2
jm
jmj sm
TA
Ls
e
k
TAD
sF
,,
,max
2
,
,
,
,,
,,2
'
4
η
2
'
)(' '
α
λ
λ
η
(
α
)
()
+
+
−−−⋅
jmcjm
jm
jmtujmjmjm
T
)
A
Ls
k
,,
,max
2
,,,,
'
4
η
62
αλλ
(
⋅
+
−
⋅−
jmcjm
jm
jm
jm
2
tujm
TA
Ls
e
k
D
,,
,max
2
,
,
3
,
(
'
4
η
2
α
λ
λ
)(
λ
)
()
[]
342'4
,,,
2
,
2
,,,,max
2
,
++−−+⋅
tujmjmjmjmjmjmcjmjmjm
kTALs
λαλαηλ
()
⋅
+
−
⋅
−
=
jmcjm
jm
jm
jm
2
tujm
jmc
k
jm
jmj sm
TA
Ls
e
TAD
sF
,,
,min
2
,
,
2
,
,,
,,2
'
4
η
2
'
)( ' ' '
α
λ
λ
η
(
α
)
()
+
+
−−−⋅
jmcjm
jm
jm tujmjmjm
TA
Ls
k
,,
,min
2
,,,,
'
4
η
62
αλλ
()
⋅
+
−
⋅+
jmcjm
jm
jm
jm
2
tujm
TA
Ls
e
k
D
,,
, min
2
,
,
3
,
(
C
'
4
η
2
α
λ
λ
)(
λ
(
2
)
()
[]+++−−+⋅
342'4
,,,
2
,
2
,,,, min
η
2
,
tu jmjmjmjmjmjmcjmjmjm
kTALs
λαλα
[
ηλ
()
(
λ
))
]
32
'
,,,,
,min, max
4
,
,,,
+++⋅
−
+
tujmtujmjmjm
jmjmtu jm
jm
L
cjmjm
kk
Lk
TA
λλα
λ
)(' ' ')(' ')(')(
,,3,,3,,3,,3
jmjmSjmjmSjmjmSjmjmS
sFsFsFsF
++=
where:
()
⋅
−−
+
⋅=
jm
'
cjm
jmjmj
η
m
jm
jm
2
tujm
jm
'
c
2
jm
λ
jmjsm
TA
PLL
e
k
TAD
sF
,,
,max,min,max2
,
,
,
,,
,,3
'
4
2
)(
α
λ
η
(
α
)
()
+
−−
+−−⋅
jm
'
cjm
jmjmj
η
m
jm tujmjmjm
TA
PLL
k
,,
,max ,min, max
2
,,,,
4
62
αλλ
()
⋅
−−
+
⋅−
jmcjm
jmjmj
η
m
jm
jm
2
tujm
TA
PLL
e
k
D
,,
,max ,min, max2
,
,
3
,
'
4
2
(
L
α
λ
λ
)()
()
[]
342'4
,,,
2
,
2
,,,,max
P
,min
L
,max
2
,
++−−−−−⋅
tujmjmjmjmjmjmcjmjmjmjmjm
kTA
λλαλαηλ
()
⋅
+
−
⋅
−
=
jmcjm
jm
jm
jm
2
tujm
jmc
k
jm
jmj sm
TA
Ls
e
TAD
sF
,,
, min
2
,
,
2
,
,,
,,3
'
4
η
2
'
)(' '
α
λ
λ
η
(
α
)
()
+
+
−−−⋅
jmc
)
jm
jm
jmtujmjmjm
TA
Ls
k
,,
,min
2
,,,,
'
4
η
62
αλλ
(
⋅
+
−
⋅−
jmcjm
jm
jm
jm
2
tujm
TA
Ls
e
k
D
,,
, min
2
,
,
3
,
'
4
η
2
α
λ
λ
(
)(
λ
)
()
[]
342'4
,,,
2
,
2
,,,,min
2
,
++−−+⋅
tujmjmjmjmjmjmcjmjmjm
kTALs
λαλαηλ
and
()⋅+−⋅
−
⋅
−
=
sPL
TA
P
e
k
D
sF
jmjm
jmcjm
jm
jm
jm
2
tujm
jmjsm
, max,max
,,
,max
2
,
,
,
,,3
'
4
2
)(' ' '
η
α
λ
λ
(
α
)
−+−−⋅
jmcjm
jm
jmjmtujmjmjm
TA
P
k
,,
, max
2
,,,,,
'
4
22
η
αλλλ
2. T’m,j ≥ 0
()
⋅
+
−
−
⋅−=
jmcjm
jm
jm
jm
tujm
jmc
2
jm
jmjms
TA
Ls
e
k
TAD
sF
,,
,max
2
,
,
,
,,
,,1
'
4
η
2
2
'
)(
α
λ
λ
η
(
α
)
()
+
+
−−−⋅
jmcjm
jm
jmtu jmjmjm
T
)
A
Ls
k
,,
, max
2
,,,,
'
4
η
62
αλλ
(
⋅
+
−
−
⋅−
jmcjm
jm
jm
jm
tujm
TA
Ls
e
k
D
,,
, max
2
,
,
3
,
(
'
4
η
2
2
α
λ
λ
)()
()
[]+++−−
k
+⋅
342'4
,,,
2
,
2
,,,,max
L
2
,
tujmjmjmjmjmjmcjmjmjm
kT
)
A
1
s
C
λλαλα
)(
j
(
j
,
ηλ
(
L
(
)++
−
[
+
−
jm
mjmtujm
tujmjm
Ls
Lk
,max
,min,max
2
,
,,
λ
λ
)(
λ
)
]
()
jmjm tujm
tujmtu
L
j
−
mm
L
jm
4
jm
λ
jmcjm
k
kkTCA
,min,max,
,,,,,,
322'
+++
−
λλαη
)(' ')(')(
,,2,,2,,2
jmjsmjmjsmjmjsm
sFsFsF
+=
where:
()
⋅
−
T
−
−
⋅=
jmcjm
jmj
A
m
jm
jm
tujm
jmc
2
jm
jmjsm
L
'
L
e
k
TAD
sF
,,
,min,max
η
2
,
,
,
,,
,,2
4
2
2
'
)('
α
λ
λ
η
(
α
)
()
+
−
T
−−−⋅
jmcjm
jmjm
jmtujmjmjm
A
L
'
L
k
,,
,min,max
η
2
,,,,
4
62
αλλ
Page 9
9
()
⋅
−
T
−
−
⋅+
jmcjm
jmjm
jm
jm
tujm
A
LL
e
k
D
,,
,min,max
η
2
,
,
3
,
'
4
2
2
α
λ
λ
()(
λ
)
()
[]+++−−−⋅
34
)
2'4
,,,
2
,
2
,,,, min
L
,max
L
2
,
tujmjmjmjmjmjmcjmjmjmjm
kTA
λαλαηλ
(
⋅
+
−
−
⋅−
jmcjm
jm
jm
jm
tujm
jmc
2
jm
TA
Ls
e
k
TAD
,,
,max
2
,
,
,
,,
'
4
η
2
2
'
α
λ
λ
η
(
α
)
()
+
+
−−−⋅
jmcjm
jm
jmtu jmjmjm
T
)
A
Ls
k
,,
,max
2
,,,,
'
4
η
62
αλλ
(
⋅
+
−
−
⋅−
jmcjm
jm
jm
jm
tujm
TA
Ls
e
k
D
,,
,max
2
,
,
3
,
'
4
η
2
2
α
λ
λ
()(
λ
)
()
[]
342'4
,,,
2
,
2
,,,,max
2
,
++−−+⋅
tujmjmjmjmjmjmcjmjmjm
kTALs
λαλαηλ
and
()
⋅
+
−
−
⋅=
jmcjm
jm
jm
jm
tujm
jmc
2
jm
jmj sm
TA
Ls
e
k
TAD
sF
,,
,min
2
,
,
,
,,
,,2
'
4
η
2
2
'
)(' '
α
λ
λ
η
(
α
)
()
+
+
−−−⋅
jmcjm
jm
jm tujmjmjm
T
)
A
Ls
k
,,
, min
2
,,,,
'
4
η
62
αλλ
(
⋅
+
−
−
⋅+
jm
'
cjm
jm
jm
jm
tujm
TA
Ls
e
k
D
,,
, min
2
,
,
3
,
4
η
2
2
α
λ
λ
()(
λ
)
()
[]+++
(
λ
−−
'
−
+⋅
3
]
42'4
,,,
2
,
2
,,,,min
η
2
,
tujmjmjmjmjmjm
[
cjmjmjm
kTALs
λ
)
2
αλαηλ
()
()
32
,,,,
,min
L
,max
L
4
,
,,,
+++⋅
tujm tujmjmjm
jmjmtujm
jmjmcjm
kk
k
TAC
λλα
λ
)()()(
,,3,,3,,3
' ''
jmjmSjmjmSjmjmS
sss
FFF
+=
where:
()
⋅
−−
+
−
⋅−=
jmcjm
jmjmj
η
m
jm
jm
tujm
jmjmc
jmjsm
TA
PLL
e
k
TDA
sF
,,
,max,min,max
2
,
,
2
,
,,
,,3
'
'
4
2
2
'
)(
α
λ
λ
η
(
α
)
()
+
−−
+−−⋅
jmcjm
jmjmj
η
m
jmtujmjmjm
TA
PLL
k
,,
,max,min,max
2
,,,,
'
4
62
αλλ
()
⋅
−−
+
−
⋅−
jmcjm
jmjmj
η
m
jm
jm
tujm
TA
PLL
e
k
D
,,
,max,min,max
2
,
,
3
,
'
4
2
2
α
λ
λ
()(
λ
)
()
[]+++−−−−−⋅
342'4
,,,
2
,
2
,,,,max
P
,min
L
,max
L
2
,
tujmjmjmjmjmjmcjmjmjmjmjm
kTA
λαλαηλ
()
()
(
α
)
[]+−−⋅
+
−⋅
+
−
−
⋅+
62
'
4
η
'
4
η
2
2
'
,,,
,,
, min
T
2
,
,,
,min
T
2
,
,
2
,
,,
tujmjmjm
jmcjm
jm
jm
jmcjm
jm
jm
jm
tu jm
jmjmc
k
A
LsA
Ls
e
k
TDA
λλα
α
λ
λ
η
()
⋅
+
−
−
⋅+
jmcjm
jm
jm
jm
tujm
TA
Ls
e
k
D
,,
,min
2
,
,
3
,
'
4
η
2
2
α
λ
λ
()()
()
[]
342'4
,,,
2
,
2
,,,,min
L
2
,
++−−+⋅
tujmjmjmjmjmjmcjmjmjm
kTAs
λλαλαηλ
()
⋅
−
−
⋅+−⋅=
jmcjm
jm
jm
jm
jmjm
tujm
jmj sm
TA
P
esPL
k
D
sF
,,
,max
2
,
,
,max, max
,
,,3
'
4
2
2
)(' '
η
α
λ
λ
(
λ
)
++−−⋅
12
'
4
,,,
,,
,max
A
2
,,
tujmjmjm
jmcjm
jm
jmjm
k
T
P
λα
η
αλ
Case C:
L
jmjm
,min,max
L
=
1. T’
m,j < 0
()
⋅
+
A
−
⋅
⋅
=
jmcjm
jmjm
jm
jm
2
jmtu
jm
jm
2
j
2
(
m
jmjSm
T
Ls
η
e
k
eC
sF
,,
,max
'
,
2
,
,
2
,
,
,
,
,,
4
)(
α
λ
λ
α
λ
)
(
λ
) +
1
+
+−+
+
A
−⋅
12
'
4
,,,
,,
,max
T
,
2
,,
tujmjmjm
jmcjm
jmjm
jmjm
k
Ls
η
λααλ
(
λ
)
2
,
,,
jmtu
tujmj
k
m
kC
λ
−
2. T’
m,j ≥ 0
()
⋅
+
A
−−
⋅
⋅
=
jmcjm
jmjm
jm
jm
2
jmtu
(
s
η
jm
jm
2
j
2
m
jmjSm
T
Ls
η
e
k
eC
sF
,,
,max
'
,
2
,
,
2
,
,
,
,
,,
4
)(
α
λ
λ
α
λ
)
(
λ
) +
++−
+
A
−⋅
12
'
4
,,,
,,
,max
T
,
2
,,
tujmjmjm
jmcjm
jmjm
jmjm
k
L
λααλ
(
λ
)
2
,
,,
1
jmtu
tujmj
k
m
kC
λ
+
−
Page 10
10
VI. REFERENCES
[1] W. A. Brainard, “Analysis of the Economics of Photovoltaic – Diesel –
Battery Energy systems for Remote Applications” US DOE Tech. Rep.
CONF830622PT5, pp.591596, 1983.
[2] T. M. Calloway, Design of Intermedite Sized, Autonomous Photovoltaic
– Diesel Power Plant”, US DOE Tech. Rep. SAND852136, pp.129,
1986.
[3] J. M. Gordon, “Optimal Sizing of StandAlone Solar Power Systems”,
Solar Cells, Vol. 20, pp. 295313, 1987.
[4] R. Ramakumar, I. Abouzahr, “Loss of Power Supply Probability of
StandAlone Photovoltaic Systems: a Closed Form Solution Approach”,
IEEE Trans. Enegy Conversion, Vol. 6, No. 1, pp. 111, March 1991.
[5] R. Ramakumar, I. Abouzahr, An approach to assess the performance of
utilityinteractive photovoltaic
Conversion, Vol. 8, No. 2, pp. 145153, June 1993.
[6] M. Yaramanoglu, R. B. Brinsfield, R. E. Jr. Muller, “Estimation of
Solar Radiation Using Stochastically Generated Cloud Cover Data”,
Energy in Agriculture, Vol. 4, pp. 227242, 1985.
[7] J. Duffie and W. Beckman, Solar Engineering of Thermal Processes,
WileyInterscience, 1991.
[8] F. M. Butera, S. Farruggia, R. Festa, C. Ratto, “Calcolo dei Valori Medi
Mensili della Radiazione Diffusa”, Energie Alternative HTE, Vol. 50,
Nov.  Dec. 1987 (in Italian).
[9] A. Papuolis, “Probility Random Variables, and Stochastic Processes”,
Second Edition, New York: McGrawHill, 1982.
[10] K. G. T. Hollands, R. Huget, “A Probability Density Function for the
Clearness Index with Applications”, Solar energy, Vol. 30, No. 3, pp.
195209, 1983.
[11] R. Billinton, R. N. Allan, Reliability Evaluation of Power Systems,
London: Pitman Advanced Publishing Program, 1984, Chapter 2.
systems”, IEEE Trans. Enegy