Solar Cells, 25 ( 1 9 8 8 ) 127 - 1 4 2
127
A NOVEL METHOD FOR DETERMINING THE OPTIMUM SIZE OF
STAND-ALONE PHOTOVOLTAIC SYSTEMS
C. S O R A S and V. M A K I O S
Laboratory of Electromagnetics, Department of Electrical Engineering, University of
Patras, Patras (Greece)
(Received July 8, 1988; accepted August 8, 1988)
Summary
A method is presented to select the optimum tilt angle, photovoltaic
array area and battery storage capacity of stand-alone photovoltaic systems.
This method uses monthly average meteorological data and easily acquirable
system parameters in order to determine possible photovoltaic system sizes,
capable of supplying any given monthly average hourly load profile. The
optimum system selected is that with the minimum life-cycle cost while
ensuring a desired reliability level. In the life-cycle cost computations a
battery-life model has been used to determine the number of battery bank
replacements. The reliability criterion used is the loss-of-energy probability.
The method can be implemented on a personal computer and is applied to
an illustrative example, where the optimum system size proposed by this
methodology is compared with that of a newly installed system on a Greek
island.
1. Introduction
The existence of analytical methods for selecting the optimum size of
stand-alone photovoltaic (SAPV) systems, is the basic requirement for their
proliferation. Depending on the approach used to estimate the system
performance, which is the prerequisite in any system sizing process, two
solutions related to the system sizing problem can be identified, namely
detailed simulations and simplified design methods.
The models which perform detailed photovoltaic system simulation,
e.g. ref. 1, calculate deterministically the energy flow into the system for
each hour during the whole period of the analysis, using historical data of
solar radiation and ambient temperature. Although these models provide
an adequate solution to the system sizing problem, they feature some
significant disadvantages, i.e. large computing time, not easily acquirable
system parameter values and the need of long-term hourly meteorological
data, which are not available for most locations worldwide.
0379-6787/88/$3.50
© Elsevier S e q u o i a / P r i n t e d in T h e N e t h e r l a n d s
128
The difficulties of the above models have led to the development of
simplified methods that, without giving the variety of information of the
detailed models, are able to select the optimum system size, though with
differing degree of success (Section 2). In this paper a new methodology
is described (Section 4) which overcomes the limitations of the existing
methods. The proposed method is used in an illustrative example in Section
5 where the system size proposed is compared with a newly installed system
on a Greek island.
2. Review of the existing system sizing methodologies
Most of the simplified methods which have been developed so far
for stand-alone systems [ 2 - 6 ] are based on the comparison between the
m o n t h l y average daily photovoltaic system o u t p u t for an array of a given
size and tilt angle and the average daily load. The year round energy deficits
dictate the long-term storage capacity which added to the short-term storage
determines the effective capacity of the battery and consequently its rated
value. By varying the tilt angle and size of the array and computing the corresponding storage capacity, a number of candidate photovoltaic system
sizes are created, among which the most economical is selected as the
optimum. The simplicity and applicability to a wide range of configurations
are the main advantages of these methods. However, t h e y ignore the effect
of supply reliability on system size and hourly variation of both solar radiation and load demand on the calculation of system performance.
An extremely simple method for the selection of system size based on
a predetermined probability of system outage due to weather variations,
has been proposed by Evans e t al. [7]. This method, however, proposes only
one system size which naturally is not the most economical.
Barra e t al. [8] have proposed another analytical method for the determination of the optimal size of SAPV systems, which is based on a predetermined uncovered energy load fraction. Uncovered load fraction, however,
is not a measure of reliability but only an indication, stating that smaller
uncovered fractions result in higher reliability. Further drawbacks of this
m e t h o d are: the assumption that all the energy produced by the photovoltaic array passes through the battery, and the use of initial investment as
the economic criterion, which does not take into account the effect of the
number of battery replacements.
A method which takes into account the effect of reliability on system
size has also been developed [9, 10]. The method is limited to one value
of loss-of-energy probability (LOEP) and two array tilts. Furthermore this
m e t h o d assumes that all the energy from the photovoltaic array passes
through the battery, it is based on daily (not hourly) values of solar radiation
and load, and does not consider the effect of system performance on battery
lifetime.
129
3. Definition of the problem
SAPV systems exhibit the following contradiction: although they are
the first systems put into use and have the highest arithmetic percentage of
applications of all photovoltaic systems, they are the most difficult to determine their o p t i m u m size, due to the use of batteries as the sole back-up
source which influences immensely the reliability of the whole photovoltaic
system. The o p t i m u m size, therefore, cannot be determined independently
of the load supply reliability which it ensures. The sizing problem could be
finally summarized as follows.
Which is the combination of the size A of the photovoltaic array, its
tilt angle s and the rated battery capacity B, that satisfies a predetermined
reliability goal and is the most economical?
In this paper an analytical method is described for the selection of the
o p t i m u m combination, which uses the LOEP concept for the reliability
calculation and the life-cycle costing methodology (LCC) for the economic
calculations. In Fig. 1 a typical SAPV system configuration is shown, which
will be used in our analysis.
SOLAR
PV ARRAY
RADIATION/
/
VOLTAGE
REGULATOR
/
EB
CABLES
OR
CONVERTER
OR
INVERTER
ED
"BATTERY i Ec
Fig.1. Typicalconfigurationof a stand-alonephotovoltaicsystem.
4. Description of the methodology
The analytical method developed is presented in the flow chart (Fig. 2),
while the necessary input data are given in Table 1.
For each value of the array tilt the following calculations are executed
step-by-step. Initially the monthly average hourly radiation on the tilted
surface
calculated according to well known formulae [11 ]. The monthly
average daily insolation on the tilted surface/TT is then calculated as
i Tis
130
INPUT DATA
I
FOR s=q~-lO° TO s=(p+20'
STEP 5°
FOR MONTH=I TO 12
FOR HOUR= I TO 24
CALCULATION OF l'r
CALCULATION OF H'r' °ll'I' ' n
ESTIMATION OF A .
AND A
mln
m&x
FOR U=O TO 0.1 STEP AU
FOR A=Amin TO Amax STEP ~A
FOR MONTH= I TO 12
FOR HOUR=I TO 24
CALCULATION OF EA,i ,$ ,E~3,i , ED,I
CALCULATION OF
N
~
c~
YES
ESTIMATION OF Be, m
SELECTION OF Be, Br
CALCULATION OF LOEP, LEC
I
I
SELECTION OF THE OPTIMUM SYSTEM
Fig. 2. Flow chart of the sizing methodology.
24
/qW=~/T
(1)
1
For the calculation of the standard deviation of the daily insolation, O~HT, the
method presented in Section 11 of the Monegon handbook [9] has been
used. The calculation of the monthly average daffy array efficiency ~/e is
executed via the simplified method of Evans [12] (see Appendix A).
131
TABLE 1
The input data required for the sizing methodology
Input data
Location data
Latitude ¢, ground reflectance p, monthly average daily isolation on a horizontal surface H, monthly average ambient temperature Ta
Load data
Monthly average hourly load Li, uncovered energy load fraction U
PV-array data
Efficiency at reference conditions of insolation and temperature ~TR,temperature coefficient for cell efficiency ~, nominal
operating cell temperature (NOCT), degradation factor 77d, incremental array step AA
Battery data
Average energy efficiency 77B, maximum allowed depth of
discharge DODs, battery-type coefficient Bc, average battery
life N A
Power conditioning
unit data
PV-array wiring efficiency ~w, maximum power point tracking
efficiency ~TT, voltage regulator efficiency WVR, cabling efficiency r/c
Economic data
Component costs, structure and foundation cost, economic
evaluation period Y, general inflation rate g, discount rate d,
escalation rate for O & M e 0
A m i n i m u m a r r a y area Amin, b a s e d o n t h e f a c t t h a t t h e p h o t o v o l t a i c
a r r a y m u s t b e c a p a b l e o f covering t h e load plus t h e p o w e r c o n d i t i o n i n g u n i t
losses o n a y e a r l y basis, can b e r o u g h l y e s t i m a t e d f r o m t h e f o r m u l a
Ami n = I N T E G E R [ -I
-~
[AA
~'~T~end
1
J
(2)
where Ld is the montly average daytime load, L is the monthly average daily
load (L = Z24
~- A A is the incremental step in the array area, which de~=l~J,
pends on the array configuration for each application (e.g. it can be the area
of an additional module or subarray) and the other factors are explained in
Table 1. At the other extreme a m a x i m u m array area Am~x can be estimated
as a multiple (say 3 times) of the array area required to cover the load plus
losses for the worst case month:
Amax =
3[max{(f'a+(L--~d)/~B)/(~W~T~V~C)}]
I'IT~eT~d
(3)
132
For a given value of array area A, between Ami n and Amax, it is possible
to compute the energy flow in the photovoltaic system. The monthly
average hourly energy output from the photovoltaic array ]~A,i, is
EA,i = A/w~eT~W7~d
(4 )
The monthly average hourly energy /Ts, i which is available to charge the
battery is then
EB, i = /~A, iT~T?TVRfl)
(5)
where (P is the hourly radiation utilizability function calculated via an
analytical formula developed by Clark e t al. [13] (see Appendix B). The
m o n t h l y average hourly energy ED, i which flows directly from the photovoltaic array to the load is then:
/~D,i = EA, iT~T~/VR(I -- CI~)~/C
(6)
The m o n t h l y average daffy energy flow in the photovoltaic system is then
obtained by summing the hourly values:
24
EA = E gA, i
i=l
24
EB = ~ E~,i
(7)
i=l
24
ED = 2: ~D,i
i=l
and the daffy solar radiation utffizability c~ is taken from the equation:
24 IT~
= i =El tTT
(8)
The critical level of the monthly average daffy insolation /~c on the tilted
array at which the energy supplied by the photovoltaic system is equal to the
load/~ is
H c = Y,/A~?s
(9)
where ~, is the m o n t h l y average efficiency of the entire photovoltaic system
and is given by the equation
~s = ~Ter/W~T?/VR~TCr/d( 1 -- ~ + ?/B c~)
(I0)
The condition /~c ~>/~T implies an inadequate array. Otherwise it is
possible to appoint suitable values to the rated battery capacity B~ in order
to create candidate system sizes of the form (s,A, Br). This was done by
varying the uncovered energy load fraction U, which is defined as
133
U = I -- f
(11)
where f is the energy load fraction covered by the photovoltaic system and
which for stand-alone systems tends to unity. For given values of array tilt,
c o m p o n e n t efficiencies and climatic data, f depends on the size of the
array and the battery. In the extreme case where no dumping losses occur
in the battery, all the energy produced by the photovoltaic array reduced
only by c o m p o n e n t inefficiencies is transfered to the load and thus f is
equal to its m a x i m u m value fm calculated as
(12)
fm ----(ED "{- J~B?~B~C)/j~
This occurs either for every small array areas or for very large storage capacities (infinite) relative to the load, provided an adequate array size. In all
other cases owing to dumping losses, f is smaller than fro. Figure 3 shows the
relationship of f with its m a x i m u m value fm for a given storage capacity
along with the asymptotes for infinite capacity. This curve has the form of
a hyperbola and can be formulated using the following equation proposed
by Bartoli e t al. [14] :
(I -- f)(fm -- f) = ~/
(13)
where the parameter ~/ determines the adherence level of the curve to its
asymptotes and is essentially a function of storage capacity. Thus, for given
values of array size and tilt, fm is determined from eqn. (12) and for each f
value the parameter ~/ from eqn. (13). Finally for a given set of the form
(s, A, U) the effective storage capacity Be,~ for each month is calculated
according to ref. 14 as
aDKTL
Be,m ~B(b+ D/~T)2
(14)
where D is the daylight fraction of the day,/~T is the monthly average clearness index, a = 0.695 exp(--71.2"y) + 1 and b = 0.274To'la6.
1.0
f
0.5
0
I
2
fm
3
Fig. 3. Energy load fraction f covered by a SAPV system
case of no dumping losses [14].
vs.
its maximum value fm for the
134
The month with the maximum storage requirements dictates the effective battery capacity which corresponds to the given values of s, U and A :
Be = max(Be.m)
(15)
The rated battery capacity is then:
(16)
B~ = B e / D O D s
where DODs is the maximum allowable depth of discharge of the battery
during its seasonal operation.
For each candidate photovoltaic system size of the form (s, A, B~) the
supply reliability it ensures is calculated in terms of the yearly LOEP which
b y definition [15] is the sum of the monthly values LOEPM:
12
LOEP = ~
LOEPM
(17)
M= I
The method used for the LOEPM calculation was that°described in ref. 9.
For each candidate size besides LOEP, its levelized energy cost, LEC, is also
computed using the m e t h o d o l o g y of Appendix C, where a model of battery
life is used for the estimation of the number of battery replacements through
the system lifetime.
5. A sizing example
Let us consider the case of a remote house for which we intend to
cover the basic energy needs in terms of lighting, radio, television and refrigeration. The load characteristics assumed are presented in Table 2 and the
resulting load profile in Fig. 4, The house is located on a Greek island,
Antikythira, with ¢ = 36 °, p = 0.2. The m o n t h l y average values of insolation
/7 on a horizontal surface, ambient temperature T~ and load/~ are presented
in Table 3.
TABLE 2
R e m o t e h o u s e l o a d characteristics
Appliance
Nominal power
Hours in operation
(W)
Winter
Summer
2 fluorescent lamps
2 × 20
18.00 - 24.00
20.00 - 24.00
Refrigerator
( 1 1 5 l)
30 ( W i n t e r )
40 ( S u m m e r )
0.00 - 24.00
0.00 - 24.00
T V (17 in)
Colour
37
18.00 - 24.00
18.00 - 24.00
135
Ci(w)
Summer--~F . . . . .
!
120
100
I
Wint
1
80
60
40
20
6
12
HOUR OF THE DAY
18
24
Fig. 4. Average hourly load profile for the winter and summer months.
TABLE 3
Input data for the sizing example
Month
/~ (W h m -2 day -1)
Ta (°C)
L (W h day -1)
1
2
3
4
5
6
7
8
9
10
11
12
2032
2679
3806
5100
5226
7233
7387
6774
5333
3677
2533
1935
13.5
13.8
14.6
17.5
21.7
26.3
29.3
29.3
26.1
22.1
18.8
15.1
1182
1182
1182
1342
1342
1342
1342
1342
1342
1182
1182
1182
T h e p h o t o v o l t a i c m o d u l e s t o b e used in this a p p l i c a t i o n have t h e
f o l l o w i n g characteristics: ~?R = 11% at TR = 25 °C a n d GT = 1 kW m -2, ~ =
0 . 0 0 4 °C -1 , N O C T = 45 °C a n d ~d = 0.90. Since t h e n o m i n a l o p e r a t i o n
v o l t a g e o f t h e s y s t e m has b e e n d e t e r m i n e d t o b e 2 4 V d.c., t h e i n c r e m e n t a l
a r r a y step results in AA = 0 . 7 8 2 m 2 (i.e. t h e area o f t w o a d d i t i o n a l series
modules).
T h e c h a r a c t e r i s t i c s o f t h e flat-plate b a t t e r y w h i c h is a s s u m e d t o b e
used are ~?B = 0.80, D O D s = 0 . 8 0 , Bc = 0 . 0 3 a n d NA = 8 0 0 cycles.
T h e e n e r g y efficiencies o f t h e s y s t e m c o m p o n e n t s are ~?w = 0.97,
7~T 0 . 9 5 , ~VR = 0 . 9 9 , ~C = 0.98.
T h e c o s t values o f t h e c o m p o n e n t s a n d t h e e c o n o m i c p a r a m e t e r s w h i c h
are presented in Table 4 are typical for Greece t o d a y (U.S$1 = 140 drch).
=
136
TABLE 4
Cost and e c o n o m i c a s s u m p t i o n s
Cc~ = 15 $ Wp - l
Cb=200$k
W 1h 1
Array cost
Battery cost
Structure and f o u n d a t i o n c o s t
V o l t a g e regulator cost
Electrical system cost
E c o n o m i c evaluation period
General inflation rate
Escalation rate for O & M
D i s c o u n t rate
mo °
.W
~<
L,J
k'--
15
14
13
12
11
10
9
1
Ce = 700 $
Y = 20 yrs
g = 0,15
e 0 = 0.12
d = 0.17
s=45°
01[~I i
~
LOEP
......................
8
7
C s = 0 . 9 $ Wp
Cr = 850 $
O- 3
6
10
5
~I, I ~.:'2'....
.......... .......................................................................................................
".,.. ....
.~
4
'-
3 26,10-2
2
..............................................................................................
~
..........................
" ......................................................................
..
. .
0,1
1
0
7-/4 860 946 1032 ' 1204 1376 1548 1720
PHOTOVOLTAICARRAYSIZE (Wp)
1892 ' 2064
Fig. 5. E f f e c t i v e battery capacity B e ( k W h ) v e r s u s P V a r r a y size A (Wp) w i t h U and L O E P
as parameters for s = 45 °.
The results of the proposed m e t h o d o l o g y for the above described
r e m o t e h o u s e are s h o w n in Fig. 5 and Table 5 for tilt angle s = 45 °, w h i c h
was f o u n d t o result in t h e m o s t e c o n o m i c a l system. The p h o t o v o l t a i c array
size A (Wp) is simply:
A (Wp) = A (m 2) 1 0 0 0 17R
The
installed
poration.
effective
(18)
results o f our m e t h o d o l o g y are n o w being compared w i t h a system
o n the same island, Antikythira, b y the Greek Public Power CorThis system has an array o f A = 6 9 0 Wp, tilted at s = 45 ° and has
battery capacity, Be = 7.7 kW h. Calculating the supply reliability
137
TABLE 5
The optimum system for different LOEP values
LOEP
s (deg)
A (Wp)
Be (kW h)
LEC ($ kW-1 h-1)
0.2
0.1
0.01
0.001
45
860
4.6
7.9
45
860
5.9
8.0
45
946
7.9
9.2
45
860
14.2
9.3
of this system size, using our method and under the same assumptions
mentioned above, we found LOEPM values greater than one for December
and January, i.e. the installed system is likely to fail for a period of two
consecutive months, something undesirable. Repeating LOEP calculations
for the installed system with the assumption that the o u t p u t of the photovoltaic array is not degraded with time, i.e. ~d = 1, we found a value of the
yearly LOEP of 0.24, which is translated to an outage probability of 7 days
per year equally distributed between December and January. This value of
7 days per year is acceptable for such systems (taking into account that t h e y
are installed on a location w i t h o u t electricity to date), but this value will
increase substantially as degradation effects rise with time, resulting in unacceptable values, as we mentioned above for a degradation of only 10%
0?d = 0.90). For this reason, we believe, that it would have been more
suitable to install a larger array from the beginning although having a smaller
battery capacity as can be seen from Table 5. The very high LEC values
(Table 5) reflect the high initial cost of photovoltaic systems in Greece,
because of high import d u t y , t o d a y up to 70%, and also the large values of
inflation and discount rates.
6. Discussion
For the estimation of performance of the photovoltaic system, two
simplified methods of calculating the m o n t h l y average array efficiency Re
and the hourly utilizability dp have been used in this paper as discussed in
Section 4. The use of these methods, the accuracy of which has been compared with detailed simulation programs using long-term historical data,
secures an increased accuracy in energy flow calculations, especially taking
into account the uncertainty in solar radiation data.
Further, the use of an empirical correlation for the estimation of the
battery capacity Be (eqn. {14)) is not critical, since it gives only an indicative
value to create a series of candidate systems. This could be done also by
varying Be in the range of, say, 2 to 30 daffy loads, although a much larger
computing time is required.
138
In conclusion, the novel m e t h o d d e v e l o p e d in this paper to select
t h e o p t i m u m size o f SAPV systems has the advantages o f all simplified
m e t h o d s , i.e. use o f m o n t h l y average m e t e o r o l o g i c a l data, easily acquirable
system p a r a m e t e r values and small calculation t i m e in a PC system. Moreover, it possesses the following characteristic advantages w h i c h in previous
m e t h o d s are n o t usually t a k e n into a c c o u n t .
(a) It uses the L O E P criterion, a w i d e l y used reliability c o n c e p t in
p o w e r systems, having the advantage o f taking into a c c o u n t the cost vs.
reliability t r a d e - o f f in system sizing. LOEP also provides a c o m m o n base
f o r c o m p a r i s o n w i t h o t h e r systems e.g. a diesel generator.
(b) It is applicable f o r a n y m o n t h l y average h o u r l y load profile.
(c) By using the utilizability c o n c e p t , it raises t h e a c c u r a c y o f calculat i o n o f s y s t e m p e r f o r m a n c e , because it does n o t use t h e w r o n g a s s u m p t i o n
t h a t all e n e r g y passes t h r o u g h the b a t t e r y .
(d) It uses a m o d e l f o r t h e b a t t e r y lifetime based o n its average daily
d e p t h o f discharge and t h e specific characteristics o f t h e t y p e o f b a t t e r y .
(e) It uses as the e c o n o m i c criterion the life cycle cost m e t h o d o l o g y ,
w h i c h is m o s t a p p r o p r i a t e f o r solar e n e r g y systems.
Acknowledgment
T h e a u t h o r s w o u l d like to t h a n k Dr. J. Milias-Argitis f o r m a n y f r u i t f u l
discussions and his valuable assistance for t h e graphical p r e s e n t a t i o n o f the
results.
References
1 E. Hoover, SOLCEL-II: An improved photovoltaic system analysis program, Sandia
National Laboratories Rep. SAND 79-1785, 1980 (Sandia National Laboratories,
Albuquerque, NM).
2 Solar Power Corporation, Solar Electric Generator Systems; Principles of operation
and design concepts, 1979.
3 C. Soras and V. Makios, Feasible stand-alone photovoltaic systems in Greece, Proc.
5th Eur. Photovoltaic Solar Energy Conf, Athens, Reidel, Dordrecht, 1983, p. 485.
4 M. Buresch, Photovoltaic Energy Systems; Design and Installation, McGraw-Hill,
New York, 1983.
5 PRC Energy Analysis Company, Solar photovoltaic applications seminar: design,
installation and operation of small stand-alone photovoltaic power systems, U.S.
DOE/CS/32522-T1, 1980 (Department of Energy, Washington, DC).
6 Monegon Ltd., Designing small photovoltaic power systems, No. M l l l , 1981.
7 D. L. Evans and F. T. C. Barrels, Battery sizing criteria for stand-alone photovoltaic
power systems, AS/ISES, Philadelphia, PA, 1981.
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CR-165352, (NASA Lewis Research Center), 1981.
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139
11 J. A. Duffie and W. A. Beckman, Solar Engineering of Thermal Processes, WileyInterscience, New York, 1980.
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Energy, 27 (1981) 555.
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Appendix A: Calculation of ~ [A1]
The calculation of the monthly average daily photovoltaic array efficiency ~e is executed using the formula
a +T~+3
~, = ~Ta 1 - - /3(h(/~T)F UL
- - TR)]
(A1)
where
(°C -1) -
1 - - ~72/7~1
(A2)
Tc~ - - Tel
~71,772 are t w o array efficiencies at t h e corresponding cell temperatures T c,,
Tc: under the same i n d o l a t i o n value
h(/~T) (kW m -z) = 0 . 2 1 9 + 0.832/~T
(A3)
F is a correction factor for n o n - o p t i m u m tilted arrays:
F = 1 -- 1.17 × 10 -4 (Sopt - - s ) 2
(A4)
The o p t i m u m array tilt Sopt, for each m o n t h as a f u n c t i o n o f latitude
¢ is given in Table A 1 . The ratio o f thermal loss coefficient U L (kW m -2 °C-I)
TABLE A1
Monthly optimum tilt angles
Month
Sopt (deg)
January
February
March
April
May
June
July
August
September
October
November
December
~b+ 29
~b+ 18
~b+ 3
~ b - 10
~ -- 22
~b-- 25
~ b - 24
~ b - 10
¢- 2
¢ + 10
~b+ 23
¢ + 30
140
to solar cell a b s o r b a n c e a can be e s t i m a t e d f r o m t h e installed n o m i n a l
o p e r a t i n g cell t e m p e r a t u r e (INOCT) which is d e t e r m i n e d f r o m the N O C T
test d a t a and the w a y t h e array is installed, f r o m the following e q u a t i o n :
UL
GT,NOCT
-
(A5)
a
( I N O C T -- T~,NOCT)
T h e values o f solar r a d i a t i o n GT, NOGT and a m b i e n t t e m p e r a t u r e Tc~,NOCw are
k n o w n f r o m the N O C T test, while for r a c k - m o u n t e d arrays I N O C T =
N O C T - - 2 [A2].
References for Appendix A
A1 D. L. Evans, Simplified method for predicting photovoltaic array output, Sol.
Energy, 27 (1981) 555.
A2 M. Fuentes, Thermal characterization of flat-plate photovoltaic arrays, Proc. 18th
IEEE Specialists Conf., 203 (1985).
A p p e n d i x B: Calculation o f ~ [B1]
T h e m o n t h l y average h o u r l y utilizability 4p is calculated f r o m the
equation:
if X c ~ X m
or=
i°
(1--XdXm) 2
ifXm = 2
( I ] ~ ] -- {~2 + (1 + 2 a ) ( 1 - - X e / X m ) 2 ) I / 2 [
(B1)
otherwise
%
where
(B2)
o~ = ( X m - - 1)/(2 -- Xm)
X m is a dimensionless variable d e f i n e d as
X m = 1.85 + 0.169/~/k 2 -- 0 . 0 6 9 6 ( c o s
S)/k 2 --
0 . 9 8 1 k / ( c o s 5) 2
(B3)
/~ is the ratio o f t h e m o n t h l y average h o u r l y insolation o n t h e tilted array
to t h a t o n a h o r i z o n t a l surface
R = Iw/I
(B4)
is t h e m o n t h l y average h o u r l y clearness i n d e x
k = I/I o
(B5)
5 is t h e d e c l i n a t i o n o f t h e sun and X c is t h e dimensionless critical h o u r l y
r a d i a t i o n level
Xe = f c / i T
(B6)
w h e r e i c is t h e h o u r l y critical r a d i a t i o n level at w h i c h the h o u r l y e n e r g y
p r o d u c t i o n is equal to t h e h o u r l y load
141
References for Appendix B
Bl
D. R. Clark, S. A. Klein and W. A. Beckman, Algorithm for evaluating the hourly
radiation utilizability function, J. Sol. Energy Eng., 105 (1983) 281.
Appendix
C: Calculation
of photovoltaic
system economics
The life cycle cost LCC of a photovoltaic
system consists of three
items: the initial capital investment Cr, the present value of operation and
maintenance
costs OM,.,. and the present value of battery replacement costs
R p.v. [Cl1
LCC = Cr + OM,.,. + R,.,.
(Cl)
is a function of the operation and maintenance
0%“.
first year of operation OM, and the economic environment:
cost during the
(3
0Mp.v. =
ifd=eO
OM,Y
where OM, is assumed to be [C2] :
+ B,C,,)
OM, = O.Ol(AC,
(C3)
R P.v. is mainly a function of the number of battery replacements
II,
over the system life and, assuming no salvage value of the replaced batteries,
is given by
R P.V.
u 1 +g
=B&
z
j=l
vilv+l
-
i l+d
(C4)
I
The battery
life N, (cycles) in real operation is dominated
by the daily
depth of discharge DODd and depends on the specific battery characteristics,
i.e. average life NA (cycles) at a specified DOD,, (usually in laboratory tests
DOD,, = 0.8) and value of the battery coefficient
B,, which for flat-plate
batteries lies in the range 0.02 - 0.03 and for tubular batteries from 0.01 to
0.02 [C3]:
N, = 0.5NA exp{-B,lOO(DODd
where
- DODo)}
(C5)
142
T h e n , v is c o m p u t e d as
v = INT{Y/(NR/365))
(C7)
T h e levelized e n e r g y cost f r o m t h e p h o t o v o l t a i c s y s t e m is t h e n
LCC × C R F
LEC -
12
(C8)
E Nd/
M=I
w h e r e Nd is t h e n u m b e r o f d a y s per m o n t h a n d C R F the capital r e c o v e r y
f a c t o r given b y
C R F = d / { 1 - - (1 + d) - y }
(C9)
References for Appendix C
c1 H. L. Macomber, J. B. Ruzek, F. A. Costello and staff of Bird Engineering, Photovoltaic Stand-Alone Systems: Preliminary Engineering Design Handbook, N A S A
CR-165352, (NASA Lewis Research Center), 1981.
C2 PRC Energy Analysis Company, Solar photovoltaic applications seminar: design,
installation and operation of small stand-alone photovoltaic power systems, U.S.
DOE/CS/32522-T1, 1980 (Department of Energy, Washington, DC).
C3 L. Thione, R. Buccianti, L. Dellera and P. Ostano, A contribution to the assessment
of specific problems of large photovoltaic generation plants in view of the improvement of their reliability, CESIoMilano, EEC Contr. No. ESC.P.052.I(S), 1984.