Evolutionary techniques applied to the optimal short-term scheduling of the electrical energy production

E olu iona y echniques applied o he op imal sho - e m scheduling o he elec ical ene gy p oduc ion Alicia T oncoso , Jose ´ C. Riquelme , Jesu ´s S. Aguila -Ruiz , Jesu´s M. Riquelme San os Abs ac This pape p esen s an e olu iona y echnique applied o he op imal sho - e m scheduling (24 h) o he elec ic ene gy p oduc ion. The equa ions ha define he p oblem lead o a non-con ex non-linea p og amming p oblem wi h a high numbe o con inuous and disc e e a iables. Consequen ly, he esolu ion o he p oblem based on combina o ial me h- ods is a he ha d. The equi ed heu is ics, in oduced o assu e he easibili y o he cons ain s, a e analyzed, along wi h a b ie desc ip ion o he p oposed gene ic algo i hm (GA). The GA is used o compu e he op imal on/off s a us o he mal uni s and he fi ness unc ion is ob ained by sol ing a quad a ic p og amming p oblem by means o a s anda d non-linea In e io Poin (IP) me hod. The esul s om eal-wo ld cases based on he Spanish powe sys em a e epo ed, which show he good pe o mance o he p oposed algo i hm, aking in o accoun he complexi y and dimensionali y o he p oblem. Finally, an IP algo i hm is adap ed o deal wi h disc e e a iables ha appea in his p oblem and he ob ained esul s a e compa ed wi h ha o he p oposed GA. Keywo ds: Gene ic algo i hms; Scheduling; Op imiza ion; Feasibili y; In e io poin algo i hms 1. In oduc ion The op imal sho - e m scheduling o he elec i- cal ene gy p oduc ion [1] aims a de e mining which gene a ing uni s should be online and he co e- sponding op imal gene a ion o he mal and hyd o uni s along he scheduling pe iod, usually 24 h, in o de o minimize he expec ed o al cos sa is ying he o ecas ed sys em load. The scheduling ask leads o a non-linea mixed-in ege p og amming p oblem. Mo eo e , his p oblem is coupled in ime by he maximum speed ha gene a ing uni s, spe- cially he mal uni s, a e able o change he p oduced ene gy (known as up and down amps), and also by he opology o he hyd oelec ic powe plan s, wi h a delay in hou s be ween he wa e o a ese oi being used and he a ailabili y o ha wa e in he * Co esponding au ho . E-mail add esses: [email p o ec ed] (A. T oncoso), iquelme@ lsi.us.es (J.C. Riquelme), [email p o ec ed] (J.S. Aguila -Ruiz), [email p o ec ed] (J.M. Riquelme San os). ese oi s downs eam. A eally la ge numbe o a iables, bo h con inuous and disc e e a iables, is needed o p ope ly model his p oblem. Many app oaches ha e been p oposed o he esolu ion o his op imiza ion p oblem, anging om Dynamic P og amming o Linea Mixed-In ege P og amming o Lag angian Relaxa ion [2], he la - e being he mos widely used op imiza ion me hod in comme cial p og ams. Gene ic Algo i hms (GAs) [3,4], a gene al-pu pose s ochas ic sea ch me hod based on he mechanics o na u al selec ion, ha e also been success ully applied o he elec ical ene gy scheduling p oblem since he adap a ion is qui e s aigh o wa d due o he combina o ial na - u e o his p oblem. In he las ew yea s, he In e- io -Poin (IP) o Loga i hmic-Ba ie class o me hods has become he p e e ed nume ical app oach o sol e non-linea op imiza ion p ob- lems, as a consequence o i s enhanced capabili y o deal wi h inequali y cons ain s [5,6]. Since he ea ly de elopmen s, in ended o Linea P og am- ming p oblems, many imp o emen s ha e been p o- posed in o de o ex end he applica ion o IP me hods o con ex and non-con ex non-linea p oblems. These efinemen s ha e o do wi h s ep- size con ol [7], line-sea ch echniques o o ce con- e gence o a local minimum om a bi a y s a ing poin s [8,9], use o us egions [10], among o he s. Howe e , despi e hese imp o emen s, i s pe o - mance on mixed-in ege p oblems is equen ly dis- appoin ing because he solu ion is a om he global op imum [11]. This has aised he in e es in compu a ionally expensi e algo i hms aken om he a ificial in elligence a ea such as e olu iona y me hods [12], abu sea ch [13], pa icle swa m op i- miza ion [14], simula ed annealing [15] o an col- ony op imiza ion [16]. Al hough con e gence o he global op imum canno be heo e ically gua an- eed, he abili y o escape om local minima makes hem an a ac i e choice o many applica ions [17,18]. In his pape , a GA applied o he op imal sho - e m (24 h) elec ical ene gy p oduc ion scheduling is p esen ed. Some heu is ics a e included in o de o assu e he easibili y o he cons ain s ha appea in he p oblem. Resul s om eal-wo ld cases based on Spanish powe sys em a e epo ed. The elec ic ene gy p oduc ion scheduling p esen s a la ge numbe o a iables and cons ain s, a non-con ex non-linea objec i e unc ion and in e- ge and con inuous a iables. Thus, he main diffi- cul y is o find easible solu ions and he no el y o he pape is add essed o imp o e he easibili y. The main con ibu ions o he encoded GA can be s a ed as: a p ocedu e o gene a e he ini ial popula- ion aking in o accoun he amp cons ain s; dynamic cons ain s o minimum limi s on he hou ly ene gy p oduc ion o he he mal uni s o conside he s a ing and s opping pe iods as an al e na i e o he inclusion o o he bina y a iables o model hese s a es; a c osso e ope a o adap ed o he ea u es o his p oblem leading o an ade- qua e pe cen age o easible indi iduals; and spa - si y echniques and op imal o de ing used o educe he compu a ional o e head. In spi e o he spa si y echniques, a po en ial limi a ion o he GA is a he ela ed o he CPU ime when a com- plex opology o hyd oelec ic powe plan s is analyzed. The pape is o ganized as ollows: Sec ion 2p e- sen s he equa ions used o model he scheduling p oblem, leading o a non-linea mixed-in ege p og amming p oblem wi h a la ge numbe o bo h con inuous and disc e e a iables. In Sec ion 3a b iefly desc ip ion o he p imal-dual IP algo- i hm is made. Sec ion 4in oduces he p oposed GA, and se e al implemen a ion issues ha a e c ucial o ob ain easible solu ions a e discussed. Finally, Sec ion 5 epo s some esul s ob ained om ealis ic cases based on he Spanish powe sys em, and he main conclusions o he pape a e ou lined. 2. Fo mula ion o he p oblem The objec i e o he scheduling p oblem is o de e mine he on/off s a e and he ene gy p oduc- ion o he mal and hyd o uni s a each hou o he scheduling pe iod, in o de o minimize he o al cos o he sys em sa is ying he o ecas ed hou ly demand and he echnical cons ain s o he mal and hyd o powe plan s. The s anda d no a ion used o he scheduling o he elec ical ene gy p oduc ion p oblem is summa- ized in Table 1. This no a ion desc ibes fixed pa ame e s o he he mal and hyd o uni s, indexes, numbe o elemen s and a iables. 2.1. Objec i e unc ion The o al ene gy p oduc ion cos o he schedul- ing pe iod is defined by CT¼X n ¼1X ng i¼1 ½CiðPi; ÞþSUiUi; ð1Ui; 1Þ þSDið1Ui; ÞUi; 1;ð1Þ whe e n is he numbe o hou s o he scheduling pe iod, n g he numbe o he mal uni s, each ha ing a quad a ic cos unc ion, C i (P i, ), o he ene gy p o- duc ion, P i, ;SU i and SD i a e, espec i ely, he s a - up and shu -down cos o he he mal gene a o i, and U i, is a bina y a iable ep esen ing he on/off s a e o he he mal gene a o ia hou . I can be obse ed ha he o al p oduc ion cos is a sum o quad a ic unc ions o he ene gy o each he mal gene a o i he s a e o each gene a o was p e iously s a ed by he GA. This is he case o he p oposed echnique because he on/off s a es a e managed by he GA. No ice ha he p oduc ion cos is only due o he p oduc ion o he mal gene - a o s P i, , i.e., gene a o s ha p oduce ene gy by bu ning a uel o by a omic means. Hyd o uni s p o ide ee-o -cha ge ene gy PH h, ha is only sub- jec o he a ailabili y o wa e in he co esponding ese oi s. 2.2. Cons ain s The minimiza ion o he objec i e unc ion is subjec o echnical cons ain s, wa e balance in hyd oelec ic powe plan s and he associa ed ese - oi s, and o he sys em ene gy demand and ese e balances: •Maximum and minimum limi s on he hou ly ene gy p oduc ion o he he mal and hyd o gene a o s, Pm i6Pi; 6PM i;i¼1;...;ng; ¼1;...;n ;ð2Þ PHm h6PHh; 6PHM h;h¼1;...;nh; ¼1;...;n ; ð3Þ Table 1 Defini ion o he da a and a iables o he p oblem Da a o fixed pa ame e s SU i The s a -up cos o he he mal uni i(€) SD i The shu -down cos o he he mal uni i(€) C i (Æ) Quad a ic cos unc ion o he he mal uni i(€) Pm iLowe bound o he hou ly ene gy p oduc ion o he he mal uni i(MWh) PM iUppe bound o he hou ly ene gy p oduc ion o he he mal uni i(MWh) PHm hLowe bound o he hou ly ene gy p oduc ion o he hyd o plan h(MWh) PHM hUppe bound o he hou ly ene gy p oduc ion o he hyd o plan h(MWh) VHm hLowe bound o he wa e le el o he ese oi hin e ms o ene gy (MWh) VHM hUppe bound o he wa e le el o he ese oi hin e ms o ene gy (MWh) UR i Uppe bound o he up a e o he he mal uni i(MWh/h) DR i Lowe bound o he down a e o he he mal uni i(MWh/h) W h Inflow o he ese oi hin e ms o ene gy (MWh) D Ene gy demand a hou (MWh) R Gene a ing capaci y in ese e a hou (MWh) DT i Numbe o hou s ha he uni imus be shu -down a e s opping UT i Numbe o hou s ha he uni imus be unc ioning a e s a ing d(k) Wa e delay ime be ween ese oi kand he nex ese oi downs eam (in h) n(k) Nex ese oi downs eam ega ding he ese oi k Indexes and numbe o elemen s iThe mal uni index hHyd o plan index Hou index n g Numbe o he mal uni s n h Numbe o hyd o plan s n Numbe o hou s o he scheduling pe iod Va iables P i, Ene gy p oduc ion o he he mal uni ia hou (MWh) U i, On/off s a e o he he mal gene a o ia hou PH h, Ene gy p oduc ion o he hyd o plan ha hou (MWh) VH h, S o ed ene gy o he ese oi ha hou (MWh) whe e n h is he numbe o hyd o plan s, PH h, he ene gy p oduc ion o hyd o plan ha hou , and Pm i,PM i,PHm hand PHM ha e he limi s on he hou ly ene gy p oduc ion o he he mal uni i and hyd o plan h, espec i ely. Eq. (2) canno be ulfilled when he mal gene a- o s a e ei he s a ing o s opping, as s a ing and s opping pe iods begin, espec i ely, when he co esponding s a e changes o ON o OFF. In o de o a oid his p oblem, his equa ion is modified o he mal uni s ha a e ei he being s a ed-up o shu -down, 06Pi; 6PM i;i¼1;...;ng; ¼1;...;n :ð4Þ Mo eo e , he ene gy p oduced by he mal uni s du ing pe iods o shu ing-down (U i, = 0) is ou o he op imal scheduling. Consequen ly, penal y e ms p opo ional o his ene gy a e added o he objec i e unc ion as ollows: C0 T¼CTþX n ¼1X ng i¼1 CpPi; ð1Ui; Þ:ð5Þ •Maximum up and down amps o he mal uni s. The he mal uni s can no inc ease o dec ease he p oduc ion o ene gy a consecu i e hou s by mo e han a gi en maximum a e, DRi6Pi; Pi; 16URi;i¼1;...;ng; ¼1;...;n ;ð6Þ whe e UR i yDR i a e, espec i ely, he maximum up and down a es o he he mal gene a o i, usually known as amp limi s. •Limi s on he a ailable wa e . The hyd o uni s use wa e o gene a e elec ical ene gy and wa e is a limi ed esou ce. Thus, he ene gy p oduced by a hyd o uni is limi ed by he olume o a ail- able wa e in he associa ed ese oi . In conse- quence, ese oi le els a e subjec o capaci y limi s, VHm h6VHh; 6VHM h;h¼1;...;nh; ¼1;...;n ;ð7Þ whe e VH h, is he s o ed ene gy o ese oi ha hou , co esponding o he hyd o uni h;VHm h and VHM ha e espec i ely he minimum and max- imum limi s on he s o ed ene gy imposed by he maximum and minimum possible wa e le el o ese oi h. •Hyd aulic coupling be ween ese oi s. Time cou- pling exi s due o cascaded ese oi s, since he wa e used o p oduce ene gy in a hyd o uni will be a ailable la e o he nex hyd aulic uni down- s eam wi h a ce ain delay, ob iously when he wa e has a i ed o he co esponding ese oi . VHh; ¼VHh; 1PHh; þX nðkÞ¼h PHk; dðkÞþWh; ð8Þ whe e d(k) is he wa e delay ime in hou s be- ween ese oi kand he nex ese oi down- s eam, n(k), ha is supposed o be ese oi h, and W h is he na u al inflow o ese oi h. •The o al hou ly ene gy p oduc ion mus be equal he o al ene gy demand a ha hou , D , which has been p e iously o ecas ed. X ng i¼1 Pi; Ui; þX nh h¼1 PHh; ¼D ; ¼1;...;n : ð9Þ •The o al ene gy ha can be p oduced a each hou mus exceed he o ecas ed demand by a specified amoun , R , i.e., he gene a ing capaci y in ese e o be used i an unexpec ed e en such as he ailu e o a plan o a la ge e o on he o ecas ed demand happens. X ng i¼1 PM iUi; þX nh h¼1 PHM hPD þR ; ¼1;...;n :ð10Þ •Minimum up and down imes o he mal uni s. The minimum up ime, UT i , is he minimum numbe o hou s ha he uni imus be unc ion- ing a e s a ing. Besides, he minimum down ime, DT i , is he minimum numbe o hou s ha he uni imus be shu -down a e s opping. X DT i1 k¼0 ð1Ui; þkÞPDT i i uni iis shu -down a hou ð11Þ X UT i1 k¼0 Ui; þkPUT i i uni iis s a ed a hou :ð12Þ S a -up and shu -down cos s o ealis ic cases end o educe he numbe o shu -downs and s a -ups o a minimum, making he minimum- ime con- s ain s useless in mos cases. Mo eo e , he inclu- sion o hyd aulic gene a ion acili a es he ulfillmen o he he mal uni cons ain s because he hyd o uni s a e as e in esponse and p oduce ene gy a no cos , i.e., he hyd aulic ene gy will be s a egically dis ibu ed among he hou s o he scheduling ho izon in o de o a oid he s a ing o mo e he mal uni s han he s ic ly equi ed. As an example, Table 2 shows he numbe o cons ain s, bina y and con inuous a iables o he abo e p oblem o a es sys em comp ising 49 he - mal uni s, wo hyd o uni s and he scheduling ho i- zon emb acing 24 h. 3. P imal-dual IP algo i hm Among he dis inc i e ea u es o he abo e op i- miza ion p oblem, he mos impo an a e: la ge numbe o a iables and cons ain s in p ac ical cases; non-con exi y o he objec i e unc ion; and p esence o in ege and con inuous a iables. The IP me hods only can be applied when all he a i- ables o he p oblem a e con inuous. An al e na i e equen ly used in p ac ice consis s in elaxing he disc e e na u e o U i, and imposing ins ead he nex cons ain : 06Ui; 61:ð13Þ This leads o a simplified model wi hou disc e e a iables which equi es ha a heu is ic p ocedu e be applied du ing o a he end o he i e a i e p o- cess in o de o de e mine he bes in ege alue o e e y U i, . Thus a mixed-in ege p og amming p ob- lem is conside ed om a con inuous pe spec i e. In addi ion, e e y disc e e a iable can be handled as a con inuous a iable p o ided ha he ollowing quad a ic cons ain is added: Ui; ð1Ui; Þ¼0:ð14Þ Howe e , his p ocedu e inc eases he non-linea i y and non-con exi y o he ini ial p oblem. Based on he abo e commen s and p ac ical expe ience, he ollowing p ocedu e has been chosen o sol e he op imal sho - e m scheduling o he elec ical ene gy p oduc ion. (1) The p oblem is sol ed adding he cons ain (Eq. (13)). (2) Using he ob ained solu ion in he be o e s ep o ini ialize he IP algo i hm, sol e he p ob- lem including he cons ain (Eq. (14)). A b ie desc ip ion o a classical IP algo i hm is p o ided nex . This me hod in oduces auxilia y posi i e slack a iables in o de o u n inequali y es ic ions in o equali y cons ain s: xj6xj,xjþsj¼xjsjP0;ð15Þ whe e x j ep esen s any a iable subjec o a limi , and s j is he co esponding posi i e slack a iable. In o de o gua an ee he posi i eness o he slack a iables, loga i hmic penal y e ms a e included in he objec i e unc ion by means o a penal y ac o l ha is p og essi ely educed h oughou he i e - a i e p ocess [5]. 0ðxj;sj;lÞ¼ ðxjÞlX j ln sj:ð16Þ The main s eps o he IP algo i hm a e he ollowing: (1) Ini ialize he a iables so ha he slack a i- ables a e posi i e. (2) Ini ialize he penal y ac o lso as o make he loga i hmic e ms domina e o e he o iginal objec i e unc ion. (3) The minimiza ion o he co esponding Lag angian unc ion is pe o med by sol ing he non-linea op imali y equa ions using an one-s ep New on’s algo i hm, and he op imal inc emen o p imal and dual a iables is compu ed. (4) The s ep-leng h ais educed, i necessa y, so ha he slack a iables emain posi i e. Lag ange mul iplie s associa ed o equali y cons ain s a ising om he in oduc ion o auxilia y slack a iables mus also emain posi i e because op imali y condi ions lead o equa ions o he o m: sjzj¼l;ð17Þ whe e z j is he Lag ange mul iplie associa ed wi h he slack a iable s j . (5) Upda e p imal and dual a iables aking in o accoun he necessa y s ep-leng h limi a ion. (6) Reduce he penal y ac o l. The p opo ional ela ionship be ween he penal y ac o and he duali y gap (dugap) defined by Eq. (17) Table 2 Dimension o he p oblem o a es sys em Numbe o cons ain s Numbe o a iables Bina y Con inuous (2 Æn g +3Æn h +2)Æn +2Æn g n g Æn (n g +2Æn h )Æn 2642 1176 1272 p o ides he mos common app oach o educe his penal y ac o : l¼cPnl j¼1sjzj nl ;ð18Þ whe e c61andn l is he numbe o inequali y con- s ain s o he o iginal p oblem. S eps 3–6 a e i e a- i ely epea ed un il op imali y condi ions a e sa isfied and he penal y coefficien l, and conse- quen ly he a e age duali y gap, is small enough. Mo e sophis ica ed e sions o s eps 3 and 4, includ- ing line sea ches, modified Hessians, e c. [8–10] could be needed in he non-con ex case, in o de o a oid di e gence o con e gence o unaccep able poin s. Howe e , such efinemen s ha e no been ac ually implemen ed because he beha iou o he IP me hod has p o en good enough o he applica- ion es ed. The applica ion o New on’s me hod o sol e he non-linea op imali y equa ions yields a e y la ge, spa se linea sys em, specially when amp and hyd aulic couplings a e conside ed. Consequen ly, spa si y echniques and op imal o de ing [19] mus be used o educe he compu a ional o e head. Fig. 1 shows he fill-ins gene a ed when sol ing a small example (fi e hyd o plan s, fi e he mal plan s and a 5-hou scheduling ho izon), wi h and wi hou op imal o de ing. I can be no ed ha he fill-ins a e educed a 50% app oxima ely when an op imal o de ing is made. Table 3 shows ela i e execu ion imes o he IP algo i hm o wo ealis ic p oblems (73 he mal plan s, 24 h, 8 and 30 ese oi s, espec- i ely). As can be no iced, execu ion imes g ow conside ably wi h he numbe o a iables, specially when s anda d o de ing is pe o med. Finally, Fig. 2 shows he ela i e o e head o he diffe en p ocesses comp ising he IP algo i hm. 4. The p oposed gene ic algo i hm As p esen ed in he p e ious sec ion, he op imal scheduling o he elec ic ene gy p oduc ion is a non-linea , non-con ex, combina o ial, mixed-in e- ge and e y la ge p oblem. Hence, he e is no ech- nique ha would always lead o he op imal solu ion o he p oblem o ealis ic cases. In he las yea s, echniques based on heu is ics, dynamic p o- g amming, linea mixed-in ege p og amming and lag angian elaxa ion ha e been applied o his pa - icula p oblem. Techniques based on heu is ics ely on simple ules ha depends on he knowledge o powe plan ope a o s. Cons ain s o ealis ic p ob- lems a e no p ope ly modelled by dynamic p o- g amming app oaches, and he numbe o equi ed s a es inc eases exponen ially, hus leading o excessi e compu a ion imes. Linea p og am- ming app oaches canno p ope ly model nei he he non-linea objec i e unc ion no he non-linea cons ain s, and c ude app oxima ions a e equi ed. Finally, he use o heu is ic echniques is equi ed by lag angian elaxa ion app oaches o calcula e easible solu ions, de e io a ing he quali y o he ob ained solu ions. Consequen ly, new me hods a e s ill needed o ob ain mo e op imal solu ions o ealis ic p oblems. In his pape , a GA [20,21] has been used o sol e he scheduling p oblem due o i s abili y o deal wi h non-linea unc ions and in ege a iables. The p oposed GA algo i hm is used o compu e he op imal on/off s a es o he mal uni s, i.e., he bina y a iables, while he op imal con inuous a i- ables, i.e., he hou ly ene gy p oduc ion o hyd o and commi ed he mal uni s, a e calcula ed sol ing a ypical quad a ic p og amming p oblem by a clas- sical IP op imiza ion algo i hm in which he on/off s a es o he mal uni s a e known. Con e gence cha ac e is ics o GA depend on se e al key implemen a ion issues ha a e discussed in he es o his sec ion. 4.1. Codifica ion o he indi iduals Each indi idual is ep esen ed by he on/off s a es o he mal gene a o s du ing he scheduling pe iod. Thus, indi iduals a e ep esen ed by 0/1 ma ices, wi h columns co esponding o ime scheduling in e als and ows associa ed wi h he mal uni s. I he elemen (i,j) is equal o one, he s a e o he - mal uni idu ing ime in e al jis on. Simila ly, i he elemen (i,j) is equal o ze o, he s a e o he mal uni idu ing ime in e al jis off. Fig. 3 shows he ep esen a ion o a ce ain indi- idual o he popula ion. I can be obse ed ha he he mal uni 1 is on om 1am o 4am and he es o hou s is off; he he mal uni 2 is off om 1am o 6am and he es o hou s is on; he he mal uni 3 is on du ing all scheduling ho izon; he he mal uni 4 is only on om 11am o 4pm, e c. 4.2. Ini ial popula ion Up and down amp cons ain s o he mal uni s (Eq. (6)) a e a key ac o in he con e gence o he GA: i he ini ial popula ion is s ic ly andomly selec ed, amp cons ain s lead o many in easible indi iduals in he ini ial gene a ion, which makes successi e gene a ions suffe om poo di e si y, and he GA may con e ge p ema u ely. To assu e ha he ini ial popula ion con ains an adequa e 0 20 40 60 80 100 0 20 40 60 80 100 Columns Rows 0 10 20 30 40 50 60 70 80 90 100 0 10 20 30 40 50 60 70 80 90 100 Columns Rows Fig. 1. S uc u e o he linea sys em including fill-ins. pe cen age o easible indi iduals, ini ial on/off schedulings a e andomly selec ed bu modified o accoun o he minimum s a -up and shu -down imes imposed by amp cons ain s. Fo example, i gene a o g, wi h a maximum down amp equal o 100 MWh, is on a hou 3 p oducing an ene gy o 400 MWh, his gene a o would equi e 4 hou s o shu -down and, consequen ly, he gene a o a hou s 4, 5 and 6 should be on. The s a e U g,3 is s ic ly andomly gene a ed bu he s a es o he ollowing hou s, U g,4 ,U g,5 and U g,6 , a e gi en by Ug;3¼1)Ug;4¼Ug;5¼Ug;6¼1:ð19Þ 4.3. Fi ness unc ion The fi ness unc ion e alua es he quali y o an indi idual o he popula ion. In his case, he unc- ion is he in e se o he o al p oduc ion cos o he indi idual. The o al p oduc ion cos is ob ained sol ing a quad a ic p og amming p oblem by using a non-linea In e io Poin me hod [22,23].An ex a-high-cos fic i ious gene a o is included o sa is y he sys em demand (Eq. (9)). This fic i ious gene a o gene a es he necessa y ene gy ha he es o gene a o s canno p oduce o sa is y he demand o he cus ome s. A penal y e m p opo - ional o he defici in ese e equi emen s is added in he cos unc ion aiming a sa is ying he ese e cons ain . Penal y e ms only apply o in easible indi iduals, which a e consequen ly elimina ed h oughou he e olu iona y p ocess. 4.4. Selec ion ope a o To p oduce a new gene a ion, pa en s a e an- domly selec ed using a oule e wheel selec ion ech- nique ha selec s he bes indi iduals o ep oduc ion. The p obabili y o a pa icula indi- idual being selec ed is in p opo ion o i s fi ness unc ion, aking in o accoun ha he o al gene a- ion cos , including possible penaliza ions, is being minimized. The indi iduals chosen o be pa en s a e included in he ollowing gene a ion. 4.5. C osso e ope a o Offsp ing is ob ained by adding he bina y s ings ha esul s om andom pa i ions o each ow, as shown in Fig. 4a. A column-pa i ioning p ocedu e may also be applied (Fig. 4b). This c osso e ope - a o is a pa icula case o he mul i-poin c osso e ope a o whe e he numbe o poin s is equal o he numbe o ows o columns, espec i ely. As ows a e associa ed wi h he he mal uni s, he fi s app oach yields mainly he in easibili y o new indi iduals in e ms o minimum up and down imes (Eqs. (11) and (12)), while he second app oach has an effec bigge on he cons ain o he demand (Eq. (9)) and he ese e (Eq. (10)). This abo e s a e- men is shown in he nex example. Le be a es Table 3 Rela i e execu ion ime and i e a ion numbe . Numbe o a iables Na u al o de ing Op imal o de ing I e a ions Time I e a ions Time 2376 33 4.00 34 1.00 3960 27 140.98 34 2.45 O he s 23% Op imal o de ing 6% Linea sys em building 10% Linea sys em solu ion 61% Fig. 2. Rela i e o e head o he diffe en p ocesses. Hou s o he Schedulin g Ho izon The mal Uni s 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 0 0 0 0 0 0 0 0 ..... 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 Fig. 3. Rep esen a ion o an indi idual o he popula ion. sys em comp ising ou he mal uni s and a 4-hou scheduling ho izon. The minimum up and down ime a e 2 hou s o all he mal uni s. Two indi id- uals selec ed o be pa en s a e ep esen ed in Fig. 5. Fig. 6 shows he child en ob ained using a c osso e ope a o by ows om he andom pa i ion (1,2,1,2). I can be obse ed ha he child en do no sa is y he Eqs. (11) and (12) (gene a o h ee) while he Eqs. (9) and (10) can be easible. Fig. 7 shows he child en ob ained using a c osso e ope - a o by columns om he same pa i ion. I can be obse ed ha he indi idual on he igh does no sa is y he Eqs. (9) and (10) since all gene a o s a e off a hou s 3 and 4. Howe e , in his case he Eqs. (11) and (12) a e ulfilled. The c osso e p obabili y has been se o one, i.e., wo indi iduals ha ha e been selec ed o be pa en s a e always combined o ob ain a new indi idual. In he final e sion o he GA, he c osso e by ows has been chosen because s a -up and shu - down cos s o ealis ic cases, along wi h he inclu- sion o hyd aulic gene a ion, end o educe he numbe o shu -downs and s a -ups o a minimum, making he minimum- ime cons ain s useless in mos cases. All he ows a e always combined o ob ain a new indi idual, hough p obabili ies migh ha e been used o de e mine which ows should be combined. 4.6. Mu a ion ope a o A e he c osso e p ocess, he indi iduals o he popula ion a e mu a ed o in oduce some new gene ic ma e ial acco ding o a p e-defined mu a- ion p obabili y p. Consequen ly, he pe cen age o mu a ed indi iduals o a gene a ion is equal o 100p%. The mu a ion o an indi idual means he mu a ion o an only gene. The gene o be mu a ed 1 2 3 1 2 3 11 22 33 .. .. .. gg 1 2 3 1 2 3 11 22 33 .. .. .. gg PARENTS CHILDREN 123 123 11 22 ... ... gg 123 123 11 22 ... ... gg CHILDREN PARENTS ab Fig. 4. C osso e Ope a o : (a) andom pa i ions o ows and (b) andom pa i ions o columns. h1 h2 h3 h4 h1 h2 h3 h4 g1 1 1 1 1 g1 1 1 0 0 g2 1 1 0 0 g2 1 1 0 0 g3 0 0 0 0 g3 1 1 1 1 g4 1 1 0 0 g4 0 0 1 1 Fig. 5. Pa en s. h1 h2 h3 h4 h1 h2 h3 h4 g1 1 1 0 0 g1 1 1 1 1 g2 1 1 0 0 g2 1 1 0 0 g3 0 1 1 1 g3 1 0 0 0 g4 1 1 1 1 g4 0 0 0 0 Fig. 6. Child en by pa i ions o ows. h1 h2 h3 h4 h1 h2 h3 h4 g1 1 1 1 1 g1 1 1 0 0 g2 1 1 0 0 g2 1 1 0 0 g3 1 1 1 1 g3 0 0 0 0 g4 0 0 1 1 g4 1 1 0 0 Fig. 7. Child en by pa i ions o columns.