Sensitivity analysis of stand alone PV systems

WREC 1996

SENSITIVITY ANALYSIS OF STAND ALONE PV SYSTEMS A . M. A I - A S H W A L D e p a r t m e n t of Electrical E n g i n e e r i n g Sana'a University P.O. B o x 1 2 1 5 3 Sana'a ; Yemen

Abstract Daily solar insulation is stochastic quantity. Therefore there is always a risk for a stand alone PV system not to meet the load. loads fed by PV systems widely vary by the importance. Therefore there should be different risk level for different loads, eg the loss of load risk (LOLR) level for rural house could be much less than that for telecommunication installation. Risk reduction is always accompanied by cost increase. This paper introduces a simple method to assess LOLR with least costs which can lead to better PV system design.

Keywords: Photovoltaic, Loss of Load Risk, Probability Distribution, Least Costs 1. I n t r o d u c t i o n : The use of photovoltaic (PV) as an electrical energy source is in continuous increase. More over PV is sometimes the only feasible alternative to supply remote loads, eg telecommunication and TV transmission stations in mountainous areas [1]. Other fields of PV applications have become highly competitive for conventional sources, eg rural electrification of remote areas [2]. The amount of electrical energy generated by PV is completely dependant on solar insulation which is a stochastic quantity. Therefore there will be a Loss of Load Risk (LOLR). Various loads would require different level of risk, ie a specific reliability level of supply by PV has to be satisfied. Such a level may be achieved by : • increasing the number PV panels, or • increasing the storage capacity of Batteries, or • increasing both. The problem is how to achieved required LOLR level with minimum cost which is the goal of this work. This goal is realized by introducing a simple method based on probability function evaluation to assess LOLR with least costs.

2. Method of Analysis As mentioned earlier the method is based on probability function technique. T w o approaches are used. one for one day storage capacity of the batteries and the other is for more than one day storage capacity.

432

One day storaae caoacitv Measured data of daily sunshine hours of a year is processed. Data processing results in a matrix of two columns. First column consist of number of occurrences. The second column consist, of shun shine hours stating with maximum sunshine hours and ending with minimum. Using this matrix, the probability density function and cumulative probability function of sunshine hours can be evaluated [3] as follows:

Ni

Pi = --~ . . . . . . . . .

Pci = ~ Pi . . . . . . . . .

(i)

(2}

where : N~ - number of occurrences of the ith interval of sunshine hours. Pi - is the probability of ith interval to take place. N - Total number ofoccurrence. P= - Cumulative probability. Then the energy which could be produced by PV according to time intervals of sunshine hours is evaluated as follows: [W] = Ppv [T] . . . . . (3) Where : [W] - Vector of expected Energy produced by PV in Wh. Ppv - average daily power Generated by PV

WREC 1996 T - Vector of sunshine hour intervals. For a given number of PV panels the probability of PV system success to meet the daily energy demand is the cumulative probability (P=) which corresponds to the energy produced by PV being greater or equal to that demand, ie: W~ ;~ Wd

........

(4)

Then LOLR is found as [4]: LOLR = 1 -Po . .

. . (5)

More than one day storaQe caoacitv Usually storage Batteries is designed for more than one day. In this case there is a possibility for Batteries to be charged to a level of energy higher than the daily demand. Therefore it is necessary to take an account for the residual energy of the battery. Therefore the energy stored in the battery is calculated every day of the year, ie: _

[W b] = [ T j P ~ + W,-Wd . . . . . . (6) where : W b - is vector of daily stored energy in the batteries, W r - residual energy from previous day. W d - daily energy demand. T d - is vector of daily sunshine hours of the y e a r The elements of [W b] - vector are limited by storage capacity of the batteries. Equation 6 shows that the load has been less influenced by daily sunshine hour intervals due to the presence of residual energy. Therefore the vector of daily stored energy [W b] is used as an argument of probability functions but the evaluation is performed according to equations 1 and 2 (see Appendix 2). The success of the system to meet daily demand is determined as: Wb(i) > W d . . . . . . .

(7)

corresponding cumulative probability (Po) indicates probability of this success. Then LOLR is found by means of Equation 5. Eventually LOLR is evaluated for different cases of number of PV panels and storage capacities as will be demonstrated next. 3. Application

example

Fig. I shows the lay out of PV system to be studied. Let the load be in Sana'a region and has a daily energy demand of 1000 Wh, A daily records of sunshine hours data for this region of year 1988 was

433

Fig. 1 PV System Block Diagram obtained [5]. This data was processed resulting in a number of occurrences and related time intervals starting with 12 h ending with zero hour and increment of 0.1 hour. Further probability density function is evaluated using Equation 1 hence cumulative probability function is evaluated using Equation 2 (see Appendix 1). Fig. 2 and Fig. 3 show graphs of these functions. The minimum number of PV panels to meet such a demand is assumed five with peak power equals to 40 Wp each. The cost of PV given in [6] is used here, ie US$5 per 1 Wp. The batteries and controller cost is assumed US$50 for each half day storage capacity. One - day storage In this case the capacity of the batteries is equivalent to 1000 Wh. Therefore one should expect that the batteries will almost be completely discharged. Thus LOLR calculation is based the conditions expressed by inequality 4 and Equation 5. LOLR was calculated for different number of PV panels. More Than One Day Storaqe In this case probability functions are evaluated as explained earlier, ie their argument is [W b] vector Appendix 2 shows results of such calculations for two days storage capacity. Fig. 4 and Fig, 5 show plotted curves of these functions, Thus LOLR calculation is based the conditions expressed by Equation 5 and Inequality 7, Here also LOLR was calculated for different number of PV panels and different storage capacities. Evaluation results are shown in Fig. 6.

W R E C 1996 4. Result Discussion Probability

4.1 Probability Function Distribution Curves expressing probability functions s h o w n in Fig. 2 and Fig. 3 indicate that such functions may be at

Enelgy

Density

Function

of

( 2 Day•

Storage

Capacity)

Expected

Problblllty

o.1

Probability Desity Function of Sunshine Hours

.

.

.

.

O 1 PrOMblllty

0.09 -- -:--- !-- -:----:+--'-- -:-+-;-+ i " : - " ~ ":"-~.' -:o.o, i - i - + 4 - . i i i - i - : - : - - ~ -:-~ : o.o~-+-i--+i---i---i---i---:---i---i---:.---i-:-..:+.-:--o.o°

- - -:- - - : - - -',- - - : - - - '. - - -:- - - : + - -:- - - : - .

-'~

---:---',

0.09 ---:.+-:---:---~---!---i---~---:-.-~,---: o.oa

o.oi

- - ~ - • - '~ - - -:- - - ', - - -' . 0

1

2

,,

. . . . . . . . . :. . . . . 8

Fig. 2

4

.

°

. °

7

Sunsine

8

9

10

11

12

200O

25OO

(Wh)

4.2 Required LOLR With Least Costs Curves s h o w n in Fig, 6 express evaluation results of LOLR against the costs. These curves show clearly

13

14

(/1)

Cumulative Energy

Probability Cumulative Function (Shunshine Hours)

Probability ~ 2 Days

of

Storage

Expeoted Capacity

)

Ptoblbllll¥ 1,2,

PrOblblllly

I ~

....... ; ...... ! ...... ! .............. o . . . . . . . . . . . . . . . . . . . . i ..... ! ...... i ....

lsoo Energy

,.

:- - - -:- - -

Intorva/s

looo Expected

Fig. 4

i+"i "

.

-: 500

-.-:---: ....

- - -',+ - - , - - -',-+ - -3- - -~ - - -;- - - ~ - - -;- - - -, . . . . .

o.o= " : " : " ! i ' :

OOOl

....

+

-

o.e~- . . . . . . . .

~

:

; ',. . . . . . . . .

;

-

.

',. . . . . . . . .

~

........ ~ ........

! .......

i I

o.,,+++ . . . . . . . .

:

',

:

:

!. . . . . . . . .

!. . . . . . . . .

! ........

! ........

~=~ ........ !......... ! ........ : . . . . . . . : . . . . . . . 0.4

......

', . . . . . .

'. . . . . . .

" ......

: ......

' ...........

° 0"

0.2 o [ ...... ~....... !....... ~ '+...... :+...... =". . . . . . . . . . 0

2 Sun•hlne

8 Intervals

10 (h

)

12

: 800

: 1000 ExpeQted

: 16o5 Energy

i 2050

56oo

(Wh)

Fig. 5

14

Fig. 3 that: • Increasing number of PV panels has led to LOLR reduction down to 0.017. Further increase does not lead to LOLR reduction, ie LOLR keeps constant, • Increasing storage capacity has led to LOLR reduction with less cost increase. • Increasing both number of PV panels and storage capacity has led also to LOLR reduction. However each individual case has an LOLR reduction limit where LOLR becomes constant. • From these curves one can derive a table of least cost for specified LOLR (see table 1 ). Table 1 s h o w s also cost (C, US$) components: number of PV panels (No,PV) and storage capacity (St,D).

best fitted to Gamma function distribution [3]. Such fitting would have led to a great simplification in PV system design. H o w e v e r this will held true just for one day storage capacity which is practically inadequate because standard design would include more than one day storage capacities. In this case probability distribution functions s h o w n in Fig, 4 and Fig. 5 can't be fitted to any standard distribution. Consequently one may conclude that the PV system design should be performed using measured data rather than standard probability distribution application.

434

WREC 1996 LOSS of Load Risk with Different PV and Storage Capacities LOEB

o.1 , : : , : : , : : ' : : , : : ' : : , : ~ : ~ : ~ : ~ : ; : ~ : ~ : ~ ! ~ i ' i ! ' : i ' i ! ' ! : ' ! i ' : !

List o f R e f e r e n c e s

costs (US$) --lOxy

Fig. 6

5.

Conclusions

This paper has introduced a simple method based on probability function technique to evaluate LOLR of stand alone PV system with least costs. The standard probability distributions were looked at in order to be applied to this problem. However non of them could possibly be well fitted, A table enabling to determine the least costs for specified LOLR was derived.

1. P,D. Maycock, International PV Market, Developments and Trend Forecast, Journal of Renewable Energy Vol. 6, No 5-6, 1995. 2. Ba Haj, Bin Gadhi, A. AI-Ashwal and Shamsan, Status of Solar Energy Application in Yemen, The European Photovoltaic Solar Energy Conference, 8-12 April 1991 Lisbon, Portugal. 3. R. Billinton and R. Allan, Reliability Evaluation of Engineering Systems, Pitman Ltd, 1983. 4. A.M. AI-Ashwal and I. Moghram, Proportion Assessment of Combined PV- Wind Generating Systems, To be published in Journal of Renewable Energy, code 633 [in a press]. 5. Civil Aviation and Meteorology Authority Data Records in Yemen. 6. A. Derrik, Solar PV for Development: Progress and Prospects, WRC, 11 - 16 September 1994, Reading, UK.

Acknowledgement The author would like to acknowledge the technical assistance of his son Natheir in performing this work. Table 1 LOLR

Io.o9 I o o5 Io.o,2 Io.o33

0.03

10.019

I 0.011

[0.003

No,PV

5*40

5*40

5*40

6*40

6*40

6*40

6*40

7*40

St, D

1

1.5

2

1.5

2

2.5

3

3

C,US$

1100

1150

1200

1350

1400

1450

1500

1700

435

WREC 1996 ADOe'~X 2 De~. & ~ :

,~oerd~x 1 Der~. & etrn. P r d ~ b Fur~onl No. of Oc, 1 2 6 5 7 5 7 9 21

Denst. prob. 0.OO27397 0.0054795 0.0164384 0.013~N~ 0.0191781 0.0153086 4.0191781 0.024~75 0,O575342

Cure. Prob. 0,5027337 0.0082199 O,0246575 0.0383~ 2 0.0573342 0,0712020 0,O(104110 0,115O685 0.1728027

Bunsh IntrO. 12,00 11.80 11,70 11,50 11,50 11,30 11.20 11,10 11.00

Expetd, Enrgy Wh 1020.0 1553.0 1872.0 1536.0 1840.0 1508,0 1792,0 1776,0 1780,0

~1

0.07071~3 6.0849315 0.3304110 0.O876712 0.0273673 0,0184384

0.24~3151 0.33424538 O,4246575 0.5120288 0,5397280 0.55(~644

10.80 10,70 16.53 14.50 10,30 14.20

1720,0 1712,0 1~,0 1530,0 1645,0 1832.0

0.013~86 0.~01370 0.01~53 0.013O589 4.0082152 0.013(~53 0.0O54795 0,53547O5 0.5054753 0.0191781 0,0532192 0,002"/397 O.010958g 0,0273973 0,3O82162 0.01005~ 0.01~6 0.010053o O.013R4R~I 0,O054795 0,OO52192 0.0t91751 0.0054795 0.0054753 0,013~ 0.0246575 0.0027397 0.0(~4795 0,0215178 0,5082192 0.0101~o89 0.0104384 0,00547~3 0.2054725 0.0089~2 0.3O273O7 0,5027397 0,0062192 0.3O273O7 0.002T3~7 0.3O273O7 0.0054705 0,0027337 0.0054705 0.3O2"/337 0,5027337 0.O527507

0.5353530 0,53O5000 0,513O553 0.5243375 0,8320767 0.6465753 0.5320548 0.5375342 0,5330137 0.8821215 0.59O4110 0.(~201507 0.7041095 0.73153~ 0,73O7250 0,750534Q 0.7(N3~6 0,7753425 0.7890411 0.76452O5 O,802731]7 0.8212175 0.6273373 0,8320767 O,8465753 0,871 2020 0.57~7~ 0.8794521 0,2013539 O,gOOS~O 0,2205479 0 . ~ 0.R4248~8 0,9479452 0.~J~11844 0.9589041 0,5318438 0.~0 0.9728027 0.9753425 0,9755322 o.gtk.~15 0.~0t4 0,~17808 0.2~1523O 0.~72203 1,0000000

10,10 10,3O 0.50 9.70 0,60 0.50 6.30 0.20 6,10 9.50 8.80 6,70 5.80 5.50 5,3O 6,25 8.10 8.50 7.84 7,7O 7.60 7,50 7.30 7.~ 7,10 7.00 5.B0 8.50 6,50 5,3O 5.20 5.0O 5,70 5,60 5.50 5.30 5.10 5,00 4.50 4.5O 4,30 4.10 3,50 2.50 1,60 1,50 0.50

1616.0 1500.0 1553.0 1532,0 1535.0 1520.0 1453.0 1472,0 1456.0 1440.0 1408,0 1392.0 1376.0 1530.4 1328.0 1312.0 1206.0 123O.0 1248,0 1232.0 1215,0 1200,0 1153,0 1152.0 1t36.0 1120.0 1086.0 1536,4 1040,0 1536,0 202,0 ~0.0 912,0 8~6.0 530.0 848,0 815.0 800,0 738,0 725.0 888.4 8,56,0 460.0 400.0 ~J~,O 240,0 80.0

33 32 : 51 5 4 3 5 2 2 2 7 3 I 0 3 4 5 4 5 2 3 7 2 2 5 9 1 2 5 3 4 5 2 2 3 1 1 3 1 1 1 2 1 2 1 1 1

No. C ~ c . "~ 1 5 2 1 2 2 1 1 2 1 1 1 1 1 I 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 t 1 1 1 I 1 1

436

Prot~'='h'~v FuncHon of Ene'm (2 ~

dr.)

Dent,Prob. 0,5770~50 0,OO27203 o,5053799 0,0O55850 O,0O27933 0,2055886 0,2055366 0.0027033 0,0027633 0.2055553 0,0027933 0.G027933 0.C027933 0,OO27933 0,0027933 0.0027933 0.2055865 0,502723O 0,0027933 0,G027933 0.0027633 0.0027933 0,0027933 0.0027233 0,0027933 0,0027g33 0.0027933 0,0027233 0,0027933 0.0027933 0,0027203 0,0027933 0,0027633 0,5027933 0.0027633 0.50.2703O 0.0027633 0.0027633

Expct, Energy Wh 2000.0 1984.0 1338,0 1952,0 1204.0 1850.0 1672.0 1508.0 1792.0 1712,0 1696.0

Cure. Prob. 0.5770~50 0,8795383 0.8882682 0.5938,548 O,8966481 0.9022347 0,9O78213 0,9106146 0,0134078 0.9189944 0,9217577 0.9245810 0,9273743 0,03o1676 0,9329809 0,9357542 0,0413405 0,9441341 0.9469974 0.9497907 0,9525140 0,9553073 0.9561006 0,9508939 0.0636872 0.9654805 O.geo;ff35 0.9720871 0.9748504 0.9776537 0,9804470 0.9832403 0,9860335 0.9888268 0,3316201 0,9944134 0.207~067 1.0200000

1610,0 1200,0 1584,0 1552,0 1536.0 1¢05.0 1280,0 1264.0 1216.0 1088,0 1056.0 1040.0 B48.0 800.0 736.0 672.0 624,0 592,0 544.0 528.0 448,0 33&O 336,0 288.0 240.0 160.0 112,0