MULTI-CRITERIA DECISION-MAKING BASED ON COMBINED VAGUE SETS IN ELECTRICAL OUTAGES PROBLEMS

Transcription

1 MULTI-CRITERI DECISION-MKING BSED ON COMBINED VGUE SETS IN ELECTRICL OUTGES PROBLEMS KH. BNN TBRIZ UNIVERSITY & HREC IRN S. KHNMOHMMDI TBRIZ UNIVERSITY IRN S. H. HOSEINI TBRIZ UNIVERSITY IRN BSTRCT The naure of elecrcal dsrbuon syses s changng fro sple arkes owards copeve arkes. The odern dsrbuon syse us sulaneously be relable flexble and cos conscous. One of he an facors ha affec he relably of power syses s ouages. Wh all effors ha have been done he exsence of ouages s a realy. Manageen of elecrcal ouages s a necessy and n ouage es we need soe approprae decson akngs. In hs paper we propose an negraed quanave ehodology based on vague ses o asss he dsrbuon syses n akng hese decsons. Ths ehodology allows nforaon on he supplers and cusoers o be expressed eher qualavely or quanavely o use a ulcrera decson akng odel and provde new funcons o easure he degree of accuracy n he grades of ebershp of each alernave wh respec o a se of crera represened by vague values. KEYWORDS: Vague se; Mul-crera decson akng elecrcal ouages. INTRODUCTION Mos of he exsng approaches n ul-crera decson akng (MCDM) conss of wo phases []: () The aggregaon of he judgens wh correspondng auhor and () he rank orderng of he decson alernaves accordng o he aggregaed judgens. However a few of hese approaches refer o he aspec of an explc odelng of relaonshps beween goals []. In addon achevng o real odelng of MCDM n he real world s he case wh nerdependen crera [3]. Snce he heory of fuzzy ses was proposed n 965 has been used for handlng fuzzy decson-akng probles [4]. However curren relaonshp analyss approaches (e.g. fuzzy ulple objecve progras (FMOP) [3] and decson akng based on relaonshp beween goals (DMRG) [ 5 6]) usually resul n denfyng relaonshps. Based on fuzzy se heory nroduced by Zadeh; fuzzy se approach o ul-objecve decson akng s llusraed by Zeran; soe approaches o solve ul-arbue decson probles based on fuzzy se heory are copared and a fuzzy ul-arbue decson-akng ehod usng crsp weghs s presened [7]. The srucure of he knowledge based of fuzzy rule based syses n a herarchcal way was exended n order o ake ore flexble [8]. lso an ordered weghed aggregaon operaor s nroduced and nvesgaed he properes of he operaor n [9]. Roughly speakng a fuzzy se s a class wh fuzzy boundares [0]. The fuzzy se n he unverse of dscourse U U ={u ; u ; : : : ; u n } s a se of ordered pars {(u ; µ (u )); (u ; µ (u )); : : : ; (u n ; µ (u n ))} where µ [0;] s he ebershp funcon of he fuzzy se ; and µ (u ) ndcaes he grade of ebershp of u n. When he unverse of dscourse U s a fne se hen he fuzzy se can be represened by Eq. (). = µ ( u ) / u + µ ( u ) / u µ ( u n ) / u n = µ ( u ) / u () When he unverse of dscourse U s an nerval of real nubers beween a and b hen a fuzzy se s ofen wren n he for Eq. (). n =

2 = b µ a ( u ) / u () Where u [ a; b]. I s obvous ha u U he ebershp value µ (u ) s a sngle value beween zero and one. Ths sngle value cobnes he evdence for u U and he evdence agans u U whou ndcang how uch here s of each and he sngle nuber ells us nohng abou s accuracy []. The concep of vague ses s presened n []. In vague ses ruh-ebershp funcon and false-ebershp funcon f are used o characerze he lower bounds on µ. These lower bounds are used o creae a subnerval on [0; ] naely [ (u ); f (u )] o generalze he µ (u ) of fuzzy ses where (u )< µ (u ) < f (u ). For exaple le be a vague se wh ruh-ebershp funcon and false-ebershp funcon f respecvely. If [ (u ); f (u )] = [0:5; 0:8] hen we can see ha (u )=0:5 f (u )=0:8 and µ (u )=0:. I can be nerpreed as he voe for resoluon s: 5 n favor agans and 3 absenons. Recenly soe new echnques are presened for handlng ul-crera fuzzy decsonakng probles based on vague se heory where he characerscs of alernaves are derved by vague ses [0]. In he proposed echnques a score funcon S s used o evaluae he degree of suably o whch an alernave sasfes he decson-aker's requreen. new funcon o evaluae he degree of accuracy of vague ses s also proposed []. Ths proposed funcon s a new concep on vague se heory and can provde useful way o effcenly help he decsonaker o ake hs decsons. In os real world probles ndeed he characerscs of alernaves he porance of crera are also vague ses. In hs paper a new cobned vague ses ul-crera decson akng (CV-MCDM) ehod s presened where boh characerscs of alernaves and he degree of porance of crera are consdered as vague ses.. MULTI-CRITERI FUZZY DECISION-MKING PROBLEM Ths secon revews soe easures o handle ul-crera fuzzy decson-akng probles [0 ]. Le be a se of alernaves and le C be a se of crera where: =... } C = C C... C }. { n { ssue ha he characerscs of he alernave are presened by he vague se presened by Eq. (3). = {([ c ( fc )][ ( f )])([ c( fc)][ ( f)])...([ cn( fcn)][ n( fn)])} (3) Where cj and f cj are j h creron ruh and false ebershp funcons respecvely and j [0] j + f j j n. In he case of f j = j and -f c = c can be rewren as Eq. (4). = {([ c c ][ ])([ c c][ ])...([ cn cn][ n n])} (4) In hs case he characerscs of hese alernaves can be represened by able. C C. C n [ c c] [ c c] [ cn cn] [ ] [ ] [ n n] [ ] [ ] [ n n] : [ ] [ ] [ n n] : [ ] [ ] [ n n] Table. The characerscs of he alernaves In hs case he degrees o whch alernave sasfes and does no sasfy he decsonaker's requreen can be easured by he evaluaon funcon E as shown by Eq. (5).

3 E( ) = ([ ] [ ]... [ n n ]) = [ Mn(... n ) Mn(... n )] (5) = [ ] = [ f ] Where denoe he nu operaor of he vague values and E() s a vague value. Le x =[ x ; f x ] be a vague value where x n [0; ]; f x n [0; ]; and x +f x <. The score of x can be evaluaed by he score funcon S by Eq. (6). S( x) = x f x (6) Where S ( x) [ + ]. Based on he score funcon S he degree of suably o whch alernave sasfes he decson-aker's requreen can be easured by Eq. (7). S ( E( )) = + (7) The larger he value of S(E( )) he ore he suably o whch alernave sasfes he decson-aker's requreens. Le S ( E S ( E S ( E ( ( ( )) )) )) = = s s (8) L If S(E( ))=s and s be he larges value aong he values s ; s ; : : : ; and s hen alernave s he bes choce. ssue ha he degree of porance of crera C ; C ; : : : and C n consdered by expers are vague ses. Then he degree of suably o whch alernave sasfes he decson-aker's requreens can be easured by he cobned vague funcon CV by Eq. (9). CV ( ) = S([ ]) S([ c c ]) + S([ ]) S([ c c]) S([ n n]) S([ cn cn]) (9) By usng Eq. (7) Eq. (9) can be rewren by Eq. (0). CV ( ) = [ + ] [ c + c ] + [ + ] [ c + c ] [ n + n ] [ cn + cn ] (0) Le = CV ( ) = p CV ( ) = p CV ( ) = p M If CV( )=p and p s he larges value aong he values p p... and p hen alernave s he bes choce. For evaluaon of he degree of accuracy of vague ses an accuracy funcon H s defned [0]. In hs paper slar defnon s used for cobned vague ses. The degree of accuracy of x can be evaluaed by he accuracy funcon H represened by Eq. (). H ( x) = x + f x () Where H ( x) [0; ]. The larger he value of H(x) ndcaes ore degree of accuracy he grade of ebershp of vague value. The relaon beween he score funcon S and he accuracy funcon H s slar o he relaon beween ean and varance n sascs. Then based on Eq. (5) and Eq. () we can oban accuracy funcon by Eq. (3). H ( E( )) = + (3) Where H ( E( )) [0; ]. The larger he value of H(E()) ndcaes ore degree of accuracy n he grades of ebershp of he alernave. fer calculang he value of H he choce of alernave ay depend on he decson-aker's nd or polcy. conservave person gh choose he alernave wh hgh H bu an aggressve person ay choose he alernave wh low H [0]. Ths easure provdes addonal useful nforaon o effcenly help he decson-aker o ake hs decsons. s ()

4 In he followng we presen a cobned vague ses echnque for handlng ul-crera fuzzy decson-akng probles. We have Eq. (4). T l=... ncl ( ) = H ([ ]) H ([ c c ]) + H ([ ]) H ([ c c]) H ([ n n]) H ([ cn cn]) (4) By applyng Eq. (3) we can ge Eq. (5). T... ncl ( ) = ( + ) ( c c + ) + ( + ) ( c c + ) ( n n + ) ( cn cn + ) (5) We hen defne he range of sasfacon of crera C ; C ; : : : ; and C n by alernave as: T l=... ncl ( ) T l=... ncl ( ) R l=... ncl ( ) = [ CV l=... ncl ( ) CV l=... ncl ( ) + ] (6) Where: R c ( ) [ ]. Le R( )=(R n ( ); R cener ( ); R ax ( )) where: l= j k... p l R R R ax n cener ( ) = CV ( ) = CV = CV... n l... n l... n l T c ( ) + T c ( ) c ( )... n l... n l c ( ) c ( ) There are any easures of rank. For exaple ax-n ax-ax and ax-cener are coon ehods. Le l c r R( ) = ( p p p ) l c r R( ) = ( p p p ) R( L ) = ( p l p c Max-n ehod: If R( )=(p l ; p c ; p r ) and p l s he larges value aong he values p l ; p l ; ; and p l hen alernave s he decson-aker's bes choce. Max-ax ehod: If R( )=(p l ; p c ; p r ) and p r s he larges value aong he values p r ; p r ; ; and p r hen alernave s he decson-aker's bes choce. Max-cener ehod: If R( )=(p l ; p c ; p r ) and p c s he larges value aong he values p c ; p c ;.. ; and p c hen alernave s he decson-aker's bes choce. I s noed ha ax-cener ehod s he sae as ha of proposed n []. 3. CSE STUDY Soe power elecrc uly cusoers experence sgnfcan dsasers when elecrcal power s nerruped and a vas ajory of uly cusoers experence relavely lle nconvenence or cos as a resul of elecrcal ouages. In hs case sudy he proble s o ake a decson abou he dsrbuon of elecrcal energy beween dfferen regons of an elecrcal dsrbuon nework n he eergency condons where here s a shorage of elecrcal energy. IEEE 4-bus sandard es syse of Fgure s consdered for hs purpose. p r ) (8) (7) Fgure. IEEE 4-bus sandard es syse

5 The proble s o ake a decson o dsrbue elecrcal energy beween dfferen regons of a cy n he eergency condons where here s a shorage of power generaon beween 40 up o 50 MW caused by naural dsasers or oher accdens. For splcy and whou loosng generaly he nework s consdered loss-less wh only real power deands. s s seen n Fg. here are hree alernaves for cung of supply. lernave : Cung of supply o regon (47.8 MW) lernave : Cung of supply o regon 5 (43.0 MW) lernave 3: Cung of supply o regons 3 and 4 ( =45.7 MW) The goal s o provde a suable and opzed decson for elecrcal energy dsrbuon consderng dfferen crera. Several crera can be consdered. In hs paper soe poran crera are consdered [3] as shown n able. Eneranen area Toal e beng conneced o he power dsrbuon nework 3 Populaon 4 Hgh buldngs 5 Educaon ceners 6 r pors 7 Coercal ceners 8 Resdenal area 9 Indusral area 0 Eergency acvy ceners Hospals Table. Consdered crera survey s pleened o fnd he weghs of crera. The noralzed resuls of survey fro expers knowledge abou hese crera are shown n able 3. Ths able shows he uly values and weghs of crera as vague ses. C C C 3 C 4 C 5 C 6 C 7 C 8 C 9 C 0 C [..9] [.4.5] [.5.7] [.4.9] [.6.7] [.7.8] [.5.9] [.7.7] [.7.8] [.7.9] [.8.9] [..9] [.3.8] [.4.9] [.6.8] [..8] [.5.6] [.6.7] [.7.9]] [.4.6] [.8.9] [.3.7] [.3.8] [.5.7] [.6.7] [.4.7] [..8] [.5.7] [..9] [..8] [.3.7] [.4.6] [.5.5] 3 [.3.5] [.4.5] [.4.5] [..6] [.7.9] [.8.9] [.7.8] [.5.9] [.4.8] [.7.7] [.6.7] Table 3. Crera weghng funcons and ules By usng Eqs. (5) o (8) he evaluaon score weghng funcons and rankng values are derved as presened by able 4. 3 E() [..6] [..5] [..5] S(E()) CV() H(E()) T Rax Rn Rcener Rank ( ) ( ) ( ) Table 4. Evaluaon score and cobned vague funcons

6 Trough he above resuls he bes choces can be deerned. Table 5 shows he bes choces based on dfferen ehods. nd fnally we can drve decson values for each alernave hrough he calculaed funcons. 3 Score Mehod CV Mehod Max-n Mehod Max-ax ehod Max-cener ehod Table 5. The bes choces based on dfferen ehods I sees ha alernave 3 ay be a good choce. 4. CONCLUSION In hs paper we proposed an negraed and quanave ehodology based on cobned vague ses o asss decsons akers. lso we have presened a new ehod o easure he degree of accuracy n he grades of ebershps of each alernave wh respeced cobned vague values for handlng ul-crera fuzzy decson-akng probles. The proposed ehod can provde ore useful daa o help he decson-aker o ake hs decson ore effcenly. n llusrave exaple used o deonsrae he applcaon of he proposed funcon and ake a decson abou he dsrbuon of elecrcal energy beween dfferen regons of an elecrcal dsrbuon nework n he eergency condons. 5. REFERENCES [] T. J. Ross Fuzzy Logc wh Engneerng pplcaons Hoboken NJ John Wley & Sons June 004. [] R. Felx Relaonshps beween goals n ulple arbue decson akng Fuzzy Ses and Syses 67 (994) [3] C. Carlsson R. Fuller Inerdependence n fuzzy ulple objecve prograng Fuzzy Ses and Syses 65 (994) 9-9. [4] L.. Zadeh Fuzzy ses Infor. and Conrol 8 (965) [5] R. Felx Fuzzy decson akng based on relaonshps beween goals copared wh he analyc herarchy process n: proceedngs of 6h Inernaonal Fuzzy Syses ssocaon World Congress 995 pp [6] R. Felx S. Reddg. delhof Mulple arbue decson akng based on fuzzy relaonshps beween objecves and s applcaon n eal forng n: Proceedngs of he nd IEEE Inernaonal Conference on Fuzzy Syses 993 pp [7] H.J. Zerann Fuzzy Ses Decson Makng and Exper Syses Kluwer cadec Publshers Boson 987. [8] O. Cordon F. Herrera I. Zwr Lngusc Modelng by Herarchcal Syses of Lngusc Rules IEEE Trans. On Fuzzy Syses Vol. 0 No. Feb. 00. [9] R.R. Yager On ordered weghed averagng aggregaon operaors n ulcrera decson akng IEEE Trans. Syses Man Cyberne. 8 (988) [0] D. H. Hong C. H. Cho "Mul-crera fuzzy decson-akng probles based on vague se heory" Elsever Fuzzy se and Syses Vol pp 03-3 [] W.L. Gau D.J. Buehrer Vague ses IEEE Trans. Syses Man Cyberne. 3 (993) [] S.M. Chen J.M. Tan Handlng ul-crera fuzzy decson-akng probles based on vague se heory Fuzzy Ses and Syses 67 (994) [3] S. Khan Mohaad I. Hasanzadeh R. M. Mahur K. V. Pal New Fuzzy Decson Makng Procedure ppled o Eergency Elecrc Power Dsrbuon Schedulng PERGMON Engneerng pplcaons of rfcal Inellgence 3 (000)