Optimization of Wind Farm Turbine Layout Including Decision Making Under Risk

This a icle has been accep ed o inclusion in a u u e issue o his jou nal. Con en is inal as p esen ed, wi h he excep ion o pagina ion. IEEE SYSTEMS JOURNAL 1 Op imiza ion o Wind Fa m Tu bine Layou Including Decision Making Unde Risk Ja ie Se ano Gonz´ alez, Manuel Bu gos Pay´ an, and Jes´ us M. Riquelme-San os Abs ac —This pape p esen s a new con ibu ion o op imal wind a m design, including he main isk managemen aspec s. The objec i e o he algo i hm is o op imize he expec ed p o i s o he wind a m by aking in o accoun ha he wind da a used o design he wind a m in ol es some deg ee o unce ain y ha a ec s he inal e u n o he p ojec . Ne p esen alue (NPV) will be used as a igu e o he p o i abili y in he p oposed me hod. The maximiza ion o he NPV means he maximiza ion o he cumula i e ne cash lows (by maximizing he gene a ion o ne ene gy) and minimiza ion o he in es men . Bo h e ms mainly depend on he numbe and ype o wind u bines, owe heigh , and geog aphical posi ion, among o he ac o s. The e o e, he ool de eloped in his pape is in ended o de e mine he wind a m con igu a ion mos sui able in he p esence o isk due o unce ain y in he wind da a. Index Te ms—Expec ed alue, gene ic algo i hm, isk managemen , u ili y heo y. I. In oduc ion THE USE OF wind ene gy o gene a e elec ici y is becoming mo e and mo e impo an in mos coun ies. Cu en in e es in enewable ene gy esou ces, such as wind powe , is mainly due o double suppo . On he one hand, i is d i en by he economic and poli ical aspec s, such as he upwa d end in ossil uel p ices and insecu i y o supply. On he o he , he e a e social and en i onmen al aspec s, esul ing om he e e -inc easing social awa eness abou he ha m ul impac s o he emission o g eenhouse gases esponsible o global clima e change. Wind powe ins alled wo ldwide by he end o 2010 will amoun o a o al o 196.63 GW, o which 85.98 GW co e- sponds o Eu ope, and o hese, 20.68 GW, o Spain [1]. The g ow h a e o o al ins alled capaci y in he wo ld was 19% in 2010. This alue has emained ela i ely cons an du ing he las ew decades, and no hing seems o indica e ha much will change in he coming yea s. A wind a m is made by a clus e o wind u bines (WTs), mo e o less packed. This con igu a ion o e s some economic Manusc ip ecei ed Sep embe 15, 2010; e ised May 12, 2011; accep ed May 25, 2011. This wo k was suppo ed in pa by he Minis y o Sci- ence and Technology, Spain, unde Resea ch P ojec s ENE207-66072/ALT, ENE200768032-C04-02, and ENE2011-27984, and in pa by he Go e nmen o Andalusia, unde P ojec s P06-TEP-01882 and TEP-5170. The au ho s a e wi h he Depa men o Elec ical Enginee ing, School o Enginee ing, Uni e si y o Se ille, Se ille 41092, Spain (e-mail: ja ie se - [email p o ec ed]; mbu [email p o ec ed]; [email p o ec ed]). Colo e sions o one o mo e o he igu es in his pape a e a ailable online a h p://ieeexplo e.ieee.o g. Digi al Objec Iden i ie 10.1109/JSYST.2011.2163007 ad an ages ela ed o he in es men , plan ope a ion, and main enance cos s. Howe e , he WT compac ness deg ee is limi ed by spacing cons ain s due o he wake e ec s ( he sc eening e ec p oduced by he o o o a u bine on hose loca ed behind i , downs eam). As a consequence, he layou o speci ic indi idual WT posi ions de e mines he o e all e iciency o ex ac ion o wind ene gy in a wind a m. The design o a wind plan aimed a gene a ing elec ici y and i s p ope ope a ion, o e he acili y li e span, is a com- plex and mul idisciplina y ask ha in ol es many di e en a eas o expe ise, om enginee ing o o he sciences. I is no only a complex en e p ise om a echnological s andpoin bu also a deg ee o unce ain y, in e ms o e u n o p o i abili y, ha is highe han desi able. The e a e many ac o s ha in luence he unce ain y in he e u ns on in es men . Among hese, he main a e as ollows. 1) Fu u e p ices and cos s: The u u e p ices o goods, such as he p ice o he ene gy o discoun a e, a e ob iously unknown. Bu , in o de o es ima e he p esen e u n om selling he elec ici y p oduced, i is necessa y o know he selling p ice o he ene gy and discoun a e, h oughou he whole span li e o he wind a m ( ypically 20 yea s). This could also include he cos s o a ious ac o s ha in luence he no mal de elopmen o he p ojec du ing cons uc ion (such as ci il wo k o implemen a ion delays), ope a ion ( u bine una ail- abili y, losses in he dis ibu ion ne wo k, wake e ec losses, o main enance cos ), and in he inal phase o decommissioning o possible u u e egula o y changes ha could a ec he economic o inancial scena io. 2) Wind: The sale o gene a ed elec ic ene gy is he sou ce o income o he wind a m. The e o e, he andom na u e o wind (speed dis ibu ion and di ec ions) in- oduces some deg ee o unce ain y in annual ene gy p oduc ion. In his sense, i is wo h no ing ha he op imum posi ioning o each o he u bines wi hin a wind a m is one o he majo ac o s ha in luence he p o i abili y o he ins alla ion. This is due o he u bine wake e ec . Wake e ec losses a e he esul o in e ac ion o wo main ac o s: he wind (wind ose and speed, no con ollable) and layou o he u bines in he wind a m (con ollable a he p ojec s age). In his pape , he p oblem o he op imal design o wind a ms (selec ion o he u bines loca ion, u bine ype, and hub heigh ), aking in o accoun he unce ain y in he s a is ical 1932-8184/$26.00 c 2011 IEEE This a icle has been accep ed o inclusion in a u u e issue o his jou nal. Con en is inal as p esen ed, wi h he excep ion o pagina ion. 2 IEEE SYSTEMS JOURNAL Fig. 1. Example o annual a ia ion in he compass ose. Wind ose om Lis /Syl (Ge many) o he yea s 1969 and 1972 [4]. Fig. 2. Change in Weibull pa ame e s, mean wind speed, and es ima ed yea ly gene a ion o ene gy in Hong Kong since 1968 un il 1997 [5]. cha ac e iza ion o he wind, is analyzed. The unce ain y om he wind in o ma ion, in bo h wind di ec ion and in ensi y (speed), becomes an unce ain y in he es ima ion o he yea ly gene a ed ene gy. I is, he e o e, he ac o ha mos di ec ly a ec s he p o i abili y o he wind a m. Bu , o una ely, a he same ime, i is he mos con ollable ac o in he design s age (op imizing he u bine layou ). The objec i e is o de e mine he con igu a ion o he wind a m, so ha he unce ain y o he p o i abili y is se o a le el o isk accep able o in es o s. Mo eo e , he s udy should allow he in es o o know he maximum and minimum le els o p o i abili y o he p ojec , which is an essen ial piece o in o ma ion in in es o decision making when deciding whe he o unde ake o ejec he p ojec [2], [3]. As an example o unce ain y o wind da a, Fig. 1 shows changes in he wind ose a Lis /Syl (Ge many), measu ed in 1969 and 1972 [4]. Fig. 2 shows he a ia ions in Weibull dis ibu ion pa ame e s (scale, C, and shape, K), mean wind speed, and es ima ed annual ene gy p oduced along 30 yea s in Hong Kong (adap ed om [5]). To da e, he e a e se e al pape s ha use a ma hema ical model o op imize he solu ion o he loca ion p oblem (po- si ioning o mic osi ing) o WTs in a wind a m. In [6]–[9], he au ho s p oposed a a he simple wind a m cos model because hey we e mainly in e es ed in demons a ing he e ec i eness o he op imiza ion algo i hm. The ou pape s use he same wake decay, simpli ied cos models, and e y simila objec i e unc ions, and hey analyze he same cases. They only di e in he op imiza ion echnique. While [6] and [8] used a gene ic algo i hm (GA), [7] used a g eedy algo i hm and [9] used a Mon e Ca lo simula ion. A mo e ealis ic wind a m cos model is de eloped in [10]– [12] using a GA as he op imiza ion algo i hm. In ela ion o he use o decision me hods applied o he planning o wind a ms, he mos signi ican wo k was done by [13], whe e he objec i e was o de e mine he mos app op ia e gene a ion capaci y unde unce ain y. This pape in oduces a new app oach o he p oblem o op imal posi ioning o u bines in a wind a m, including de- cision making unde isk. As men ioned abo e, he economic pe o mance o a wind a m has a high deg ee o unce ain y. The e o e, in his pape , a p obabilis ic op imiza ion me hodol- ogy has been de eloped by aking in o accoun a se o possible scena ios and hei p obabili y o occu ence. This p obabilis- ic app oach allows ob aining solu ions wi h a beha io unde isk be e han he pe o mance o con igu a ions ob ained by he de e minis ic app oaches de eloped o da e. A e his in oduc ion, he pape is o ganized as ollows. Sec ion II desc ibes he p oblem app oach and p oposed me hodology. Sec ion III p esen s he economic model o he wind a m. Tes cases and conclusions a e p o ided in Sec ions IV and V, espec i ely. II. Me hodology The p oposed ool combines an e olu iona y algo i hm, as he op imiza ion echnique, wi h a decision me hod unde isk (Fig. 3). An ini ial popula ion wi h a se o possible solu ions (indi iduals) is andomly gene a ed by he e olu- iona y algo i hm. This popula ion e ol es, gene a ion a e gene a ion, owa d he op imum by means o he c osso e and mu a ion ope a o s. The i ness o each indi idual is e alua ed by he decision me hod ha assesses he economical sui abili y [based on ne p esen alue (NPV)] o each o he indi iduals composing he popula ion (al e na i es) o he se o scena ios (s a es o na u e). A. Op imiza ion Algo i hm The complexi y o he p oblem o op imal posi ioning o he WTs in a wind a m a ises no only om a echnical poin o iew, due o s ong links be ween i s a iables, bu also om a pu ely ma hema ical poin o iew. The p oblem consis s o bo h disc e e and con inuous a iables, being, he e o e, an in ege -mixed ype p oblem. The p oblem exhibi s mani old op imal solu ions (con exi y) and canno be comple ely desc ibed in an analy ical o m; some a iables ha e a ange o non-allowed alues (solu ions space no simply connec ed) and o he s a e in ege s. This ac makes he p oblem non-de i able, p e en ing he use o classical analy ical op imiza ion echniques. GAs ha e been used success ully in p e ious wo k o op imize he p oblem o mic o-posi ioning o he u bines o a wind a m and demons a ed hei sui abili y o he complexi y o his p oblem [6], [8], [10]–[12]. GA a e obus op imum This a icle has been accep ed o inclusion in a u u e issue o his jou nal. Con en is inal as p esen ed, wi h he excep ion o pagina ion. GONZ ´ ALEZ e al.: OPTIMIZATION OF WIND FARM TURBINE LAYOUT INCLUDING DECISION MAKING UNDER RISK 3 Fig. 3. Op imiza ion algo i hm inco po a ing unce ain y ( isk) scena ios. sea ch echniques ha ind he minimum o maximum o a unc ion based on p inciples inspi ed om he na u al gene ic and e olu ion mechanisms obse ed in he na u e [14]–[16]. The ype and heigh o he WTs a e disc e e a iables ha a e no easily managed by adi ional nume ical algo i hms due o he nondi e en iable na u e o he p oblem. The e o e, in ege codi ica ion has been used in he algo i hm implemen a ion, which has been also applied o he loca ions o he WTs. The in ege codi ica ion used ep esen s e e y possible solu- ion o he p oblem by means o a ma ix, whe e he columns e e o he u bines o an indi idual and e e y ow codi ies he cha ac e is ics o each u bine: posi ion o he wind gene a o in Ca esian coo dina es (Xi,Yi), ype o wind gene a o (Ti), and owe heigh (Hi). The ype o u bine is codi ied wi h a numbe , which will be he index in he gene a o da abase ha uses he algo i hm as an inpu . The a o emen ioned da abase con ains all he necessa y in o ma ion o he wind gene a o s ha could be ins alled in he wind a m (i.e., maximum and minimum heigh o he owe s, u bine and owe cos s, ounda ion cos , and powe –wind speed cu e). The e o e, hese ma ices ha e a a iable numbe o columns, depending on he numbe o gene a o s equi ed by he codi ied indi idual solu ion [17]–[19]. GA mainly makes use o wo kinds o ope a o s o gene a e new indi iduals (po en ial solu ions): c osso e and mu a ion. The c osso e ope a o is applied on wo selec ed indi iduals, called pa en s, o gene a e new indi iduals, called sons, wi h a mix o ch omosomes (cha ac e is ics) om he pa en s. The selec ion me hod used is known as oule e wheel, whe e he indi iduals wi h highes i ness (objec i e unc ion) a e mo e likely o be selec ed. Fi e special ypes o c osso e ope a o s ha e been de eloped o imp o e he algo i hm pe o mance [17]–[19]. The mu a ion ope a o is applied on one indi id- ual o gene a e ano he by andomly changing one o mo e ch omosomes. When he popula ion is con ined in a local maximum, his ope a o leads o he c ea ion o indi iduals ou o his zone o local a ac ion. This way he algo i hm can e ol e owa d he global maximum. The op imiza ion algo i hm manages se e al kinds o cons ain s ha makes TABLE I Resul s Ma ix: NPV o Each Indi ual and Scena io Scena io S1... Sj... SmEV (NPV) Scena io pS1(E1)... pSj (Ej)... pSm(Em) EV1= m  j=1 NPV1jpj p obabili y Indi idual 1s NPV11 NPV1jNPV1mEV2= m  j=1 NPV2jpj ... ... ... ... ... ... ... Indi idual i h NPVi1... NPVij ... NPVim EVi= m  j=1 NPVij pj ... ... ... ... ... ... ... Indi idual n h NPVn1... NPVnj ... NPVnm EVn= m  j=1 NPVnj pj he p ocess o designing a wind a m mo e lexible, such as ollows. 1) Se ing up o bidden a eas whe e, o di e en easons, i is no possible o place a u bine. When an indi idual shows a wind gene a o in a o bidden a ea, he genes co esponding o he posi ion o his ae ogene a o a e mu a ed ill hey become loca ed in an allowed a ea. 2) Tu bines on he same posi ion o ou side he e ain. Usually, a e he ope a ion o c osso e o mu a ion, non- easible solu ions can be c ea ed. In his case, a egene a i e algo i hm goes h ough he indi iduals, emo ing he u bines ha a e w ongly placed. 3) Limi a ion o he maximum in es men o be done i he in es o has a limi ed a ailable capi al o s a he p ojec . This cons ain will be aken in o accoun by penalizing he indi idual in case o o e in es men . 4) Limi a ion in he maximum numbe o gene a o s. This es ic ion is con olled by limi ing he numbe o he indi iduals h oughou he gene a ion o he ini ial pop- ula ion and c ossing ope a ion. B. Me hods o Decision Making When add essing he wind a m op imiza ion p oblem wi hin he amewo k o a de e minis ic app oach, once he inpu a iables alues a e se ( he de e minis ic scena io), an op imiza ion algo i hm, based on GA echniques, de e - mines he op imal con igu a ion o he wind a m Fig. 4(a). Bu , when add essing he p oblem wi h a isk app oach, he unce ain y in he inpu a iables mus be conside ed Fig. 4(b). Now, he inpu a iables a e cha ac e ized by a se o scena ios (S1,S 2,...,S N) and i s p obabili y o occu ence (p1,p 2,...,p N). In his case, Table I shows he co espond- ing ma ix esul s. As can be seen, he NPVij elemen o his ma ix shows he NPV ela ed o he i h po en ial wind a m con igu a ion (indi idual i h), conside ing he j h scena io (Sj). In his pape , wo di e en objec i e unc ions we e used de- pending on he decision c i e ion adop ed: maximum expec ed alue (MEV) o maximum expec ed u ili y (MEU), based on he u ili y heo y (UT). Wi h he MEV app oach, he objec i e is o ind he wind a m con igu a ion wi h MEV o NPV. To each ha goal, he expec ed alue (EV) o NPV o each indi idual mus be This a icle has been accep ed o inclusion in a u u e issue o his jou nal. Con en is inal as p esen ed, wi h he excep ion o pagina ion. 4 IEEE SYSTEMS JOURNAL Fig. 4. Op imiza ion algo i hm. (a) De e minis ic app oach. (b) Inco po a - ing unce ain y ( isk) scena ios. calcula ed as [20] EVi= m  j=1 NPVij pj.(1) The MEU c i e ion is based on he UT. The UT models he beha io o he decision make by means o a unc ion ( he u ili y unc ion) exp essing he p e e ence o he decision make o each one o he al e na i es. The u ili y unc ion used in his pape is an exponen ial ype [2] de ined by u(NPV(x))=⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ 1−e−(NPV(x)−NPVmin)/ρ 1−e−(NPVmax−NPVmin)/ρ ,i ρ=∞ NPV(x)−NPVmin NPVmax −NPVmin ,i ρ→∞ (2) whe e uis he u ili y alue co esponding o NPV(x), xis he con igu a ion o he wind a m, and ρis he pa ame e o isk ole ance. The u ili y unc ion eaches he maximum alue, equal o one, o NPVmax, and he minimum alue, equal o ze o, o NPVmin. In his pape , NPVmin has been aken as he minimum alue o NPV ha he decision make would be willing o accep o unde ake he p ojec , and NPVmax is he maximum alue o NPVij co esponding o he ma ix o esul s h oughou he e olu ion o he GA. Fig. 5 shows, as an example, he amily o u ili y cu es depending on he isk ole ance pa ame e . The isk ole ance pa ame e , ρ, allows modeling he a i ude o he decision make . I ρ>0, he decision make is isk a e se because, as he NPV inc eases, he slope o he u ili y cu e dec eases. I ρ<0, he decision make is isk seeking ( he decision make inc easingly app ecia es NPV inc eases). In bo h cases, he highe he isk ole ance, he highe he isk p e e ence o he decision make . The ex eme case is eached when he alue o isk ole ance is in ini e (ρ→±∞), ep esen ing a neu al decision make a i ude owa d isk because he u ili y cu e is a s aigh line. The e o e, he decision make uni o mly alues each inc emen o NPV. Finally, in his app oach, he objec i e Fig. 5. Decision-make a i ude depending on he isk ole ance pa ame e . will be achie ed as he MEU is calcula ed by EUi= m  j=1 u(NPVij )·pj.(3) Bo h c i e ia (MEV and MEU) is necessa y o calcula e he NPV o each scena io and po en ial solu ion (indi idual) aking in o accoun he economic model o he wind a m de ailed in he hi d sec ion. III. Wind Fa m Economic Model A wind a m wi h a ce ain u bine con igu a ion ( u bine a ed capaci y, ype, heigh , and loca ion), x, equi es an ini ial capi al in es men o build and pu he acili y in o p oduc ion, IWF(x). This ini ial in es men is necessa y mainly o a o d he WT acquisi ion cos s, as well as he ci il and elec ical in as uc u e cos s. The wind a m, once in ope a ion, deli e s a s eam o bo h inancial bene i (p o i s om he gene a ed elec ic ene gy selling), NESk(x), and o dina y ope a ion and main enance cos s, CO&Mk(x), yea a e yea , o e he li e span o he p ojec , LT. A inal p esen cos o he ins alla ion decommissioning, CD(x), and a p esen esidual alue, VR(x), a e he p oduc ion pe iod, mus also be conside ed. This way, he NPV o he wind a m, NPV(x), aking in o accoun he equi alen discoun a e, , can be w i en as NPV(x)=−IWF(x)−CD(x)+VR(x)+ LT  k=1 Nk(x) (1 + )k(4) whe e he ne cash low, Nk, ep esen s he ne incomes p oduced by he wind a m du ing he k h yea . This e m is only he di e ence be ween he income esul ing om he ene gy sale and ope a ion and main enance cos s, Nk(x)= NESk(x)−CO&Mk(x). The e o e, he maximiza ion o he NPV means a balance be ween he minimiza ion o he in es men and maximiza ion o he ne cash lows ( o maximize he ne gene a ion o ene gy). Bo h e ms depend on he numbe and ype o wind gene a o s, owe heigh , and geog aphical posi- ion, among o he s. Table II shows a ypical cos b eakdown o a wind a m, adap ed om [21]. As can be seen, mos o he ini ial in es men is o he pu chase o WTs, while he emaining in es men is aimed a he cos s o elec ical This a icle has been accep ed o inclusion in a u u e issue o his jou nal. Con en is inal as p esen ed, wi h he excep ion o pagina ion. GONZ ´ ALEZ e al.: OPTIMIZATION OF WIND FARM TURBINE LAYOUT INCLUDING DECISION MAKING UNDER RISK 5 TABLE II Typical Ini ial Cos S uc u e o a Wind Fa m I em % WTs 65–75 Subs a ion and elec ical in as uc u e 10–15 Inne elec ical dis ibu ion ins alla ion 6–9% Subs a ion and e acua ion line connec ion 4–6% Ci il wo k 5–10 Componen ins alla ion 0–5 O he 5 O e all WT cos (D/kW) 800–1200 and ci il in as uc u e. The main p oblem o be sol ed is he posi ioning and selec ion o he u bines ( ype and heigh o he owe ), and his is he p oblem ha has he g ea es in luence on he wind a m p o i abili y. The cos s o ci il wo ks a e ela i ely simple o calcula e, howe e , he design o he elec ical ins alla ion is a complex p oblem and i s exac calcula ion would in ol e a high consump ion o CPU ime. This eason, in addi ion o he ela i ely low economic weigh o he elec ical ins alla ions, allows an app oxima ion o he elec ical cos s o be conside ed equal o he ci il wo k cos . To ob ain a wind a m NPV as ealis ic as possible, he e olu ion o he p ices o he sold ene gy, as well as he inc emen o he ope a ion and main enances cos , mus be conside ed. Assuming ha Ek(x) is he annual ne amoun o elec ic ene gy p oduced and sold a yea k,pkWh is he p ice o he kilowa -hou o sold ene gy, pkWh is i s annual inc emen , and CO&M is he annual inc emen o he cos o ope a ion and main enance, hen he NPV o he cash low along he wind a m li e span yields NPV(x)=−IWF(x)−CD(x)+VR(x) + LT  k=1 Ek(x)PkWh(1+pkWh)k+1 (1+ )k − LT  k=1 CO&MK (x)(1 + CO&M)k+1 (1 + )k. (5) To p ope ly e alua e he po en ial ene gy supplied by he wind a m du ing a yea , he wake speed decay e ec mus be conside ed due o he pe u ba ion o he wind speed p o ile as a esul o he ope a ion o he u bines loca ed ups eam [22]–[24]. The ac ual ne ene gy p oduced and sold by a se o u bines in a wind a m is lowe ha o he sum o he ene gies o he u bines i hey we e isola ed. This is due o wo kinds o losses: he wake e ec p e iously men ioned and he una ailabili y o he WTs (due o main enance, epai o echnical es ic ions). A. Wind Beha io Model The s a is ical beha io o he wind a a gi en al i ude will be app oxima ed by he Weibull dis ibu ion unc ion [25] ha desc ibes equency, p( ), o a gi en wind speed, ,asa unc ion o he shape pa ame e , K, and he scale pa ame e , C, using he o mula p( )=K C CK−1 exp − CK.(6) Wind speed depends on al i ude due o he exis ing ic ion o he ai wi h he g ound su ace. Gi en he wind speed, (z ), o a e e ence heigh , z , he co esponding wind speed a a di e en heigh , z, can be calcula ed using an exponen ial unc ion (z)= (z )ln(z/z0) ln(z /z0)(7) whe e z0is he leng h o oughness o he e ain [26]. The collec ion o wind ene gy done by a u bine educes he speed o he wind h ough i , causing a educ ion in he kine ic ene gy a ailable o he u bines loca ed downs eam in he di ec ion o he inciden wind. The esul ing wind speed a a dis ance, d, downs eam o he u bine ha c ea es he wake is calcula ed by [23] U(d) U0 =1 2+1 21−2CTD0 D(d)2 (8) whe e U0is he speed o he wind in ee low, D0is he diame e o he o o , D(d) he wake diame e , and CT he dimensionless h us coe icien . Finally, he calcula ion o he elec ic ene gy gene a ed in a yea , can be ob ained by combining he long- e m dis ibu ion o he wind speed conside ing he di e en di ec ions o he wind ose, and he cu e o speci ic powe o he u bine, o each ype o gene a o conside ed in he wind a m as ollows: EWF =T N  j=1 co j  ci j kA jPGen j( )pj( )d (9) whe e Tis he numbe o hou s pe yea (T= 8760h), N is he numbe o u bines, ci j is he cu -in speed (speed a which he u bine s a s gene a ion) o u bine jand co j is he cu -ou speed ( he inal gene a ion speed) o u bine j, PGen j( ) is he powe – eloci y cu e o gene a o j, and kAV j is he a ailabili y ac o o u bine j. B. Ci il In as uc u e Cos Ci il in as uc u e cos s a e made up mos ly o ounda ion cos s and cos s de i ed om he execu ion o auxilia y oads o access he wind gene a o s. The i s en y depends on he posi ion o he wind gene a o s and is calcula ed acco ding o ypical alues, aking in o accoun an inc ease in he ounda- ion cos s o cases whe e he wind gene a o s a e loca ed in a eas wi h a educed bea ing capaci y. The cos o building auxilia y oads is made up o he ac i i ies o clea ing, illing, and compac ion o oads, which a e di ec ly p opo ional o i s o al leng h. The P im algo i hm o calcula ion o minimum spanning ee [27] has been used o calcula e he con igu a ion and leng h o he auxilia y oads. IV. Tes Cases A se o cases has been sol ed o show he sui abili y o he p oposed algo i hm. The decision me hods, MEV and MEU, ha e been combined wi h he op imiza ion algo i hm o selec he mos app op ia e design o each si ua ion and le el o isk ha he decision make is willing o assume. In bo h cases, This a icle has been accep ed o inclusion in a u u e issue o his jou nal. Con en is inal as p esen ed, wi h he excep ion o pagina ion. 6 IEEE SYSTEMS JOURNAL Fig. 6. Wind ose and p obabili y o each conside ed scena io. (a) S1. (b) S2. (c) S3. (d) S4. TABLE III Main Fea u es o WTs WTA WTB WTC Ra ed capaci y (MW) 2000 2000 1670 Minimum heigh (m) 60 60 60 Maximum heigh (m) 100 100 80 Cos (MD)2.10 2.00 1.67 Towe cos (kD/m) 1.5 1.5 1.5 Founda ion cos (kD) 80 80 80 unce ain y in he da a o he cha ac e iza ion o wind has been conside ed: he wind di ec ion in Case 1 and wind speed and di ec ion in Case 2. A. Case 1: Wind Di ec ion Unce ain y Fig. 6 shows he ou scena ios conside ed o he possible wind di ec ions. Each scena io has a p obabili y o occu ence. In addi ion, each scena io consis s o he p obabili y ha he wind comes om each o he di ec ions ha make up he wind ose. O e he whole e ain, he wind speed is de ined by he same scale ac o (C=6.5) and shape (K= 2) o he Weibull dis ibu ion. The s udy has been ca ied ou on a squa e e ain o dimensions 3 ×3 km disc e ized in 10 ×10 cells as in Fig. 7. The pa cel is c ossed by a oad om eas o wes in he no he nmos a ea. Cells c ossed by he main oad and nea he no hwes co ne o he land a e o bidden a eas whe e WT placemen is no allowed. In he no heas , he e is an a ea whe e he bea ing capaci y o soil is educed so ha he ounda ion cos s will be highe . Table III shows he ca alog o u bines om which he algo i hm can selec . Fig. 8 shows he powe –speed cu es o he WTs conside ed. Table IV shows he main echnical and economic inpu da a o he algo i hm. Fo each scena io, he de e minis ic op imal solu ion (DOS) has been calcula ed. These solu ions a e shown in Fig. 9. Table V shows o each o he DOS, he EV and NPV (MD) ha would be ob ained o o he scena ios. As can be seen, Fig. 7. Main e ain ea u es o Case 1. Fig. 8. Powe –speed cu es o conside ed WTs. TABLE IV Main Technical and Economical Inpu Da a Li e span (yea s) 20 In e es a e (%) 6 P ice o ene gy (D/kWh) 0.07 Maximum no. o u bines 8 Auxilia y oads cos (D/m) 100 O e cos in a ea o low bea ing capaci y (%) 25 Roughness leng h (m) 0.0055 A ailabili y ac o (%) 95 P esen cos o decommission (%) 3 P esen esidual alue (%) 3 Minimum NPV o unde ake he p ojec (MD) 1 o each o he scena ios, he maximum alues a e achie ed o he de e minis ic solu ion o ha scena io. Fig. 10 shows he op imal solu ion eached by he algo i hm aking in o accoun he MEV c i e ion. The selec ed WTs a e ype B (WTB) wi h a heigh owe o 100 m. In his case, he algo i hm eaches he op imal solu ion in 254 gene a ions using a unning ime o 1170 s, on a PC In el Pen ium Dual Co e 2 GHz. The second column o Table VI shows he main economical esul s ob ained by he algo i hm o Case 1. By using he MEU c i e ion, he aim is o maximize he expec ed u ili y (EU) using he exponen ial u ili y unc ion de ailed in (2). The case has been pe o med se e al imes This a icle has been accep ed o inclusion in a u u e issue o his jou nal. Con en is inal as p esen ed, wi h he excep ion o pagina ion. GONZ ´ ALEZ e al.: OPTIMIZATION OF WIND FARM TURBINE LAYOUT INCLUDING DECISION MAKING UNDER RISK 7 TABLE V Resul s o Case 1: NPV (MD) DOS o Each Scena io Scena io 1 Scena io 2 Scena io 3 Scena io 4 EV DOS (S1) 21.02 20.78 19.69 20.41 20.09 DOS (S2) 20.76 21.21 20.40 16.30 19.94 DOS (S3) 20.20 20.77 21.05 21.05 20.93 DOS (S4) 20.20 20.77 21.05 21.05 20.93 Fig. 9. DOS ob ained o each scena io. (a) Scena io 1. (b) Scena io 2. (c) Scena io 3. (d) Scena io 4. Fig. 10. Op imal layou ob ained by he MEV c i e ion o (a) Case 1 and (b) Case 2. by a ying he decision-make a i ude (by a ying he alue o isk ole ance, ρ). Fig. 11 shows he op imal con igu a ion depending on he isk ole ance pa ame e . Table VII shows he esul s ob ained by he algo i hm o Case 1 using he MEU c i e ion. Each ow shows he NPV (MD), o each scena io and i s EV, o each op imal con igu a ion depending on he pa ame e o isk ole ance, ρ( he isk p e e ence inc eases mo ing down in he able). Wi h a e si e o neu al a i udes o isk (ρ≥0), he op imal solu ion is he same as ha ob ained by he MEV c i e ion Fig. 11. Op imal layou ob ained o Case 1 by he MEU by a ying he isk ole ance o he decision make . (a) ρ=−0.01. (b) ρ=−0.008. (c) ρ=−0.005. (d) ρ=−0.002. TABLE VI Main Economical Resul s Ob ained wi h MEV C i e ion Case 1 Case 2 EV o NPV (MD) 20.96 19.06 In es men (MD) 19.91 19.79 Tu bines cos (MD)17.20 17.20 Ci il in as uc u e cos (MD)1.35 1.30 Elec ical in as uc u e cos (MD)1.35 1.30 A e age powe EV (kW) 4042 3842 (Fig. 10). Howe e , by inc easing he isk p e e ence o he decision make , he op imal solu ion leads o an NPV inc ease o a speci ic scena io. Fo example, wi h alues o he isk ole ance ρ=−0.01 and ρ=−0.008, he op imal solu ion maximizes he NPV o Scena ios 3 and 4 because hey a e he mos likely scena ios. I he decision make becomes mo e isky (ρ=−0.005) he op imal solu ion maximizes he NPV o he Scena io 2 (because i is mo e p o i able, as can be seen in Table V), despi e educing he EV due o he lowe p o i abili y o he emaining scena ios. I he decision make akes a highly isky a i ude (in his case ρ=−0.002), he solu ion ob ained coincides wi h ha esul ing om a de e minis ic app oach conside ing only he Scena io 2. In conclusion, he decision make akes isk wi h he possibili y o inc easing p o i abili y in he e en ha he occu ence o Scena io 2 is highe han expec ed. As shown, he p oposed me hods yield solu ions ha im- p o e hose ob ained by he de e minis ic app oach in he p esence o isk. On he o he hand, i is also possible o ob ain solu ions o he le el o isk ha he decision make is willing o ake. Mode a e a i udes owa d isk lead o con igu- a ions mo e insensi i e o changes in wind di ec ion; howe e , inc easing he isk p e e ence, he end is o maximize he This a icle has been accep ed o inclusion in a u u e issue o his jou nal. Con en is inal as p esen ed, wi h he excep ion o pagina ion. 8 IEEE SYSTEMS JOURNAL TABLE VII Resul s o Case 1: NPV (MD) by Va ying he Risk P e e ence Scena io 1 Scena io 2 Scena io 3 Scena io 4 EV ρ=0.005 20.84 20.94 20.99 20.99 20.96 ρ=0.1 20.84 20.94 20.99 20.99 20.96 ρ→±∞ 20.84 20.94 20.99 20.99 20.96 ρ=−0.01 20.49 20.85 21.03 21.03 20.95 ρ=−0.008 20.20 20.77 21.05 21.05 20.93 ρ=−0.005 20.81 21.17 20.53 17.26 20.16 ρ=−0.002 20.76 21.21 20.40 16.28 19.94 TABLE VIII Wind Di ec ions Scena ios Scena io (P obabili y) C(m/s) Di ec ions (P obabili y %) S1(2.25%) 7.5 ENE (20%) E (70%) ESE (10%) S2(8.25%) 6.5 ENE (20%) E (70%) ESE (10%) S3(4.50%) 5.5 ENE (20%) E (70%) ESE (10%) S4(9.75%) 7.5 NE (20%) ENE (70%) E (10%) S5(35.75%) 6.5 NE (20%) ENE (70%) E (10%) S6(19.50%) 5.5 NE (20%) ENE (70%) E (10%) S7(3.00%) 7.5 E (20%) ESE (70%) SE (10%) S8(11.00%) 6.5 E (20%) ESE (70%) SE (10%) S9(6.00%) 5.5 E (20%) ESE (70%) SE (10%) TABLE IX Resul s o Case 2: NPV (MD) by Va ying he Risk P e e ence S1S2S3S4S5S6S7S8S9EV ρ=0.1 34.81 21.10 7.45 34.81 21.10 7.45 34.81 21.10 7.45 19.06 ρ→±∞ 34.81 21.10 7.45 34.81 21.10 7.45 34.81 21.10 7.45 19.06 ρ=−0.134.87 21.06 7.29 34.87 21.06 7.29 34.87 21.06 7.29 19.00 ρ=−0.03 34.26 20.56 6.94 35.00 21.21 7.43 29.02 16.02 3.49 18.07 p o i abili y o a gi en scena io wi hou aking in o accoun he loss o p o i abili y o he emaining possible scena ios. B. Case 2: Wind Di ec ion and In ensi y Unce ain y Case 2 conside s he same condi ions desc ibed in Case 1, bu wi h di e en wind condi ions, and adds unce ain y in he scale pa ame e o Weibull dis ibu ion. The scena ios conside ed in his case a e shown in Table VIII. The solu ion eached by he algo i hm, aking in o accoun he MEV c i e ion, is shown in Fig. 10. Table VI shows he mos ele an economic pe o mance o he solu ion ob ained. Table IX p esen s he esul s ob ained wi h he MEU c i e- ion, a ying he isk p e e ence o he decision make . I can be seen ha he solu ion ob ained wi h neu al and a e si e a i udes o isk (ρ≥0) is he same as ha ob ained wi h he MEV c i e ion. Risk p e e ence a i udes (ρ<0) lead o an inc ease in he p o i abili y o he Scena io 4. Scena ios 1 and 4 p o ide he highes NPV bu he highe p obabili y o Scena io 4 leads he algo i hm o inc ease he p o i abili y o such a scena io. Fig. 12 shows he solu ions ob ained o he le els o isk p e e ence ρ=−0.1 and ρ=−0.03, espec i ely. As can be seen, by inc easing he isk p e e ence o decision make , he u bines selec ed change om WTB o WTA because his u - bine can p o ide a bi mo e powe despi e equi ing a sligh ly Fig. 12. Op imal layou ob ained o Case 2 by he MEU by a ying he isk ole ance o he decision make . (a) ρ=−0.1. (b) ρ=−0.03. highe ini ial in es men . I can also be seen how by inc easing he isk om ρ=−0.1 oρ=−0.03, he geog aphical layou o he wind gene a o changes o maximize ene gy cap u e ha would be p oduced o he Scena io 4. V. Conclusion The design o a wind a m is an ex emely complex ask ha in ol es a la ge numbe o a iables. A he p ojec s age, he beha io o many o hese a iables is di icul o cha ac e ize ei he due o ac o s, such as e o s in he es ima ion o cos s and unce ain y in economic beha io , o due o he andom na u e o some o he a iables, such as he wind. Among all o he a iables ha in luence he p o i abili y o a wind a m p ojec , he cha ac e is ics o he wind ha e he g ea es in luence on he plan con igu a ion and i s economic e iciency. The e o e, in his pape , he isk analysis and decision making has ocused on he unce ain y in wind esou ce (di ec ion and speed) cha ac e iza ion. The pe o mance o he p oposed me hod has been success- ully e i ied by analyzing a se o es cases, wi h di e en wind scena ios. As a esul , when he isk analysis was included, he op imiza ion p ocess o he wind a m led o solu ions (plan con igu a ions) less sensi i e o he unce ain y han he de e minis ic solu ion. In addi ion, he p oposed me hods allowed us o ob ain con igu a ions ha me he le el o isk ha he de elope wishes o assume. Fu he mo e, he algo i hm p o ided in o ma ion on he le els o p o i abili y unde unce ain y in he wind cha ac e iza ion. Re e ences [1] Wo ld Wind Ene gy Associa ion. (2011, Ap .). Wo ld Wind Ene gy Repo 2010 [Online]. A ailable: h p://www.wwindea.o g [2] P. R. Ga ey, Analy ical Me hods o Risk Managemen : A Sys ems Enginee ing Pe spec i e. New Yo k: Taylo & F ancis, 2009. [3] P. R. Ga ey, Sys em-o -Sys ems Risk Managemen : Pe spec i es on Eme ging P ocess and P ac ice. Bed o d, MA: MITRE Co po a ion, Jan. 2005. [4] L. Ja ass, L. Ho mann, A. Ja ass, and G. Obe mai , Wind Ene gy. Be lin, G emany: Sp inge -Ve lag, 1981. [5] I. Y. F. Lun and J. C. Lam, “A s udy o Weibull pa ame e s using long- e m wind obse a ions,” Renew. Ene gy, ol. 20, no. 2, pp. 145–153, Jun. 2000. [6] G. Mose i, C. Poloni, and B. Di iacco, “Op imiza ion o wind u bine posi ioning in la ge wind a ms by means o a gene ic algo i hm,” J. Wind Eng. Ind. Ae odyn., ol. 51, no. 1, pp. 105–16, 1994. This a icle has been accep ed o inclusion in a u u e issue o his jou nal. Con en is inal as p esen ed, wi h he excep ion o pagina ion. GONZ ´ ALEZ e al.: OPTIMIZATION OF WIND FARM TURBINE LAYOUT INCLUDING DECISION MAKING UNDER RISK 9 [7] U. 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Mo a, “Global op imiza ion o wind a ms using e olu i e algo ihms,” in Wind Powe Sys ems: Applica ions o Compu a ional In elligence. Be lin, Ge many: Sp inge , 2010. [13] C. Kongnam, S. Nuchp ayoon, S. P em udeep eechacha n, and S. Ua- ongji , “Decision analysis on gene a ion capaci y o a wind pa k,” Renew. Sus ain. Ene gy Re ., ol. 13, no. 8, pp. 2126–2133, 2009. [14] R. Spillman, “Gene ic algo i hms, na u e’s way o sea ch o he bes ,” D . Dobb’s J., ol. 8, no. 2, pp. 26–30, Feb. 1993. [15] J. J. G e ens e e, “Op imiza ion o con ol pa ame e s o gene ic algo i hms,” IEEE T ans. Sys . Man, Cybe n., ol. 16, no. 1, pp. 122-128, Jan.–Feb. 1986. [16] D. Goldbe g, Gene ic Algo i hms in Sea ch, Op imiza ion and Lea ning. Reading, MA: Addison-Wesley, 1989. [17] J. S. Gonz´ alez, A. G. G. Rod íguez, J. C. Mo a, J. R. San os, and M. B. Pay´ an, “A new ool o wind a m op imal design,” in P oc. IEEE Bucha es Powe Tech Con ., Jun.–Jul. 2009, pp. 1–7. [18] J. C. Mo a, “Op imizaci´ on global de pa ques e´ olicos median e algo i - mos e olu i os,” Ph.D. disse a ion, Dep . Elec ic. Eng., Uni . Se illa, Se ille, Spain, Sep. 2008. [19] J. R. San os, M. B. Pay´ an, J. Cale o, and J. C. Mo a, “An e olu i e algo i hm o wind a m op imal design,” Neu ocompu ing, ol. 70, nos. 16–18, pp. 2651–2658, 2007. [20] D. Luce and H. Ra ia, Games and Decisions: In oduc ion and C i ical Su ey. New Yo k: Wiley, 1957. [21] M. Junginge , A. Faaij, and W. C. Tu kenbu g, “Cos educ ion p ospec s o o sho e wind a ms,” Wind Eng., ol. 28, no. 1, pp. 97–118, 2004. [22] I. Ka ic, J. Højs up, and N. O. Jensen, “A simple model o clus e e iciency,” in P oc. EWEC, 1986, pp. 407–409. [23] S. F andsen, R. Ba helmie, S. P yo , O. Ra hmann, S. La sen, J. Højs up, and M. Thøge sen, “Analy ical modelling o wind speed de ici in la ge o sho e wind a ms,” Wind Ene gy, ol. 9, no. 1, pp. 39–53, Jan. 2006. [24] S. T. F andsen, “Tu bulence and u bulence-gene a ed s uc u al loading in wind u bine clus e s,” Ph.D. hesis disse a ion, Wind Ene gy Dep ., Risø Na l. Lab., Tech. Uni . Denma k, Copenhagen, Denma k, Risø-R- 1188(EN), 2007. [25] W. Weibull, “A s a is ical dis ibu ion unc ion o wide applicabili y,” J. Appl. Mech.-T ans. ASME, ol. 18, no. 3, pp. 293–297, 1951. [26] T. Bu on, D. Sha pe, N. Jenkins, and E. Bossanyi, Wind Ene gy Handbook. New Yo k: Wiley, 2001. [27] R. P im, “Sho es connec ion ne wo ks and some gene aliza ions,” Bell Sys . Tech. J., ol. 36, no. 6, pp. 1389–1401, 1957. Ja ie Se ano Gonz´ alez ecei ed he M.Sc. de- g ee in elec ical enginee ing om he Uni e si y o Se ille, Se ille, Spain, in 2007. He is cu en ly pu suing he Ph.D. deg ee in elec ical enginee ing wi h he Uni e si y o Se ille. His cu en esea ch in e es includes enewable ene gy. Manuel Bu gos Pay´ an ecei ed he Ph.D. deg ee in elec ical enginee ing om he Uni e si y o Se ille, Se ille, Spain, in 1994. Since 1983, he has been wi h he Depa men o Elec ical Enginee ing, School o Enginee ing, Uni e si y o Se ille, whe e he is cu en ly an Asso- cia e P o esso . His cu en esea ch in e es s include elec ical machines, enewable ene gy, and powe quali y. Jes ´ us M. Riquelme-San os ecei ed he Ph.D. de- g ee in elec ical enginee ing om he Uni e si y o Se ille, Se ille, Spain, in 1999. Since 1994, he has been wi h he Depa men o Elec ical Enginee ing, School o Enginee ing, Uni e si y o Se ille, whe e he is cu en ly a P o- esso . His cu en esea ch in e es s include powe op imiza ion and con ol, powe sys em analysis, and powe quali y. View publica ion s a sView publica ion s a s