Multicriteria Decision Making for Reactive Power Compensation in Distribution Systems

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Russell Godwin Chandler
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1 Proceedgs of he Euroea Coug Coferece Mucrera Decso Makg for Reacve Power Coesao Dsrbuo Syses HESTER J. RÚJO Deare of Pag ad Moorg of Dsrbuo Nework Maeace Mas Geras Eergy Coay ve. Barbacea Beo Horzoe - MG BRZ us@ceg.co.br h:// PETR Ya. EKE Graduae Progra Eecrca Egeerg Pofca Cahoc Uversy of Mas Geras ve. Do Jose Gasar Beo Horzoe - MG BRZ eke@ucas.br h:// RFE P. FCÃO FHO Deare of Dsrbuo Nework Pag Mas Geras Eergy Coay ve. Barbacea Beo Horzoe - MG BRZ rafae@ceg.co.br h:// Y V. KOKSHENEV Graduae Progra Eecrca Egeerg Federa Uversy of Mas Geras v. oo Caros Beo Horzoe MG BRZ ya.koksheev@ga.co h:// HENRQUE S. SCHUFFNER Deare of Eecrca Egeerg Pofca Cahoc Uversy of Mas Geras ve. Do Jose Gasar Beo Horzoe - MG BRZ herque.schuffer@ga.co h:// bsrac: - Ths aer reses resus of research reaed o foruag ad sovg he robe of caacor acee dsrbuo syses wh he fraework of ucrea decso akg odes. The acao of ucrera aroach s dreced frs of a a overcog he dffcues of suaeous observg coradcory cosras for he erssbe uer ad ower voage s a dffere buses of dsrbuo syses as we as oher ora codos oerag caacors. The souo of he robe of caacor acee he ucrera saee s based o cobg he Bea-Zadeh aroach o decso akg a fuzzy evroe wh he acao of he geerazed agorhs of dscree ozao. The aer resus are usraed by coug exeres wh a rea dsrbuo syse. Key-ords: - Dsrbuo Syses Reacve Power Coesao Mucrera decso akg Bea- Zadeh roach Dscree Ozao Geerazed gorhs of Dscree Ozao. SBN:

2 Proceedgs of he Euroea Coug Coferece roduco Caacors are wdey used dsrbuo syses for reacve ower coesao o acheve ower ad eergy oss reduco ad o rove he syse voage rofe. Tradoay he robe souo s dreced a he deerao of he ocaos szes ad yes of caacors o ze he obecve fuco of a ecooca characer whe he cosras o voage agudes a dffere oad eves are sasfed. The ehods for sovg he reacve ower coesao robe are cassfed []. The foowg grous of ehods are cassfed: aayca ehods uerca ehods heursc ehods ad arfca egece based ehods. works cosdered [] are dreced a sovg he robe wh he fraework of oocrera odes. The aory of ore rece works (for exae [2-4] s aso dreced a sovg he robe wh he oocrera fraework. The rese aer s devoed o foruag ad sovg he caacor acee robe wh he fraework of ucrera decso akg odes. s assocaed wh he foowg cosderaos. The ecessy of suaeous observg cosras o he erssbe uer ad ower voage s a dffere buses of dsrbuo syses creaes essea dffcues ( s o ucoo ha hese cosras geerae suaos whe he corresodg feasbe regos are ey. These dffcues ca be overcoe by zg he obecve fuco of a ecooca characer as we as he obecve fuco whch refecs a voue of oor eergy cosuo (eergy cosued wh voage agudes ousde of he erssbe s. The fexby of cudg he obecve fuco refecg eergy quay s cofred by he resus of [5]. However hese resus are based o ayg fuzzy ogc. does o er oe o ake drecy o accou he dscree aure of he caacor acee robe. Thus f he robe s assocaed wh he deerao of he ocaos ad szes of fxed caacors ca be aroxaed by a bcrera ode. he sae e f we ak abou he deerao of he ocaos szes ad yes (fxed or swched of caacors he uber of he obecve fucos s o be ore. arcuar oe of he os ora quesos oerag swched caacors s he observao of a desrabe (erssbe uber of her couaos er a e u [6]. However ahough he aer resus are of a geera characer cosderg s ed sze he robe foruao s reaed o he deerao of he ocaos ad szes of caacors. 2 Probe Foruao Cosderg he dscree aure of he caacor acee robe he geerazed agorhs of dscree ozao are used for s souo. These agorhs frsy have bee reseed [7]. The resus of her deveog are refeced for sace [89]. The agorhs are assocaed wh he ehod of orazed fucos ad cobe fora ad fora rocedures. arcuar he agorhs are based o ayg he deas of greedy heurscs [] whch bascay rovde he bes heursc aog ossbe heurscs wh a ror esaes ad offer a bass for effecve aroxaed aroaches. The agorhs aow oe o oba quas-oa souos afer a sa uber of ses overcog he NP-coeeess. They do o requre aayca secfcao of obecve fucos ad cosras. Ths esures he fexby ad he ossby o correcy refec dverse yes of a daa by he use of so-caed dscree sequeces x s s τs... s =... ρ v ( where ρ τ... are echca ecooca ec. s s characerscs requred for forg obecve fucos cosras ad her crees whch corresod o he sh dscree (eger Booea vaue of he varabe x. Cosderg hs s raoa o foruae he geera robe of caacor acee as foows. f caacors ca be saed o he dde ad ow voage eves he forao o caacors ca be reseed as he creasg dscree sequeces M M M Q K g δ =... (2 for caacors of he dde voage eve ad Q K g δ =... q (3 for caacors of he ow voage eve. M (2 ad (3 Q ad Q are sadard szes of he h ad h caacors for he dde ad ow voage eves resecvey; K ad K are her M oa coss; g δ ad g δ are her secfc osses. s aura ha dffere dscree sequeces ca be aed o dffere buses by roducg he bus dex =.... For sace f dffere dscree sequeces have o be aed o each dde voage bus he (2 s rasfored o M M M Q K g δ =... =.... (4 However e us cosder he ossby o ay he sae dscree sequeces for he dde voage eve ad he sae dscree sequeces for he ow voage M SBN:

3 Proceedgs of he Euroea Coug Coferece eve o sfy he robe saee. Fro he dscree sequeces (2 ad (3 s ecessary o choose Q M =... q =... ad Q =... r =... ( s suosed ha he dex s assocaed wh he dde voage buses; however f he cosdered bus corresods o a dsrbuo rasforer he caacors ca be saed a s dde as we as s ow voage buses o ze he obecve fuco of he rese vaue of he roec's fee M M M Z ( Q... Q... Q Q... Q... Q M = ( K K Ω ω = ω= r ( o& d M { ( K K = T M M c ( δ Q δ Q g g e = = M M c ( δ Q δ Q g g = M M c ( δ Q δ Q g g o o = g T ce M R ( Q Q Q 3 V c 2 f f = = = f g M R ( Q Q Q 3 V 2 f f = = f c o M R ( 3 2 f Q Q Q } (4 V f = o = f where =... T s a curre dex of oad curve ses; eas ha a se beogs o ower syse eak oad e; o eas ha a se s ou of ower syse eak oad e; f eas ha oad of bus fows hrough brach f; V s a ework oa voage; R f f =... g s a ressace of brach f; ω =... Ω s a curre year; r s a dscou rae; o & ad d are reave exeses reaed o caacor oerao ad aeace ad derecao resecvey; c e s eergy cos; c ad c o are rasorao arffs for ower syse eak oad e ad ou of ower syse eak oad e. The zao of (4 us be execued whe he cosras o voage agudes a dffere oad eves are sasfed. These cosras ca be reaed o he os reoe cosuers ad o he eares cosuers of ow voage eworks. Cosderg hs he cosras for he feror voage eves ca be reseed as V V g g re M 2 Q f Q f V = = = = f f f f J =... T (5 where J s a se of he os reoe cosuers of ow voage eworks wh voage eves V re < V ( V s he erssbe feror voage eve; f s a reacace of brach f ; f eas ha brach f beogs o he way of suyg he os reoe cosuer. The cosras for he eares cosuers of ow voage eworks ca be reseed as foows: V V g g g e M 2 Q f Q f V = = = = f f f f =... J =... T. (6 where V s he erssbe sueror voage eve. s was dcaed above he suaeous observao of (5 ad (6 ees dffcues. arcuar ay voao of he cosra (6 for ay bus sos he ozao rocess. Cosderg hs s raoa o chage he oocrera robe (2- (6 by he robe (2-(4 ad he foowg addoa obecve fuco M M M ( Q... Q... Q Q... Q... Q J = = T = = J T = =. (7 refecg a voue of oor eergy cosuo. (7 s a overa voue of eergy cosuo wh he voage eves sueror V ; s a overa voue of eergy cosuo wh he voage eves feror V ; s a voue of oor eergy cosuo wh he voage eves sueror V by he h ow voage ework for h oad curve se; s a voue of oor eergy cosuo wh he voage eves feror V by he h ow voage ework for h oad curve se. f araeers of a ow voage ework as we as s oads are avaabe he evauao of or creaes o dffcues. f hey are o avaabe s ossbe o ay so-caed ow voage ework odes reseed he ex seco. SBN:

4 Proceedgs of he Euroea Coug Coferece 3 Evauao of Poor Eergy Cosuo Suose ha acve a ad reacve oads of he h ow voage ework have a ufor dsrbuo aog he egh of s ode. Ths ode ca be defed as a fuco of a axu voage dro V ax corresodg o a axa oad ax. The esao of V ax s assocaed wh dffcues. Takg hs o accou s ossbe o uze a vaue defed by roec ors of he uy. voage dro ca be defed as V ax ax = ax r V = V V S S. (8 ax curre fow hrough a eeeary seco of he ode corresodg o a dsace fro he eares cosuer ca be defed by = ( -. (9 e ( - = The corresodg voage dro ca be cacuaed as ( V e = ( - Z = ( - Z where Z s a secfc edace. voage dro fro he eares cosuer o he o cudes wo cooes. The frs oe s assocaed wh a ufor oad dsrbuo aog. The secod oe s assocaed wh a coceraed oad whch s equa o a oa oad obaed aog of -. Thus cosderg (9 ad ( s ossbe o wre he foowg correao: V =.5 Z whch ca be reduced o ( - Z 2 ( Z ( V (.5 Z. = (2 The rao Ψ = refecs ar of cosuers of a ow voage ework wh a voage dro ess ha or equa o. Cosderg hs he V correao (2 ca be reseed as The souo of (3 s 2 ( Ψ 2Ψ V =. (3 V Ψ =. (4 V V he sae e ar of cosuers aced bewee he o ad he ed of he e s Ψ =. (5 V V Cosderg re V = V e ad ayg (4 V ad (5 s ossbe o esae eergy ( e V V / V = [ ] ad eergy as ( e V V V / V = [ ] as (6. (7 4 The Bea-Zadeh roach ad Mucrera Decso Makg he aayzg ucrera odes a vecor of obecve fucos F( = { F (... F ( } s cosdered ad he robe cosss suaeous ozg a obecve fucos.e. F ( exr =... q (8 where s a feasbe rego R. The frs se sovg (8 s assocaed wh deerg a se of Pareo souos Ω []. Ths se s usefu. However does o er oe o oba uque souos. s ecessary o choose a arcuar Pareo souo o he bass of forao rovded by a decso aker (DM. Three aroaches o usg hs forao are cassfed [2]: a ror a oseror ad adave. The os referabe aroach s he adave oe. hs aroach he rocedure of successve rovg he souo quay s erfored as a raso fro α Ω o α Ω cosderg he forao α rovded by a DM. he aayzg ucrera robes s ecessary o sove quesos reaed o orazg crera seecg rces of oay ad cosderg rores of crera. Ther souo ad herefore he deveoe of ucrera ehods are carred ou severa drecos [3]. hou dscusso of he s ecessary o o ou ha a ora queso ucrera decso akg s he souo quay. s cosdered hgh f eves of sasfyg crera are equa or cose o each oher (haroous souos whe a obecve fucos have he sae orace [4]. s o dffcu o exed hs coce for he case whe he orace eves of obecve fucos are dffere. Fro hs o of vew shoud be recorded he vady ad advsaby of he dreco reaed o he rce of guaraeed resu whch ca be reazed [24] o he bass of ayg he Bea-Zadeh aroach o decso akg a fuzzy evroe [5]. The Bea-Zadeh aroach ers oe o q SBN:

5 Proceedgs of he Euroea Coug Coferece reaze a couaoay effecve ad rgorous (fro he sado of obag souos Ω ehod of aayzg ucrera odes. s use aso aows oe o reserve a aura easure of uceray decso akg ad o ake o accou dces crera ad cosras of quaave characer. he usg he aroach each obecve fuco F ( s reaced by a fuzzy obecve fuco or a fuzzy se: = { ( } =... q (9 where ( s a ebersh fuco of [2]. fuzzy souo D wh he gve fuzzy ses (9 s ured ou as a resu of he erseco q = D = wh a ebersh fuco D ( = q (. (2 The use of he erseco (2 ers oe o oba he souo rovg he axu degree ax ( = ax ( (2 D q of beogg o he fuzzy souo D. Therefore he robe (8 s reduced o = arg ax q (. (22 To oba he souo (22 s ecessary o bd ebersh fucos ( =... q refecg a degree of achevg "ow" oa by F ( =... q. Ths codo s sasfed by he use of ebersh fucos λ F ( F ( ( = (23 ax F ( F ( for axzed obecve fucos or by he use of ebersh fucos λ ax F ( F ( ( = (24 ax F ( F ( for zed obecve fucos. (23 ad (24 λ =... q are corresodg orace facors. The cosruco of (23 ad (24 deads o sove he foowg robes: F ( (25 F ( ax (26 rovdg he souos = arg F ( ad = arg ax F (. hs aer he souo of (8 deads aayss of 2 q oocrera robes (25 (26 ad (2 resecvey. Sce he souo s o beog o Ω s ecessary o bud ( = { ( ( } (27 D q where ( π = f Ω or ( = f Ω. π The rocedures for sovg he robe (2 dscussed [2] rovde a e obag Ω accordace wh (27. Thus ca be sad abou equvaece of ( D ad ( D. ers oe o gve u he ecessy of eeg a cubersoe rocedure for budg he se Ω. The exsece of s addoa codos (dces crera ad/or cosras of quaave characer defed by gusc varabes [2] reduces (22 o = arg ax ( (28 where q s ( = q... s are ebersh fucos of fuzzy vaues of gusc varabes whch refec hese addoa codos. Takg he above o accou he souo of he ucrera caacor acee robe s reduced o ayg he geerazed agorhs of dscree ozao [7-9] o sove he ax robe (2. 5 usrag Exae The reseed resus have served for eaborag a cuso-deveoed Eecrc Power Dsrbuo ayss (EPODN sofware. Ths sofware s eeed Java/C o rovde fexbe ower fow ode o ozao agorhs whe suyg rch vsuazao ad aayss caabes o he user. odfed backward-forward swee agorh [6] s eeed aog wh he echques of arae rocessg for obag hgh erforace resus o arge scae odes (for sace eworks wh over busses aowg acao of he aer resus o he rea-word eworks o coveoa desko couers. e us cosder he resus obaed fro EPODN for he robe of acg fxed caacors a dsrbuo ework 3.8/.22 kv of oe of subsaos 38/3.8 kv of he Mas Geras Sae Eergy Coay (CEMG. Ths ework cudes 3 feeders feedg 9 rary cosuers ad 292 dsrbuo rasforers wh 9756 secodary cosuers. The oa egh of eworks s 36 k. The foowg souo aeraves are reseed π SBN:

6 Proceedgs of he Euroea Coug Coferece Tabe : a sae; oocrera souo whch zes he obecve fuco Z observg he cosras for he sueror voage eves; B oocrera souo whch zes he obecve fuco Z; C oocrera souo whch zes he obecve fuco ; M ucrera souo whch rovdes a corose bewee souos ad B; M 2 ucrera souo whch rovdes a corose bewee souos B ad C. Tabe. Souo resus erave Obecve fuco Z R$ Obecve fuco Mh B C M M Cocuso The resus reaed o foruag ad sovg he robe of caacor acee dsrbuo syses wh he fraework of ucrea decso akg odes have bee reseed. The souo of he robe s based o cobg he Bea-Zadeh aroach o decso akg a fuzzy evroe wh he acao of he geerazed agorhs of dscree ozao. The use of he ucrera aroach ers oe o overcoe he dffcues of suaeous observg cosras for he erssbe uer ad ower voage s a dffere buses of dsrbuo syses as we as oher ora codos oerag caacors rovdg fexbe souos. 7 ckowedges Ths research was suored by he Naoa Couc for Scefc ad Techoogca Deveoe of Braz (CNPq ad he Mas Geras Sae Eergy Coay (CEMG. Refereces: [] H.N. Ng M.M.. Saaa ad.y. Chha Cassfcao of Caacor ocao Techques EEE Trasacos o Power Devery Vo.5No [2] R.. Gaego.J. Moce ad R. Roero Oa Caacor Pacee Rada Dsrbuo Neworks EEE Trasacos o Power Syses Vo.6 No [3] J.P. Chou C.F. Chag ad C.T. Su Caacor Pacee arge-scae Dsrbuo Syses Usg Varabe Scag Hybrd Dfferea evouo eraoa Joura of Eecrca Power ad Eergy Syses Vo.26 No [4] J.Y. Park J.M. Soh ad J.K. Park Oa Caacor ocao a Dsrbuo Syse Cosderg Oerao Coss EEE Trasacos o Power Syses Vo.24 No [5] B.. Souza H. Nasceo ad H.. Ferrera Mcrogeec gorhs ad Fuzzy ogc ed o he Oa Pacee of Caacor Baks Dsrbuo Neworks EEE Trasacos o Power Syses Vo.9 No [6] M.B. u C.. Vafzares ad. Huag Reacve Power ad Voage Coro Dsrbuo Syses wh ed Swchg Oeraos EEE Trasacos o Power Syses Vo.24 No [7] V.V. Zor ad P.Ya. Eke Dscree-ozao Mehods for Eecrca Suy Syses Power Egeerg Vo.8 No [8] P. Eke R. Pahares. rauo M. Sva V. Poov. Bodareko ad V. Tkacheko Geerazed gorhs of Dscree Ozao ad Ther caos eraoa Joura of Couer Research Vo.2 No [9] P.Ya. Eke ad F.H. Schuffer gorhs of Dscree Ozao ad Ther caos o Probes wh Fuzzy Coeffces forao Sceces Vo.76 No [] T.H. Core C.E. eserso ad R.. Rves roduco o gorhs MT Press 99. [] V. Pareo Cours d Écooe Poque ousae Rouge 886. [2]. Pedrycz P. Eke ad R. Parreras Modes ad gorhs of Fuzzy Mucrera Decso- Makg ad Ther caos ey ress. [3] M. Ehrgo Mucrera Ozao Srger 25. [4] P.Ya. Eke Mehods of Decso Makg Fuzzy Evroe ad Ther caos Noear ayss Vo.47 No [5] R.E. Bea ad.. Zadeh Decso-akg a Fuzzy Evroe Maagee Scece Vo.7 No [6] R.D. Zera Corehesve Dsrbuo Power Fow: Modeg Foruao Souo gorhs ad ayss: Ph. Thess Core Uversy 995. SBN:

Chapter 1 - Free Vibration of Multi-Degree-of-Freedom Systems - I

Chapter 1 - Free Vibration of Multi-Degree-of-Freedom Systems - I CEE49b Chaper - Free Vbrao of M-Degree-of-Freedo Syses - I Free Udaped Vbrao The basc ype of respose of -degree-of-freedo syses s free daped vbrao Aaogos o sge degree of freedo syses he aayss of free vbrao