Application of a novel modified hybrid algorithm for solving Dynamic Economic Dispatch problem with practical constraints

POWER ENGINEERING AND ELECTRICAL ENGINEERING VOLUME: 16 |NUMBER: 4 |2018 |DECEMBER Applica ion o a No el Modi ied Hyb id Algo i hm o Sol ing Dynamic Economic Dispa ch P oblem wi h P ac ical Cons ain s Mos e a HAMED1, Belkacem MAHDAD1, Kamel SRAIRI1, Nabil MANCER2 1Depa men o Elec ical Enginee ing, Facul y o Sciences and Technology, Bisk a Uni e si y, BP 145 RP, 07000 Bisk a, Alge ia 2Depa men o Elec ical Enginee ing, Facul y o Science Technology, Cons an ine Uni e si y 1, A. Hamani Campus, Ain El Bey Road, 25000 Cons an ine, Alge ia hamed_elec 2008@yahoo. , bemahdad@yahoo. , ks ai i@yahoo. , namance @yahoo. DOI: 10.15598/aeee. 16i4.2877 Abs ac . Dynamic Economic Dispa ch (DED) is a highly complex nonlinea op imiza ion p oblem wi h p ac ical cons ain s. The aim o DED is o op imize dynamically he ac i e powe o gene a ing uni s o e ope a ing ime conside ing p ac ical cons ain s such as al e poin e ec , p ohibi ed zones, amp a e limi s and o al powe losses. In o de o o e come he d aw- back o he wo s anda d me aheu is ics such as Fi e ly Algo i hm (FA) and Time Va ying Accele a ion based Pa icle Swa m Op imiza ion (PSOTVAC), a hyb id me hod called FAPSOTVAC is p oposed o imp o e he solu ion o DED. The main idea in oduced owa ds combining FA and PSOTVAC is o c ea e a lexible equilib ium be ween explo a ion and exploi a ion du ing sea ch p ocess. The obus ness o he p oposed hyb id me hod is alida ed on many p ac ical powe sys ems (10 and 30 uni s) o minimize he o al uel cos consid- e ing all p ac ical cons ain s. The esul s ound p o e he e iciency o he p oposed FAPSOTVAC in e ms o solu ion quali y and con e gence cha ac e is ics. Keywo ds A hyb id algo i hm, dynamic economic dis- pa ch, FA-PSOTVAC, p ohibi ed ope a ing zones, amps a e cons ain s, al e poin e - ec . 1. In oduc ion Nowadays elec ic ene gy esembles a i al a e y o ou daily li e and he main engine o any economic o comme cial ac i i y. Such occupied posi ion ende s i i eplaceable due o i s c edibili y when compa ed wi h any o he na u al ene gy. The elec ic ene gy demand in ou economic has mul iplied by 3.2 in 37 yea s o each 19738 TWh in 2010. This colossal numbe in- dica es he salien posi ion i occupies in he cu en wo ld economy. The ene gy sys em is composed o powe s a ion in- e connec ed wi h ansmission lines anspo ing he p oduced ene gy o consume s a e se e al ope a ion and con ol s ages. The non-s ocked aspec o his o m o ene gy obliges us o p oduce i in he ime o con- sump ion. The balance be ween p oduc ion and con- sump ion should be espec ed in eal ime and wi hin he capaci y o powe gene a ing uni s. This p oblem is gene ally called he p oblem o S a ic Economic Dis- pa ch (SED). The main ask o elec ic powe sys em is o en- su e ins an aneously he equilib ium be ween p oduc- ion and demand. The de e mina ion o he op imal s a e o each gene a o in e connec ed wi h he elec- ic ne wo k du ing he wen y- ou hou s complica es he solu ion o he aced p oblem. Ra he han being mo e s a ic, his p oblem becomes dynamic in ime, in o he con ex s whe ein he complexi y o nowadays ne wo k inc eases is a is i s size ha holds hund eds o bus-ba s and hund eds o housands o kilome e s o ansmission lines, in addi ion o highly complica ed s uc u e o he in e connec ed ne wo k.All hese ac- o s make he op imiza ion o he o al uel cos com- plex and i al objec i e. In his con ex , new p ac ical cons ain s, a ached mainly o he cons uc ion o he mal uni s, on he one hand, and o he condi ions imposed by he s a egy o exploi a ion, on he o he hand, should be espec ed. c 2018 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING 459 POWER ENGINEERING AND ELECTRICAL ENGINEERING VOLUME: 16 |NUMBER: 4 |2018 |DECEMBER The opening o uel’s al es pe u b he quad a ic o m o cos ’s objec i e ope a ion by in oducing a highly non-linea o m. Fu he mo e, ano he new cons ain can be added o complica e he (DED) p oblem. The la e is gene a o s amp cons ain ha does no al- low he adjus men o he gene a ed ac i e powe only by a p e-imposed alue. In [1], Ramp- a e limi s ha e been conside ed in uni commi men and economic dis- pa ch inco po a ing o o a igue e ec . This s udy ex- plains how he iola ion o such cons ain s can highly minimize o o ’s li e and inc emen s he main enance cos . Since he cons ain s a e highly non-linea , hey add p ohibi ed ope a ing zones, which a e a ached di ec ly o he equilib ium o he he mal gene a ing uni s. The la e should ope a e away om ce ain in- e als called “P ohibi ed Ope a ing Zones” o a oid some dange ous ib a ion a he le el o machine’s bea ing [2]. In such si ua ion, he o m o objec i e unc ion mus be modi ied and adap ed o ake in o conside a ion he e ec s o hese p ohibi ed zones. Se e al ma hema ical me hods ha e been applied o sol ing such non-linea p oblem. Mos o hem ha e exploi ed he ma hema ical cha ac e is ics o he cos unc ion unc ion o disco e ing he con inui y and hessian de i a ion,..e c. In his sense, au ho s in [3] made a compa ison be ween he solu ion o he i e - a i e Lamda me hod and he me aheu is ic algo i hm “B en me hod” o sol ing he p oblem o dynamic economic dispa ching wi h losses and amp cons ain s. Mo eo e , au ho s in [4] ha e applied he dynamic p o- g amming in o de o ind he solu ion o he same p oblem by conside ing he p ohibi ed ope a ing zones. Whe eas, ese e cons ain s ha e been conside ed in [5] by applying Lag ange elaxa ion me hod. Besides, au ho s in [6] ha e applied he same me hod o in es- iga e uni commi men p oblem. All hese me hods a e swi and all wha hey need is one launch ei he o ind he op imum solu ion o s ay inap owa ds he di e en men ioned cons ain s. The o dina y me hods o op imiza ion canno co e he en i e space designed o esea ch in o de o ind a low cos o hey can be apped a a local a he han a global op imum ol- lowing an exagge a ed ime ha can ne e be applied in eal ime. The applica ion o a i icial in elligence me hods p esen an al e na i e o he con en ional me hods, which leads o he de elopmen and he applica ion o many echniques such as Gene ic Algo i hms (GA) [7], Pa icle Swa m Op imiza ion (PSO) [8] and hei mod- i ied e sions. Au ho s in [9] used he modi ied e sion o PSO which hey called Modula ed pa icle swa m op imiza ion o sol e he p oblem o mu i-objec i e dy- namic economic dispa ch. In addi ion, a kene ic gas molecule op imiza ion algo i hm has been p oposed in [2] o sol e he s a ic and dynamic economic dispa ch p oblems. Whe eas, au ho s in [10] applied he Modi- ied Real Coded Gene ic Algo i hm (MRCGA) o sol e he p oblem o mul i-objec i e Dynamic Economic Dis- pa ch (DED). Fu he mo e, au ho s in [11] sol ed he la ge scale p oblem DED by using he C issc oss op- imiza ion algo i hm. Meanwhile, au ho s in [12] sug- ges ed he Al e na ing Di ec ion Me hod o Mul iplie s (ADMM) o sol ing en i onmen al economic dispa ch. In [13] Di e en ial E olu ion (DE) algo i hm was ap- plied o sol e he DED p oblem conside ing amp a es cons ain s. Fo being able o co e he whole esea ch a ea, lim- i ed by an impo an numbe o cons ain s as well as he huge non-linea i y, on one hand, and o sol e he p oblem o a la ge size DED on he o he hand, many hyb id algo i hms ha e been sugges ed. These hyb id echniques ha e been de eloped o o e come he d awback o he s anda d me aheu is ic me hods by c ea ing lexible equilib ium be ween di e si ica- ion and in ensi ica ion du ing sea ch p ocess. Au- ho s in [14] used Modi ied Pa icle Swa m Op imiza- ion and Gene ic Algo i hm (MPSO-GA) o sol ing he p oblem o s a ic economic dispa ch wi h p ohib- i ed ope a ion zones, amp cons ain s and mul i uel. Besides, au ho s in [15] p oposed he hyb id me hod (MILP-MDSD) o sol e he p oblem o dynamic eco- nomic dispa ch wi h al e poin s e ec s. Au ho s in [16] ha e used he Imp o ed Dynamic P og amming (IDP), which is a ecu si e o a dynamic p og amming o sol e he p oblem o economic dispa ch wi h p o- hibi ed zones and amp a e cons ain s. In addi ion in [17] a Chao ic sel -adap i e Di e en ial Ha mony Sea ch algo i hm (CDHS) applied in o de o sol e he p oblem o dynamic economic dispa ch whe ein, he p ohibi ed ope a ion zones and amp- a e cons ain s a e aken in o conside a ion simul aneously. Au ho s in [18] p oposed me aheu is ic Two S age Mixed In- ege Linea P og amming (TSMILP) as a me hod o sol e he p oblem o dynamic economic dispa ch con- side ing he e ec s o al es and ansmission losses. On he o he hand au ho s in [19] used Fas E olu- iona y P og amming wi h Swa m Di ec ion o sol - ing DED p oblem. Whe eas, au ho s in [20] applied he hyb id echnique o C oss-En opy Me hod and Sequen ial Quad a ic P og amming o sol e he same p oblem. This a icle in ends o sol e he p oblem o mul i cons ain s non-linea dynamic economic dispa ch o in es iga e al es poin e ec s, amps cons ain s, by in oducing p ohibi ed ope a ing zones ha ha e ne e been ea ed oge he be o e acco ding o e iew o li - e a u e. The huge numbe o cons ain s and compli- ca ion p oblem obliged us o in oduce new hyb id al- go i hms such as FA-PSOTVAC and BBO-PSOTVAC o achie e he desi ed low cos by espec ing all he p ac ical ope a ion cons ain s imposed. c 2018 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING 460 POWER ENGINEERING AND ELECTRICAL ENGINEERING VOLUME: 16 |NUMBER: 4 |2018 |DECEMBER 2. Nomencla u e: BBO: Biogeog aphy-Based Op imiza ion al- go i hm. Ci : The uni ip oduc ion cos a ime . DED: Dynamic Economic Dispa ch. FA: Fi ly algo i hm. ng: The numbe o gene a ion uni s. ni: The numbe o p ohibi ed ope a ing zones in he i h gene a ing uni . Pd( ): Load demand a ime . Pmin i, P max i: The maximum and he minimum p o- duc ion o uni i. Pi : Powe ou pu o uni a ime . Ploss: T ansmission losses. PSOTVAC: Pa icles Swa m Algo i hm wi h a a iable accele a ion coe icien . T: The o al numbe o hou s in he op- e a ion pe iod. T C: To al Cos ($). URi, DRi: The amp up and he amp down a e limi ’s espec i ely. 3. Ma hema ical Fo mula ion 3.1. Objec i e Func ion The objec i e unc ion o (DED) p oblem is o mini- mize he o al p oduc ion cos o e he ope a ion pe- iod, which can be w i en as [21]: min T C = T X =1 ng X i=1 Ci (Pi ),(1) whe e Ci is he cos o i h gene a ing uni a ime ,ng is he numbe o gene a ion uni s and Pi is he powe ou pu o i uni a ime .Tis he o al numbe o hou s in he ope a ion pe iod. The uel cos unc ion o gene a ing uni s conside ing al e-poin e ec can be exp essed using he ollowing equa ion [15]: F(P gi) = ng X i=1 ai+biP gi +ciP g2 i +|eisin( i(P gmin i −P gi ))|, (2) whe e ai, bi, ci, ei, ia e he cos coe icien s o i h powe gene a ing uni s. This objec i e unc ion should be minimized conside ing he ollowing equali y and in- equali y cons ain s [22]. 3.2. The Equali y Cons ain s: ng X i=1 Pi =Pd( ) + Ploss, = 1,2, ..., T. (3) 3.3. Inequali y Cons ain s [23] and [24] Pmin i≤Pi ≤Pmax i, i = 1, ..., ng = 1, ..., T. (4) Pmin i, P max ia e he minimum and he maximum o uni ’s p oduc ion. 1) Ramp Ra e Cons ain s [25] Pi −Pi( −1) ≤URi,(5) Pi( −1) −Pi ≤DRi.(6) 2) P ohibi ed Ope a ion Zone The P ohibi ed Ope a ion Zones [26] and [27] a e ma hema ically exp essed by he ollowing equa ion: Pi∈     Pmin i≤Pi≤PL i1, Pik−1≤Pi≤PL ik, Pu izi ≤Pi≤Pmax i, (7) whe e: niis he numbe o p ohibi ed ope a ing zones in he i h gene a ing uni . kis he index o he p ohib- i ed ope a ing zones o he i h gene a ing uni . PL iK , PU iK a e he lowe and uppe bounds o k h p ohibi ed ope a ing zones o uni i. 4. Op imiza ion Algo i hms 4.1. Pa icles Swa m Algo i hm wi h a Va iable Accele a ion Coe icien PSOTVAC Pa icle Swa m Op imiza ion wi h Time Va iable Ac- cele a ion (PSOTVAC) is a dynamic a ian o he s anda d PSO algo i hm. This algo i hm p esen s a modi ied e sion o he basic algo i hm PSO, hough i somewha di e s om he s anda d algo i hm by i s cogni i e and social coe icien s ha change du ing sea ch p ocess. The dynamic beha io o hese wo coe icien s allows o c ea e equilib ium be ween explo- a ion and exploi a ion [28] and [29]. The posi ion and he speed o each pa icle a e p esen ed in he ollowing equa ions:    V( + 1) = w∗V( ) + α1 and1∗(Pi−X( ))+ +α1 and2∗(P b −X( )), X( + 1) = X( )+( + 1), (8) c 2018 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING 461 POWER ENGINEERING AND ELECTRICAL ENGINEERING VOLUME: 16 |NUMBER: 4 |2018 |DECEMBER (α1= (c1 −c1i)i e i e max +c1i, α1= (c2 −c2i)i e i e max +c2i,(9) w= (wmax −wmin)∗(i e max −i e min) i e max +wmin,(10) whe e x( )is he ini ial posi ion o he pa icle. ( ) p esen s he ini ial speed o he pa icle. ( +1) is he new speed o he pa icle. x( +1) is he new posi ion o he pa icle. P i is he bes local solu ion. P b is he bes global solu ion. wis he ine ia ac o p esen ed by 0.4≤w≤0.9.i e is he i e a ion numbe . i e max is he maximum i e a ion numbe . α1, α2a e espec i ely he cogni i e and he social ac o s. C1i, C2i, C1 , C2 ep esen s he ini ial and inal alues o he cogni i e and he social ac o s which a e espec i ely 2.5, 0.5, 0.5 and 2.5. The lowcha o he PSOTVAC is shown in Fig. 1. Fig. 1: Flowcha o PSOTVAC. Ini ialize he da a powe sys em and pa ame e s o , PSO-TVAC C ea e a andom ini ial popula ion E alua e he objec i e unc ion 1. Upda e i e a ion coun 2. Upda e eloci y and posi ion E alua e he objec i e unc ion by powe low calcula ion by he new popula ion START Yes Selec he bes solu ion END No Con e gence? Fig. 1: Flowcha o PSOTVAC. 4.2. Fi e ly Algo i hm This algo i hm is inspi ed by and based on he p in- ciple o a ac ion be ween i e lies in na u e, which gi es many simila i ies wi h o he me aheu is ic me h- ods based on g oup collec i e in elligence such as PSO algo i hm. Based on he pseudo code o he FFA shown in Alg. 1, he FA algo i hm is go e ned by he h ee ollowing ules: •All he i e lies a e unisex; hey will mo e owa ds mo e a ac i e and b igh e ones ega dless hei sex. •The deg ee o a ac i eness o a i e ly is p opo - ional o i s b igh ness which dec eases as he dis- ance om he o he i e ly inc eases. •Fi e lies luminosi y is de e mined by an objec i e unc ion (an op imized one). Algo i hm 1 Fi e ly Algo i hm. Ensu e: : Ini ialize popula ion o m i e lies, xi, i= 1,2,3, . . . m. Ensu e: : Compu e Ligh in ensi y (xi), o i= 1,2, . . . m. while s opping c i e ia is no me do o i= 1 o mdo o j= 1 o mdo i ( (xi)> (xj)) hen e u n Mo e i e ly i owa ds j(eq 13) end i end o end o Upda e Ligh in ensi y (xi) o i= 1,2, . . . m. Rank he i e lies and ind he cu en bes end while 1) A ac i eness The a ac i eness unc ion be ween i e lies is ex- p essed by he ollowing equa ion: β( ) = B0exp(γ m),wi h m≥1,(11) whe e is he dis ance be ween any wo i e lies, B0is he ini ial a ac i eness a = 0, and γis an abso p- ion coe icien which con ols he dec ease o he ligh in ensi y. 2) Dis ance Du ing he sea ch p ocess, he dis ance be ween wo i e lies iand ja loca ion xiand xjcan be de ined by he ollowing exp ession: ij =kxi−xjk= u u d X k=1 (xi,k −xj,k)2,(12) c 2018 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING 462 POWER ENGINEERING AND ELECTRICAL ENGINEERING VOLUME: 16 |NUMBER: 4 |2018 |DECEMBER whe e ij is he dis ance be ween wo i e lies and dis he dimension o he p oblem. 3) Mo emen The mo emen o a i e ly iwhich is a ac ed by a mo e a ac i e i e ly jis gi en by he ollowing equa ion: x +1 i=x i+B0exp(−γ 2 ij)∗(xi−xj) +α( and −0.5),(13) whe e he i s e m is he cu en posi ion o a i e- ly, he second e m is used o conside ing a i e ly’s a ac i eness o ligh in ensi y seen by adjacen i e- lies, and he hi d e m is o he andom mo emen o a i e ly in case he e a e no any b igh e ones. i and ja e wo a iables which e lec he ligh in ensi y ha is associa ed wi h a speci ied i ness unc ion o pa icles o be e alua ed [30]. 4.3. BBO Algo i hm BBO is ela i ely a new me aheu is ic me hod in o- duced by (Simon, 2008) [31] and [32]. This me hod is inspi ed by mig a ion o species among islands. The i ness o a geog aphical a ea is assessed by a Habi a Sui abili y Index (HSI). Habi a s which a e mo e sui - able o species o eside a e said o ha e a high HSI. Simila ly, habi a s which a e less sui able o species o eside a e said o ha e low a HIS (Bansal e al., 2016) [33]. In BBO, a solu ion is ep esen ed by an island consis ing o solu ion ea u es named Sui abili y Index Va iables (SIV), which a e ep esen ed by eal num- be s. I is ep esen ed o a p oblem wi h nd decision a iables as: island = [SIV1, SIV2, SIV3, ..., SIVnd].(14) The sui abili y o sus aining la ge numbe o species o an island can be modeled as a i ness measu e e e ed o Sui abili y Index (SI) in BBO as: SI = (island) = (SIV1, SIV2, SIV3, ..., SIVnd).(15) High SI ep esen s a be e quali y solu ion and low SI deno es an in e io solu ion. The aim is o ind op- imal solu ion in e ms o SIV ha maximizes he SI. Each island, ep esen ing a solu ion poin , is cha ac- e ized by i s own immig a ion a e λand emig a ion a e µ. A good solu ion enjoys a highe µand lowe λ and ice- e sa. The immig a ion and emig a ion a es a e he unc ions o he numbe o species in he island as well shown in Fig. 2, and de ined o he k h island as [34]. µk= k n,(16) Fig.3. Species model o an island. Emig a ion μ Species coun Ra e I E Immig a ion λ Smax Fig. 2: Species model o an island. λk=I1−k n,(17) when E=I, he immig a ion and emig a ion a es can be ela ed as: λk+µk=E. (18) 4.4. P oposed Hyb id FA-PSOTVAC In o de o exploi he bes p op ie ies o he wo well known algo i hms, he FA and PSOTVAC, a hyb id me hod is p oposed o imp o e he solu ion o DED. The mechanism sea ch o he s anda d FA is cha ac e - ized by i s possibili y o loca e he bes solu ion bu a high numbe o i e a ion. The PSOTVAC algo i hm is cha ac e ized by i s as con e gence, howe e he so- lu ion achie ed is no compe i i e in pa icula when conside ing la ge DED p oblems. The p oposed FA- PSOTVAC is adap ed and applied o sol e he DED o la ge es sys em conside ing simul aneously he p o- hibi ed zones, he al e poin e ec and amp- a e lim- i s. The lowcha o he p oposed hyb id algo i hm is shown in Fig. 3. c 2018 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING 463 POWER ENGINEERING AND ELECTRICAL ENGINEERING VOLUME: 16 |NUMBER: 4 |2018 |DECEMBER Fig. 4. Flow cha o he p oposed hyb id algo i hm based FA-PSOTVAC. YES S a Read Da a base, i e max, Pop-size, n, K Ini ializa ion o PSO-TVAC , FA Algo i hms pa ame e s Tableau (2) Solu ion de l’algo i hme hyb ide FA-PSOTVAC . Ini ialisa ion des pa amè es FA. S a Fi ly Algo i hm E alua ion O FA Mechanism sea ch I e <= i e max/k S a O PSO-TVAC Gene a ion o he i s popula ion ela ed o he same solu ion o FA. E alua ion o PSO-TVAC Mechanism I e <=I e max- i e max/k NO NO YES S o e he Bes Solu ion o Cos and Pg Vec o End Fig. 3: Flow cha o he p oposed hyb id algo i hm based FA- PSOTVAC. 0 100 200 300 400 500 600 700 800 900 1000 Gene a ion 104 105 106 107 108 109 1010 Bes Cos ($/h) BBO-PSOTVAC BBO FA FA-PSOTVAC 100101102103 3.2 3.25 3.3 3.35 3.4 3.45 3.5 104 Fig. 4: Compa ison o con e gence cha ac e is ics o he p o- posed ou algo i hms o es sys em 1. 0 20 40 60 80 100 120 140 160 180 200 Gene a ion 105 106 107 108 109 1010 Bes Cos ($/h) Fig. 5: Con e gence cha ac e is ic o BBO-PSOTVAC o es sys em 1. 20 40 60 80 100 120 140 160 180 200 Gene a ion 105 106 107 108 109 Bes Cos ($/h) BBO-PSOTVAC FA-PSOTVAC Fig. 6: Con e gence cha ac e is ics o FA-PSOTVAC and BBO-PSOTVAC o es sys em 1. 012345678910 Uni Nembe -10 0 10 20 30 40 50 60 70 80 Ramp Up Ra e Dis ibu ion Fig. 7: Dis ibu ion o Ramp Up iola ion o 50 ials o es sys em 1. c 2018 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING 464 POWER ENGINEERING AND ELECTRICAL ENGINEERING VOLUME: 16 |NUMBER: 4 |2018 |DECEMBER Tab. 1: Bes solu ion o FA o es sys em 1. H Pg1 Pg2 Pg3 Pg4 Pg5 Pg6 Pg7 Pg8 Pg9 Pg10 Cos 1 150.0000 309.5755 73.0000 60.0000 122.8639 122.4782 76.0961 47.0000 20.0000 55 28745 2 150.0000 309.5507 73.0999 60.0000 122.8885 149.3127 93.1691 47.0000 49.9658 55 30391 3 226.5853 309.6643 153.0568 60.0000 123.0489 140.4734 123.1521 47.0000 20.0000 55 33563 4 226.5508 309.5670 185.0731 100.5493 172.7406 159.9138 129.5695 47.0000 20.0159 55 36629 5 226.6134 309.5774 238.8934 120.4456 172.8045 159.9993 129.6077 47.0000 20.0537 55 38338 6 303.2963 309.6235 297.3994 120.4101 222.6227 122.9657 129.6770 47.0000 20.0000 55 41054 7 379.7484 309.5613 296.1865 120.2595 222.2314 122.4294 129.5911 47.0000 20.0000 55 42688 8 379.9163 309.5878 331.9346 120.3543 222.5364 160.0000 129.5783 47.0463 20.0414 55 44682 9 456.5803 309.5402 340.0000 170.3125 235.8224 160.0000 129.6444 47.0412 20.0000 55 48408 10 457.0786 389.4996 339.9167 220.2939 223.6654 159.9001 129.7003 47.0000 49.9592 55 52014 11 456.4682 460.0000 323.9527 241.3500 222.6343 160.0000 129.6314 76.9459 20.0000 55 53694 12 456.5319 460.0000 339.6460 291.1819 222.5952 160.0000 129.6202 85.3100 20.0792 55 55437 13 456.6231 459.9778 297.3991 241.2082 172.7557 154.0301 129.6521 85.3238 20.0016 55 51712 14 456.4936 396.7729 294.7512 191.2087 172.4486 122.5401 129.6818 85.1015 20.0000 55 47899 15 379.8663 396.7837 233.4647 180.8780 172.5976 122.5602 129.5828 85.2840 20.0000 55 44832 16 302.8660 316.7899 185.2285 130.9164 172.7174 155.4687 129.5770 85.4114 20.0461 55 40103 17 226.7340 309.5333 200.4338 120.5870 172.8071 160.0000 129.6007 85.2751 20.0000 55 38265 18 303.2308 309.5587 229.6569 120.5972 222.5535 122.5015 129.5881 115.2475 20.0000 55 41774 19 379.9181 309.4884 297.1465 119.7770 222.6016 122.4900 129.5685 120.0000 20.0000 55 44552 20 456.3717 389.4182 319.4344 169.7228 222.5600 159.8888 129.5937 120.0000 49.9539 55 51945 21 456.7414 309.6307 306.0718 181.1537 222.7390 122.9915 129.6789 120.0000 20.0000 55 48011 22 379.8686 229.6340 267.7906 131.2140 172.7559 122.2274 129.6219 119.8901 20.0000 55 41927 23 302.1687 222.1867 187.9891 81.3181 122.8433 120.9311 99.6632 119.8957 20.0000 55 35496 24 226.0974 222.2598 178.4508 60.0000 122.5708 80.0712 129.5932 89.9053 20.0000 55 32081 To al Cos ($) 1024240 5. Simula ion Resul s In his s udy a compa a i e analysis is elabo a ed o alida e he obus ness o he p oposed hyb id algo i hm in sol ing he DED conside ing se e al p ac ical cons ain s. Fou algo i hms a e in es i- ga ed, FA, BBO, PSOTVAC, FA-PSOTVAC, and BBO-PSOTVAC. Two es powe sys ems a e in- es iga ed o alida e he e icacy o he p oposed algo i hms and in pa icula he hyb id me hod named FA-PSOTVAC. 5.1. Tes Sys em 1 The i s es sys em consis s o 10 uni s, sys em da a is akem om [35] and [17]. The op imized ac i e powe o he mal uni s du ing 24 H is achie ed conside ing al e poin e ec , p ohibi ed zones and amp a e lim- i s. Fo ai compa ison be ween di e en me hods, he popula ion size o all me hods is se o 50. Table 1 and Tab. 2 show he de ails o he op imized ac i e powe o 10 he mal uni s du ing 24 H. The FA achie es he bes solu ion 1024200 $a 500 i e a ions, he co esponding execu ion ime is 41.8955 min, he con e gence cha ac- e is ics a e shown in Fig. 4, he BBO achie es he bes o al cos 1044000 $which is highe han FA, also his algo i hm equi es la ge numbe o i e a ions (1000), a a ela i ely educed execu ion ime (7.1236 min) com- pa ed o FA. 012345678910 Uni Numbe 0 10 20 30 40 50 60 70 80 Ramp Down Ra e Dis ibu ion Fig. 8: Dis ibu ion o Ramp Down iola ion o 50 ials o es sys em 1. Table 3 depic s de ails abou he pe o mances o se e al algo i hms in sol ing DED in e ms o he bes , he mean and he maximum alue. Figu e 4 shows he con e gence beha io o o al cos minimiza ion o a pe iod o 24 h o all p oposed me hods. As well shown in Fig. 5, he hyb id algo i hm named BBO-PSOTVAC allows o achie e a o al cos o 1055000 $a a compe i i e ime (0.9350 min). On he o he side, he p oposed hyb id algo i hm based on combining he FA and PSOTVAC achie es a ema k- able o al cos o 1024163 $a a easonable execu ion ime (8.4934 min), I is also impo an o con i m ha he p oposed algo i hm is ound o be be e han o he s anda d and combined algo i hms in e ms o c 2018 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING 465 POWER ENGINEERING AND ELECTRICAL ENGINEERING VOLUME: 16 |NUMBER: 4 |2018 |DECEMBER Tab. 2: Bes solu ion o FA-PSOTVAC algo i hm o es sys em 1. H Pg1 Pg2 Pg3 Pg4 Pg5 Pg6 Pg7 Pg8 Pg9 Pg10 Cos 2 150.0000 222.2977 119.9927 60.0000 172.7551 133.2690 129.6194 47.0684 20.0000 55 30220 3 226.6829 222.2649 199.8623 60.0000 172.7874 124.6362 129.7631 47.0000 20.0000 55 33106 4 303.2261 222.1884 273.8517 60.0000 172.7244 122.4197 129.5903 47.0000 20.0000 55 36316 5 379.9085 222.2685 297.4001 60.0000 122.8779 145.7868 129.7293 47.0000 20.0276 55 37912 6 379.9504 302.2682 300.9859 60.2773 172.7347 160.0000 129.6819 47.0126 20.0874 55 41268 7 379.9384 309.6140 317.8722 110.1765 172.7422 160.0000 129.6391 47.0000 20.0155 55 43089 8 379.7673 309.5442 339.5965 120.4278 222.6159 122.4767 129.6386 47.0000 49.9280 55 44764 9 456.4904 309.5325 297.2723 170.0277 222.2960 154.8207 129.5225 76.9843 52.0546 55 48421 10 468.5611 309.5358 340.0000 220.0197 223.7258 160.0000 129.8466 85.3120 80.0000 55 52581 11 456.5008 389.5146 339.9296 241.2453 236.7129 160.0000 129.7207 85.3177 52.0588 55 53679 12 456.5777 460.0000 340.0000 258.8141 222.6798 159.9631 129.5956 85.3123 52.0564 55 55608 13 456.4978 396.7024 297.2402 284.6130 222.5896 122.3767 129.6034 85.3192 22.0572 55 51452 14 456.5069 316.7464 300.2911 241.3478 222.6032 126.4383 129.7427 55.3223 20.0000 55 48255 15 456.4983 309.2706 251.9156 191.4026 222.5805 122.5846 99.7451 47.0000 20.0000 55 45245 16 379.8921 309.5309 182.2315 172.1561 172.6454 122.4838 93.0579 47.0000 20.0000 55 39961 17 303.2201 309.5034 180.3546 176.7481 172.6892 122.4267 93.0445 47.0000 20.0100 55 38312 18 303.2842 309.9226 260.3178 181.1381 172.7265 125.5862 123.0291 47.0000 49.9962 55 41938 19 379.8756 309.5352 328.4486 181.1280 172.7430 122.6920 129.5852 76.9929 20.0000 55 44915 20 456.5191 389.5333 328.8065 224.6494 222.5961 160.0000 129.5758 85.3189 20.0000 55 51778 21 379.9298 460.0000 317.9828 180.9005 172.7697 122.4799 129.6152 85.3121 20.0090 55 48296 22 303.2117 396.7645 297.3459 130.9016 122.8333 87.0071 129.6583 85.2567 20.0201 55 41694 23 226.6244 316.7646 251.9903 81.1457 73.1616 122.4624 129.5912 55.2591 20.0000 55 35449 24 150.0000 309.5334 185.1918 60.0000 73.0000 154.6855 129.5901 47.0000 20.0000 55 31501 To al Cos ($) 1024163 Tab. 3: Bes solu ion o FA, BBO, PSOTVAC, FA-PSOTVAC, BBO-PSOTVAC o es sys em 1. Me hod Pop Size Max Bes Wo s Mean Value Min-Max o Balance Time I e a ion Solu ion Solu ion Demande Viola ion (min) FA 50 500 1024200 8161000 1254900 0.0213-0.0658 41.8955 BBO 50 1000 1044000 53566000 4909400 0.0198-0.0162 7.1236 PSOTVAC 50 1000 - - - Viola ion - FA-PSOTVAC 50 200 1024163 13793000 1794400 0.0023-0.0049 8.4934 BBO-PSOTVAC 50 200 1055000 1065600 1060000 0.0222-0.0150 0.9350 speed o con e gence, s anda d de ia ion o gene a ion cos , and compu a ional ime. Figu e 6 shows he con e gence cha ac e is ics o FA-PSOTVAC and BBO-PSOTVAC. Figu e 7 and Fig. 8 show ha he cons ain s ela ed o amp up and amp down a e e i ied. Figu e 9 shows he di ibu ion o he bes cos o 50 ials, his es demons a es he obus ness o he p oposed hyb id me hod named FA-PSOTVAC. 5.2. Tes Sys em 2 In o de o demons a e he e icacy and pe o mances o he p oposed hyb id me hods such as FA-PSOTVAC and BBO-PSOTVAC a la ge scale es sys em is con- side ed. This second es sys em consis s o 30 uni s, sys em da a is akem om [11]. Fo ai compa ison wi h o he me hods ci ed in he li e a u e, only wo cons ain s a e conside ed, he al e poin e ec s and amp a e limi s. The bes o al cos achie ed using he p oposed algo- i hms a e compa ed o a ious me hods ci ed ecen ly in he li e a u e such as E olu iona y P og amming 0 5 10 15 20 25 30 35 40 45 50 Dis ibu ion Cos o 50 T ials 0 2 4 6 8 10 12 14 Bes Cos ($/h) 106 Fig. 9: Dis ibu ion o he bes cos o 50 ials o es sys em 1. (EP) [36], Di e en ial E olu ion (DE) [37], C iss C oss Op imiza ion algo i hm (CSO) [11], Ha mony Sea ch (HS) [38] and a modi ied hyb id EP-SQP app oach (MHEP-SQP) [35], as well depic ed in Tab. 4, i is ound ha by using he p oposed hyb id me hod BBO- PSOTVAC he bes o al cos achie ed is 3105700 $. c 2018 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING 466 POWER ENGINEERING AND ELECTRICAL ENGINEERING VOLUME: 16 |NUMBER: 4 |2018 |DECEMBER Tab. 4: Bes solu ion o FA, BBO, PSOTVAC, BBO-PSOTVAC and FA-BBO o es sys em 2. Me hod Pop Size Max Bes Wo s Min-Max o Balance Time (min) I e a ion Solu ion Solu ion Demande Viola ion FA 50 200 3119200 3153700 0.0982-0.0565 11.769 BBO 50 200 3192700 7297400 0.0121-0.0562 3.2391 PSOTVAC 50 1000 - - Viola ion - BBO-PSOTVAC 50 1000 3105700 3122200 0.0313-0.0360 3.4018 FA-BBO 50 200 3141960 3166200 0.1323-0.0623 20.001 CSO [11] 30 1000 3051260 3054960 - 1.797 EP [37] - - 3164531 - NA NA DE [38] - - 3163000 3173100 NA 0.52 HS [39] - - 3143254 NA NA NA MHEP-SQP [36] - - 3151445 3157738 NA NA I is also impo an o no e ha he ob ained esul s we e achie ed a a compe i i e ime. 6. Conclusion In his s udy, ou algo i hms he FA, PSOTVAC, BBO, FA-PSOTVAC, BBO-PSOTVAC ha e been adap ed and applied o sol e he DED conside ing h ee p ac ical cons ain s simul aneously such as he al e poin e ec , p ohibi ed zones and amp a e lim- i s. The pe o mances o he s anda d algo i hms such as FA and BBO in e ms o solu ion quali y and num- be o gene a ions equi ed ha e been imp o ed by hy- b idiza ion. The main idea in oduced in his s udy is o exploi he bes p ope ies o FA and PSOT- VAC, he BBO and PSOTVAC by c ea ing lexible bal- ance be ween di e si ica ion and in ensi ica ion du ing sea ch p ocess. The pe o mances o he hyb id me h- ods we e alida ed on wo p ac ical es wi h 10 uni s and 30 uni s o sol e he DED conside ing h ee p ac- ical cons ain s. The o al cos achie ed using he hy- b id me hod named FA-PSOTVAC is compe i i e in e ms o solu ion quali y and con e gence cha ac e is- ics. Due o he compe i i e aspec o he p oposed hyb id me hod, au ho s will s i e o de elop an ex- ended hyb id a ian o sol e DED o mode n powe sys em cha ac e ized by he la ge in eg a ion o a ious ypes o enewable sou ces and FACTS de ices. Re e ences [1] WANG, C. and S. M. SHAHIDEHPOUR. 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DOI: 10.1109/MEL- CON.2010.5476272. c 2018 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING 467