Applied Energ)' 18 (1984) 37-47

The Design of Photovoitaic Plants: An Optimization Procedure

B. Bartoli, V. C u o m o , F. Fontana , C. Serio and V. Silvestrini

Gruppo Energia Solare, lstituto di Fisica della Facoitfi di lngegneria, Universitfi degli Studi di Napoli, Piazzale Tecchio,

80125 Napoli (Italy)

SUMMARY

In this paper we discuss a simple analytical method which allows one to predict the fraction of the load covered by a photovoltaic plant as a .[unction of the dimensions o['its components (i.e. area O['the photovoltaic array and batter), storage capacity), of the meteorological parameters and of the user's load.

We then use this method to per[orm an economic anaO'sis Jbr stand- alone plants and.[br juel-assisted phmts.

I N T R O D U C T I O N

An interesting problem in the design of photovoltaic (PV) systems is the optimal matching of the various components of the plant (e.g. solar cells' field area and storage battery capacity). The system is optimized when it covers a given load fraction at the lowest cost.

In this paper we discuss the problem of optimizing photovoltaic plants. In our analysis we use a computer simulation program which allows us to study numerically the system behaviour as a function of the dimensions of its components.

For a PV system like that shown in Fig. 1 it has been concluded ~'2 that:

(a) The results of the simulation programs in terms of performance have a simple analytical form as a function of the size and technical properties of the system.

37 Applied Energy 0306-2619/84/$03.00 (5 Elsevier Applied Science Publishers Ltd, England, 1984. Printed in Great Britain

38 B. Bartoli et al.

k4AXIMUM POWER TRACKER

ilNVERTER ILOAD

Fig. 1. Schematic drawing of the PV system analyzed.

(b) The fluctuations of the results around their means depend on statistical fluctuations of the meteorological input data.3 (c) Statistical analysis allows us to predict the numerical values for the parameters of the model.

In this paper we present and discuss the analytical features of the model and how the model can be used to optimize the system design.

THE ANALYTICAL M O D E L

Let us define the quantity:

Zm8 Y,, = - - (m = 1 , . . . , 12) (1)

Lm,~

where L,.a is the daily energy demand (load) which we assume to be constant during each month and L,,s is the monthly average on the daily energy which the photovoltaic system supplies to the load. Therefore, Y,, represents the average fraction of the load covered by the PV system during the mth month.

As a consequence of remark (a) in the previous section we can write: 2

y,, = y ( A , C, ~Iec, qPT, qc, rll, L,,d, Dr,, E,,h, gTm) (i = 1 . . . . . 30) (2)

where: A = useful area of the PV array; C = battery storage capacity; qPc = efficiency of the PV array; qPr = efficiency of the power tracker; qc =efficiency of the battery storage; ql =efficiency of the inverter; Dm= average time duration of the day in the mth month (i.e. from sunrise

Design and optimization of photovohaic plants 39

to sunset) and E,.h = daily solar radiat ion impinging on a horizontal surface (average on mth month) and:

KTm = Emh/Em,.

where: Emc = daily extra-atmospheric radiat ion on a horizontal surface, month ly average (mth month) .

Moreover, it has been shown 2 that, more particularly, Ym can be written as:

where:

y, . = y ( x m, B.,, Kr , . , D,.) (3)

emh

where q., is defined as follows:

vls = q e d h q e r (5)

(x,. being the month ly average energy supplied daily by the PV array). and:

C B,. = -- (6) Lm a ~ c

(Bin being the storage capacity normalized to the load). The explicit analytical form for eqn (3), derived from our analysis in

reference 2, is:

0' .1 - 1 ) ( y , . - G x , . ) : 7 ~ ( 7 )

where ?m represents a free function of the model to be determined by the simulation procedure. In principle, ?,~ would be a function of Din, KT,, and Bin. However, in reference 2 we have shown that:

7,, = 7(Krm - D~, Bin) (8)

Nevertheless, for our purposes, the following expression is more useful:

Br# = B(Kr ,~ . Din, 7m) (9)

We have shown 2 that, t h roughou t Italy, B,, can be written as:

a( Kr , . . Din) B,. = (b + (KT , . .Dm) ) 2 (10)

40 B. Bartoli et al.

5.0

Daram~ te r },

4 . 0 ~ . . . .

f . . . .

/

20

qll/~ " / / ~ -~-~z~ - ~ ~ _ ~ ~ ~ -~ - -~ "

,oo_1

^ ^~

IIIY~ "////" . . . . . ~ ~ ~ - ~ ~ i

~. o f l i N - - ' - " ~ " I F ' I r

0 I

• .0 .2 Din. KT m .4

Fig. 2. The normalized storage capacity. Bin. versus the variable D,,,. Kr,,(parameter 7,).

where" a =cq exp(-Tmfl~) + 1 ( l l(a))

b = 0tzy~ 2 (1 l(b))

with the following numerical values (which are independent of the locality and its climate; see also Fig. 2):

~1 = 0.695 +_ 0.035 ~t 2 = 0.274 ___ 0.008

fll = 71.2 + 5.7 f12 = 0.136 -t- 0-006

Using eqns(7) , (10), ( l l (a)) and ( l l (b)) enables us to describe the behaviour of the PV system as a function of the parameters , A (PV field area, in square metres) and C (battery storage capacity, in kWh/day), once the values for Kr , , and Dr, are given.

On the other hand, by using eqns (7), (10), (1 l(a)) and (1 l(b)), it is possible to find the values of A and C which cover a given load fraction, i.e. it is possible to 'size' the PV system once we have chosen the 'goal' value f for y,,: y, , >_ fi (m = 1 . . . . ,12).

As shown in reference 2, the statistical consistency of eqns (7), (10), (1 l(a)) and (1 l(b)) is completely satisfactory.

Design and optimization of photovoltaic plants 41

THE 'SIZING' METHOD

The most common configurations for a PV system are:

(a) The stand-alone system. In this case, we impose the minimum fraction, f , of the load which must be covered by the PV system. Because the system is not assisted by any conventional energy source, f is usually large (typically f >_ 0.9). (b) Fuel generator-assisted plant. In this case, the objective of the sizing procedure is to analyze the cost of the energy produced by the plant as a function of the size of the PV plant. Usually, the optimal solution corresponds to smaller values (e.g. f ~ 0.5) for the fraction covered by the PV system.

The analytical approach we have previously described allows us to optimize the size of the system in a simple way in both cases.

In case (a) we want: ym>_f (12)

for each month of the year, which is automatically achieved when condition (12) holds in the month for which occurs the minimum value of the ratio: R,, = (K,,.Dm)/L,, a (which corresponds, in Italy, to December). In our numerical example, we assume f = 0.90.

TABLE ! Relevant Meteorological Data in a Sample Locality (Genova) and the Values of Lind Assumed for the Sizing Examples (see ~The

Sizing Method' section)

Month D,,, Kr,,, E,,, h (k Wh tko') L,,,a (k Wh:&o')

I 0'604 0.117 1.312 1.06 2 0.594 0-234 2.701 1.06 3 0-624 0.271 3.200 0.94 4 0"631 0.332 3.975 0.94 5 0-636 0.406 4.913 0.81 6 0.638 0"493 5.933 0.81 7 0.637 0.584 7.084 1.50 8 0-633 0.492 5.926 1"50 9 0.627 0-360 4.276 0.94

10 0.618 0.255 2.967 0.94 11 0"607 0.151 1.710 1-00 12 0.595 0.110 1"202 1.00

42

1. ,80

B. Bartoli et al.

Xm

2, 3, 4.

.90 91 92

,93

.94

.95

Ym.9 6

.97

.98

.99

Fig. 3.

7 ,001 003 005 .007 009 011 .013 .015 .017 .019 .021

The fract ion y,,, of the load covered by the PV system as a function of x,,, (parameter 7,,).

Table 1 shows the relevant meteorological data in a sample locality (namely Genova) and the values of L,,n assumed for the sizing exercise.

Using Fig. 3, we find the values of x m and 7,. corresponding to )~. There is an infinite number of couples x,., ?", which correspond to f . As an example, in Fig. 4(a) three of them are indicated by bold lines: Xm, = 1 "2, 7m, = 0"007; X,, b = 1"3, 7rob = 0"017 and x" c = 1.4, 7"c = 0"023.

Using eqn (10) we now find the values of B,, corresponding to the above values 7,.,, 7,,b, 7me and to KTm. D" = 0"065 (i.e. for December). We have: B", = 2.5, B,, b = 1.8 and B,.~ = 1.5 (see Fig. 4(b)).

From eqn(4), with E , , h = l . 2 0 2 k W h / d a y ; ~h,c=0.13, r/1=0.90; ~h,T = 1 ; L"n = 1 kWh/day, we now obtain the values for the PV array area, A, corresponding to the above values of x,,. F rom eqn (6), with r/c =0.81, we obtain the values for the battery storage capacity, C, corresponding to the above values of B,,.

Design and optimization of photovoltaic plants 43

T A B L E 2 Optimized Solut ions o f the PV System (see case (a) in 'The Sizing Method' section) as Obtained by our

Sizing Method

Area Battery storage Cost (A, m 2) capacity (C, kWh) (US$/kll'h)

8.51 2.3 13000 9.22 1.6 14000 9.93 1'3 14900

In such a way, we obtain the required values of d and C (namely of the area and battery storage capacity) which cover the load fraction anticipated. We then compute the cost of each solution (see Table 2). We suppose a current cost of US$1500 per square metre for solar cells and

1. 2.5 4. .8

.9

• 99 x xbx" Xm

Fig. 4~a). The numerical example discussed in this paper: y,, ,=0.9, 7,, , , , = 0'007, .x,,,~ = 1"3, "/,,,/, = 0 - 0 1 7 , .x,, , , - 1-4, 7 . w = 0"023.

x,,,,, : t .2.

44 B. Bartoli et ai.

.5

Bma

e~.b Bm~

.0 .0

rig. 4(b).

R m .3 .6

The numerical example discussed in this paper: Kr,,,. D,,, = 0"065 (December): B,,,,= 2.5, B,,,h= 1.8, B,,,= 1.5.

$100 per kWh for battery storage. We can therefore choose the less expensive A and C couple, i.e. we can optimize the system cost.

Figure 5 shows the optimal cost versus the efficiency of the PV system for the month with the lowest value of R m for the three Italian localities. As was expected, as the efficiency, f , and the latitude of the locality increase, the PV system cost increases.

In case (b), we are interested to know the cost per kWh for the optimized condition as a function of the yearly efficiency, Yyear, of the PV system. We define the yearly efficiency, Yyear' as the mean value of the monthly efficiency over a year:

12

Yi" Lid i=1 (13)

Yyear - - 12

Lid i = 1

Thus, we can compute Yye,r using eqns (7) and (13)" Yye,r turns OUt to depend on A and C since D m and Krm are known in the chosen locality.

Design and optimization of photovohaic plants 45

8,000

6,000

4,000

Fig. 5.

~ENOVA(LAT.44.1)

NAPOLI ( LAT.40.8

I i

I I

I

I I

I I

I

I I

I i

CAGLIARI (LAT 39.3)1

I I I I

-~. .6 .7 .8 .9 1. 9

The cost o f the PV system versus .f fo r three I ta l ian local i t ies.

S kWh

6,000

4,000

J

2,000

Fig . 6.

46 B. Bartoli et al.

I I GENOVA I l,~__ NAPOL I CAGLIARI

G ENOVA / .~.APoL,

~S ~,- CAG LiARi

i i l f l i l i I

0 .1 .2 .3 .4 .5 .6 .7 .8 .9 1. Yyear

T h e c o s t o f the P V s y s t e m ve r sus Yye.r fo r t h r e e I t a l i an loca l i t i es w i t h t w o d i f f e r en t P V cells.

Design and optimization of photovoltaic plants 47

Varying A and C over a wide range and using eqns (7), (10) and (13), we obtain a matrix which gives, for each couple (A, C), the yearly fraction, Yyear, of covered load and the corresponding cost.

This computat ion procedure has been achieved by using a simple computer program, although the corresponding manual calculation is not very long.

Using this matrix we can find, for each value of)'year, a couple (A', C') which gives a first-order opt imum and then, varying (A, C) around (A', C'), we determine the couple corresponding to the optimal sizing.

Figure 6 shows the results obtained by plotting the optimal costs versus Yye~r for three Italian localities. The upper curves refer to the current solar cell cost (i.e. US$1500 per square metre).

In order to show how the cell's cost affects the cost of the system, we have also run our program for solar cells costing, US$500 per square metre, the results obtained also being shown in Fig. 6 (lower curves): the battery cost is left unchanged, i.e. US$100 per kWh.

CO NCLUSIONS

Our statistical analysis shows that long-term performances of photo- voltaic systems can be described by means of a simple analytical function.

The explicit form of such a function has been determined and turns out to be independent of the locality (at least for the Italian climate).

Using this function, the cost analysis of PV systems can be performed in a simple way and even by manual calculation.

REFERENCES

1. L. Barra, S. Catalanotti, F. Fontana and F. Lavorante, An analytical method to determine the optimal size of a photovoltaic plant. Accepted for publication in Solar Energy.

2. B. Bartoli, U. Coscia, V. Cuomo. F. Fontana and V. Silvestrini, Statistical approach to long term performances of photovoltaic systems. Revue de Physique Appliqude, 18(5), (May, 1983).

3. B. Bartoli, S. Catalanotti, V. Cuomo, M. Francesca, C. Serio, V. Silvestrini and G. Troise, Statistical correlation between daily and monthly averages of solar radiation data, Nuovo Cimento. 2C, (2) (1978).