Page 1

1

Abstract—This paper deals with the problem of assessing the

performance of grid-connected 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 “Power-delivered-to-the-grid” has been

derived.

Index Terms—Photovoltaic Systems, Grid-Connected 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 stand-alone or grid-connected

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 (e-mail: 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 grid-connected 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 Long-Term

Performance of Grid-Connected 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 grid-connected 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 grid-connected 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 utility-interactive 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. Closed-form

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.

CONF-830622-PT-5, pp.591-596, 1983.

[2] T. M. Calloway, Design of Intermedite Sized, Autonomous Photovoltaic

– Diesel Power Plant”, US DOE Tech. Rep. SAND-85-2136, pp.1-29,

1986.

[3] J. M. Gordon, “Optimal Sizing of Stand-Alone Solar Power Systems”,

Solar Cells, Vol. 20, pp. 295-313, 1987.

[4] R. Ramakumar, I. Abouzahr, “Loss of Power Supply Probability of

Stand-Alone Photovoltaic Systems: a Closed Form Solution Approach”,

IEEE Trans. Enegy Conversion, Vol. 6, No. 1, pp. 1-11, March 1991.

[5] R. Ramakumar, I. Abouzahr, An approach to assess the performance of

utility-interactive photovoltaic

Conversion, Vol. 8, No. 2, pp. 145-153, 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. 227-242, 1985.

[7] J. Duffie and W. Beckman, Solar Engineering of Thermal Processes,

Wiley-Interscience, 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: McGraw-Hill, 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.

195-209, 1983.

[11] R. Billinton, R. N. Allan, Reliability Evaluation of Power Systems,

London: Pitman Advanced Publishing Program, 1984, Chapter 2.

systems”, IEEE Trans. Enegy