1_estimation of Global Solar Radiation Using Artificial Neural Networks

8/10/2019 1_estimation of Global Solar Radiation Using Artificial Neural Networks

1/6


2/6

180 M. MO HAN DESe t a l .

......... i ~ . . . s o ~ J ' #~ ' A L - O , u r l y y i , ~.I~te~atloaat B o ~

* T y a e ~ a ~

~ q [ I u q ~ r I H u r l l S U f l I t I A I ~ . H ~ U e ( / 1 ~ ' / . ~ 2 5 1

eAI-Madinal l ~ , f ief eAr RIY~l~h ~

. . . . . . . . . . . . . Kingdom of l........ Sa ud i Arab i~

8 a ~ ' ~8a~arshi ~ Bis~a~ As-SuLayyi l I ~ A l * H e d l

dayl

~ % ~ A l - K v a s h . / , 'i S i P ~ L i ~ , /

Fig . 1 . So la r rad ia t ion s ta tions in the K ingdo m of Saudi A rab ia . S ta t ions used fo r t r a in ing , * s t a t ions used fo r t e s t ing

M a n y e m p i r i c a l m o d e l s h a v e b e e n d e v e l o p e d f o r d i f fe r e n t g e o g r a p h i c a l a n d m e t e o r o l o g i c a l c o n d i ti o n s(Angs t rom, 1956; Swar tmane t a l . 1971; Reddy, 1971a and 1972b; Barbaroe t a l . 1978; Goh, 1979; andO g e l m a n e t a L 1984). Mo nth ly average da i ly va lues o f g loba l so la r rad ia t ion on hor izon ta l and inc l inedsur face a re repor ted fo r Ab u D habi , Uni ted A rab Em ira tes (E1-Nashar, 1991), severa l s i t e s in Can ada( M a u r e e t a L 1979 and Ha y, 1979), thir teen s ta t ions in India (Man ie t e L 1973), two loca t ions in Leso tho ,South Afr ica (Gopina than , 1990 and 1991) and many o thers . Sabbaghe t a L (1977) p resen ted an empi r ica lfo rmula ob ta ined us ing the da i ly to ta l so la r rad ia t ion , sunsh ine dura t ion , r e la t ive humid i ty, maximumtempera tu re , l a t i tude , a l t i tude , and the loca t ion re la t ive to the w ate r su r face . The au thors com pared the i rm o d e l w i t h e x i s t i n g m o d e l s f o r d i f f e r e n t l o c a t i o n s w i t h i n a n d o u t s i d e S a u d i A r a b i a a n d f o u n d b e t t e res t imates wi th the i r mode l .

T h e p r e s e n t s t u d y u s e s a n e u r a l n e t w o r k t e ch n i q u e f o r m o d e l i n g m o n t h l y m e a n d a i l y v a lu e s o f g l o b a l s o l a rrad ia t ion on h or izon ta l su r faces . The neura l ne tworks u t i l i ze l a ti tude , long i tude , a l t i tude and the sunsh inedura t ion fo r the p red ic t ion o f so la r rad ia tion va lues . The resu l t s o f the sys tem ind ica te a re la t ive ly goodagreeme nt be tween the p red ic ted va lues and the observe d ones .

A RT I F I C IA L N E U R A L N E T W O R K S

A n A r t i f i c ia l N e u r a l N e t w o r k ( A N N ) c o n s is t s o f m a n y i n t er c o n n e c te d i d e n t i c a l p r o c e s s i n g u n i ts c a l l edneurons . Each neuron com putes a we igh ted sum of i t s n inpu t s igna l s ,x j f o r j = 1 2 . . . .. nand then app l iesa non l inear ac t iva t ion func t ion to p rodu ce an ou tpu t s igna l y

A typ ica l non l inear func t ion i s the s igm oid func t ion , de f ined by


3/6

Estim ation of glob al solar radiation 181

1

q~(x) - 1 + ex p( - x) (2)

The w eigh ts o f the in te rconnec t ions be tween neurons a re ad jus ted dur ing the t ra in ing p rocess to ach ieve ad e s i r e d in p u t / o u t p u t m a p p i n g . A N N s c o m e i n m a n y d i f fe r e n t f o r m s , s o m e r e q u i r e t o p o l o g i e s w i t h t o t alin te rconnec t ion among neurons and o thers requ i re a r rangement in l ayers . A mul t i l ayer feedforwardne twork has i t s neurons o rgan ized in to l ayers wi th no feedback or l a te ra l connec t ions . Layers o f neuronso ther than the ou tpu t l ayer a re ca l l ed h idden layers . The inpu t l ayer cons i s t s o f a se t o f sensors tha t on lyprov id e inpu t s igna l s and do n o t pe r fo rm any computa t ions . Inpu t s igna l s p ropag a te th rough the ne tw ork ina fo rward d i rec t ion , on a l ayer-b y- laye r bas i s un t i l the ou tpu t l ayer. F igure 2 show s a typ ica l neuron and at w o - l a y e r f e e d f o r w a r d n e t w o r k .

X I

X W t n

x J , f w i /

I x

d e n l a y e r

Inpu ts

F ig . 2 A typ ica l neuron and a two - layer feedforward ne twork .

One of the majo r advan tag es o f neura l ne tworks i s the i r lea rn ing ab i l i ty to pe r fo rm spec i f i c t a sks . Learn in gi s a c c o m p l i s h e d b y a d j u s t in g t h e w e i g h t s o f th e c o n n e c t i o n s b e t w e e n n e u r o ns . W e i g h t s a r e a d j u s t e d t oa l low the ne two rk to p roduce ou tpu t s as c lose as poss ib le to the know n cor rec t answers o f the t ra in ing da ta .Dur ing the l ea rn ing phase , the ne twork cap tures the under ly ing ru le fo r assoc ia t ing the inpu ts wi th thedes i red ou tpu t s . Due to the gen era l i za t ion capab i l i t i e s o f the ne tworks , i t pe r fo rms s imi la r ly on da ta tha thave no t used fo r t r a in ing .

The bac k-prop aga t ion a lgor i thm (Rumelha r t e t a l . , 1986 and Hay kin , 1994) i s a superv i sed i t e ra t ivet ra in ing method fo r mul t i l aye r feedforward ne tworks . I t uses tr a in ing da ta cons i s t ing o f P inpu t -ou tpu tpa i r s o f vec to rs tha t charac te r ize the p rob lem . A sam ple f rom the t ra in ing da ta i s chosen rand om ly andp r o v i d e d t o t h e i n p u t s o f t h e n e t w o r k , w h i c h c o m p u t e s t h e o u t p u t s o n a l a y e r - b y - l a y e r b a s i s u n t i l t h e o u t p u tlayer. The d i ffe rence be tween the ac tua l ou tpu t o f the ne twork an d the cor rec t ou tpu t tha t i s p rov id ed in thet ra in ing da ta i s used to ad jus t the weigh ts , so tha t the nex t t ime tha t same inpu t i s p rov ided , the ne tworkoutpu t wi l l be c lose r to the cor rec t one . Th is p rocess i s r epea ted fo r a l l o ther inpu t -ou tpu t pa i r s in thet ra in ing da ta . Thus , the back-pro paga t io n a lgor i thm min im izes an e rro r func t ion def ined by the average o fthe sum square d i ffe rence be tween the ou tpu t o f each neuron in the ou tpu t l ay er and the des i red ou tpu t .The e r ro r func t ion can be expressed as :

1 d

p k

3)

W here p i s the index o f the P t ra in ing pa i rs o f vec to rs , k is the index of e lements in the ou tpu t vec to r,dpki s the k th e lem ent o f the p th des i red pa t t e rn vec tor, andpki s the k th e lement o f the ou tpu t vec to r whenpa t te rn p i s p resen ted as inpu t to the ne twork . M in imiz in g the cos t func t ion represen ted in equ a t ion (3 )


4/6

182 M. MOHANDES et al.

results in an updat ing rule to adjust the weights of the connections between neurons. The weightadjustment of the connection between neuron i in a layer m and n eur onj in layer m + l can be expressed as:

Aw/, = rlSjo 4)

Where i is the index of units in a layer m, r 1 is a small positive cons tant called the learning rate, o~ is theoutput of uni t i in the mth layer, and ~j is the delta error term back-propagated from the jt h uni t in layerm + l defined by:

= [ d j - o j ] o j [ 1 - o j ] , if neu ron j is in the output layer

g j = y j [ 1 - - y j ] ~ 6 k W k j , if neu ron j is in a hidden layer

Where k is index of neurons in the layer m+2) ,ahead of the layer that contains neuron j.

Choosing a small learning rate r/ leads to slow rate o f convergence, and too large 77 leads to oscillation. Asimple method for increasing the rate of learning without oscillation is to include a momentum terma Aw j i n )that determines the effect of past weight changes on the current direction of movement in theweight space, where n is the iteration number, and a is a small positive constant. Thus the weights updaterule is:

Awj, n + 1) = r/~o, + ~Sw j i n ) 5)

The iterative process o f presenting an input-ou tput pair and updating the weights continues until the errorfunction reaches a pre-specified value or the weights no longer change. In that case the t raining phase isdone and the network is ready for testing and operation. Detailed description of the multilayer feedforwardneural networks and the backpropagation algorithm may be found in Haykin, 1994).

THE DEVELOPED MODEL AND RESULTS

ANNs have been used in a broad range of applications including: pattem classification Lippmann, 1987and Bishop, 1996), func tion approximation, optimization, prediction and automatic control Pham et al.,1995). Here ANNs are used for modeling global solar radiation on horizontal surfaces in the K ingdo m ofSaudi Arabia. We used the back-propagation algorithm for training several multi-layer feed-forward neuralnetworks to estimate the monthly mean daily values of global solar radiation. The best network consists of4 inputs, 10 neurons in one hidden layer and one neuron in the output layer. The latitude, longitude,altitude and sunshine duration values have been used as inputs to the network. The latitude and longitudewere provided in degrees and the altitude in meters. The sunshine duration values were provided as a ratioof the actual values divided by the maximum possible values for each location. The output is the ratio ofmonthly mean daily value of the global solar radiation divided by extraterrestrial radiation outside theatmosphere. The data from 31 stations, 12 for each, was used for training the neural network. Thisprovided 372 input-outpu t pairs for training. The tra ining process continues until the error function definedin equation 3) approaches a prespecified minimum value. After the training is completed, the developedmodel is used for testing, where the data from the remaining 10 stations is used. Figure 1 shows thegeographical locations of these 41 stations where circles represent the stations used for training while thestations used for testing are represented by stars. The testing data was not used in the model ing to give anindication of the performance of the system in unk nown locations. The performance of the neural networkmodel in addition to the observed values for the 10 locations are shown in Fig.3. The Mean AbsolutePercentage Errors MAPE) for these locations are shown in Tablel. These results indicate the viabil ity ofthis method for global solar radiation modeling.

Table 1. Mean Absolute Percentage Errors MAPE) for ten locations used for testing.

Station Tabuk AI-UIa Unayzah Shaqra Dawdami Yabrin Turabah Heifa Kwash NajranMAPE 10.7 6.5 14.6 10.5 13.4 10.1 16.4 11.3 19.1 13.5


5/6


6/6

184 M. MO HAN DESet al.

C O N C L U S I O N

This paper in t roduced neura l ne tworks t echn ique fo r mode l ing the spa t i a l va r ia t ion o f g loba l so la rrad ia t ion . The ava i lab le da ta f rom the 41 da ta co l lec t ion s ta t ions i s d iv ide d in to 31 loca t ions fo r t r a in ing

the neura l ne tworks and 10 loca t ions fo r t e s t ing . The t es t ing da ta was n o t used in the mode l ing to g ive anind ica t ion o f the per fo rmance o f the sys tem in unknow n loca t ions . The resu l t s on these 10 loca t ionsind ica te a re la t ive ly goo d agreem ent be tween the ob served and pred ic ted va lues .

A C K N O W L E D G M E N T

T h e a u t h o r s w i s h t o a c k n o w l e d g e t h e s u p p o r t o f t h e R e s e a rc h I n s t it u t e o f K i n g F a h d U n i v e r s i t y o fPe t ro leum M inera l s , Dhahran-31261 , Saudi Arab ia .

R E F E R E N C E S

Saudi Arab ian So la r Rad ia t ion At las , SAN CST , Riyadh , Saudi Arab ia , (1983).

Angs t rom, A. (1956) . On the computa t ion o f g loba l so la r rad ia t ion f rom records o f sunsh ine .Arkiv.Geophysik 3(23) , 551-556.Swar tm an , R . K. and O. O gunlade g (1971). Co r re la t ion o f so la r rad ia t ion wi th com mo n param ete rs in

Toron to , Canada .Solar Energy 13, 345-347.Redd y, S . J . (1971) . A n em pi r ica l method fo r the es t imat ion o f the to ta l so la r rad ia t ion .Solar Energy 13,

289-290 .Redd y, S . J . (1971). A n e mpi r ica l method fo r the es t imat ion o f ne t r ad ia t ion in tens i ty.Solar Energy 13,

291-292 .Barbaro , S . , S . Coppo l ino , C . Leone , and E . S inagra (1978) . G loba l so la r rad ia t ion in I t a ly.Solar Energy

20, 431-435.Goh , T. N. (1979) . S ta t i s t ica l s tudy o f so la r rad ia t ion in format ion in an equa tor ia l r eg ion (S ingapore ) .

Solar Energy 22, 105-111.

Oge lm an , H. , A . Ecev i t , and E . Tasd emirog lu (1984). A new metho d fo r es t imat ing so la r rad ia t ion f rombr igh t sunsh ine da ta .Solar Energy 33(6) , 619-625.E1-Nashar (1991) . Sola r radiat io n charac ter is t ics in Abu D habi .Solar Energy 47(1) , 49-55.Maure and N. Ga lan i s (1979). So la r rad ia t ion da ta fo r Quebec .Solar Energy 23, 309-314.Hay, J . E . (1979) . C a lcu la t ion o f month ly mean so la r rad ia t ion fo r hor izon ta l and inc l ined sur faces .Solar

Energy 23(4) , 301-307.Ma ni , A . and O. C hacko (1973) . So la r rad ia t ion c l imate o f l nd ia .Solar Energy 14, 139-156.Gop ina than , K. K . (1990) . S o la r rad ia t ion on inc l ined sur faces .Solar Energy 45(1), 19-25.Gop ina than , K. K. ( I 991) . S o la r rad ia t ion on v ar ious ly o r ien ted s lop ing sur faces .Solar Energy 47(3), 173-

179.Sabbagh , J . A . , A . A. M . S ay igh , and E . M. A . E1-Sa lam (1977) . Es t imat ion o f to ta l so la r rad ia t ion from

meteo ro log ica l da ta .Solar Energy 19, 307-311.Rumelhar t , D . E . , J . L . McCle l land , and PDP Research Group (1986) .Parallel Distributed Processing.

T h e M I T P r e s s .Haykin, S. (1994) .Neural networks:A comprehensive oundation. M a c m i l l a n C o l l e g e P u b l i s h i n g .L ippm ann , R . P. (1987) . An in t roduc t ion to comp ut ing wi th neura l ne ts .IEEEASSP Mag. 4-22.Bishop, C. M. (1996) .Neural networks or pattern recognition. Oxford U nivers i ty Press.Pham, D. T. and Liu X. (1995) .Neural networks for identification predication and control Spr inger-

Ver lag , London .

Documents

1_estimation of Global Solar Radiation Using Artificial Neural Networks