OPTIMIZATION OF OPEN CANAL CROSS SECTIONS BY DIFFERENTIAL EVOLUTION ALGORITHM

Transcription

1 Matheatical ad Coputatioal pplicatios, Vol. 16, No. 1, pp , 11. ssociatio fo Scietific Reseach OPTIMIZTION OF OPEN CNL CROSS SECTIONS BY DIFFERENTIL EVOLUTION LGORITHM Mustafa Eka Tua, Mehet li Yudusev Depatet of Civil Egieeig Celal Baa Uivesit, 4514 Muadie, Maisa, Tuke stact- Ope caals ae ipotat wate tasfe stuctues used i wate esouces sstes. s such, the a equie sustatial aout of ivestet depedig o its legth ad coss sectio. Theefoe, coss sectio desig should e caied out o a optiizatio asis. Taditioall, optial sizig of ope caal coss sectios ae udetake oliea optiizatio techiques such as Lagage Multiplies. I this stud, optiu coss sectios of diffeet caal geoeties ae otaied usig diffeetial evolutio algoith ad the fidigs of these execises ae copaed with those of give i elated liteatue. It is oseved that diffeetial evolutio algoith ca e well applicale to the pole ad capale of givig the gloal optia. Ke Wods- Ope caal, diffeetial evolutio algoith, optiizatio. 1. INTRODUCTION Ope caals ae used i wate esouces sstes to tasfe lage quatit of wate fo a ive o aothe souce to whee it is used. The ae essetial eleets of iigatio ad watepowe sstes. The ae fee suface stuctues, which ca wate gavit. ope caal a equie sustatial aout of ivestet depedig o its legth ad coss sectio, akig the optial sizig essetial. Optial sizig is to fid the optial coss sectio diesios at iiu costuctio cost. Noliea optiizatio techiques such as Lagage Multiplies have taditioall ee used to udetake optial sizig of ope caal coss sectios. Seveal eseaches ude diffeet coditios have ee pefoed ito optial coss sectio desig. Majoit of this eseach deals with assessig optial caal sectios of diffeet geoeties fo uifo flow coditios [1], [], [], [4]. Thee ae elativel fewe ue of studies fo this issue cosideig o-uifo flow coditios [5][6]. Swaee et al. () have otaied the paaetes of a optial caal coss sectio ased o the iiu cost of eathwok, which iceases with a icease i excavatio depth [6]. Diffeetial evolutio algoith (DE) is oe of the evolutioa algoiths that ca e used fo a optiizatio pocess. Thee ae seveal DE studies fo diffeet optiizatio poles icludig the idetificatio of stuctual sste paaetes [7], ultiple ojective esevoi opeatio pole [8] ad ass iiizatio tuss pole [9].

2 78 M.E. Tua ad M.. Yudusev I this stud, optiu coss sectios of diffeet caal geoeties ae otaied usig diffeetial evolutio algoith. The diffeet geoeties cosideed iclude tiagula, cicula, ectagula ad tapezoidal caal sectios. The ojective fuctio is i the fo of cost iiizatio. The cost figues take ito accout ae the costs of caal coveig ad excavatio. The fidigs of optiizatio execises caied out diffeetial evolutio algoith ae copaed with those of give i elated liteatue. Based o the fidigs otaied, it ca e stated DE is capale of fidig the optial diesios fo caal coss-sectios.. DEFINITION OF THE PROBLEM The solutio to a optiizatio pole ais to fid gloal optia. I the paticula pole of caal coss sectio pole, the optiizatio pole copises a ojective fuctio i the fo of iiu cost suject to the flow equieets to cove a specific dischage i the caal cosideed. The decisio vaiales, the diesios of caal coss sectios, ae side slope, otto width, flow depth, ad adius. Geoetic popeties of a geeic caal coss sectio ae give i Figue 1. η d ψ d η a Figue 1. Geeic Tpe of Caal Sectio Give the tpe of caal liig ad slope, the cost fuctio fo a lied caal ca e witte as follows: C P L E a d whee L : uit liig cost i TL/L, E : uit excavatio cost i TL/L, : additioal cost of excavatio pe uit depth i TL/L 4, a: flow aea at height η, P: wetted peiete, : flow aea ad L: legth. Such a total cost expessio fo a caal assues that the caal is costucted fo the sae soil coditios ad cost of excavatio iceases with the icease i excavatio depth. ssuig the uifo flow coditios appl i the caal, Maig s uifo flow equatio is used to defie uifo flow as follows: (1)

3 Optiizatio of Ope Caal Coss Sectios Diffeetial Evolutio lgoith 79 1 / 1/ Q R S () whee : Maig oughess coefficiet, : flow aea, R: hdaulic adius ad S: otto slope of caal The optiizatio pole fo a caal coss sectio is as follows: iiize C P L suject to E 1 / 1/ a d Q R S (4) Seveal diesioless paaetes ca e defied to exaie the effects of the vaiales ad to copae with the esults give i elated liteatue usig the legth scale / 8 1/ Q S The diesioless paaetes defied ae as follows: C C / E L L / E (7) / / P P / / / E / (1) Usig these diesioless paaetes, the optiizatio poles stated i () ad (4) ca e ewitte as follows: iiize () (5) (6) (8) (9) (1) (11) (1) C P L suject to 1 5 / / P a d (14) (15)

4 8 M.E. Tua ad M.. Yudusev The optiizatio pole give i (14) ad (15) is aaged fo tiagula, ectagula, tapezoidal ad cicula caal coss sectios, of which geoetic diesios ae give i Figue,, 4 ad 5 as follows: Optiizatio pole fo tiagula coss sectio: 1 1 Figue. Geoetic Diesios fo Tiagula Coss Sectio iiize C L suject to 1 5/ 1 / 1 (16) (17) Optiizatio pole fo ectagula coss sectio: Figue. Geoetic Diesios fo Rectagula Coss Sectio iiize C L (18)

5 Optiizatio of Ope Caal Coss Sectios Diffeetial Evolutio lgoith 81 suject to 1 / 5 / (19) Optiizatio pole fo tapezoidal coss sectio: Figue 4. Geoetic Diesios fo Tapezoidal Coss Sectio iiize 1 L C () suject to 1 1 / 5 / (1) Optiizatio pole fo cicula coss sectio: Figue 5. Geoetic Diesios fo Cicula Coss Sectio 1 1

6 M.E. Tua ad M.. Yudusev 8 iiize acsi acsi C L acsi () suject to acsi acsi 1 / 5 / (). THE DIFFERENTIL EVOLUTION LGORITHM (DE) The diffeetial evolutio algoith (DE), which was poposed Sto ad Pice [1], is a evolutioa optiizatio algoith like geetic algoiths. DE executes populatio asis. The idividuals costitutig populatio geeate ew populatios though utatio, cossove ad selectio opeatos. Each idividual i the populatio is a vecto of D diesio. The diesio of the vecto, D, is the sae ue as the ue of vaiales i optiizatio pole [11]. DE eaches to optia the followig steps; 1. Iitializatio Lowe ( ) ad uppe ( ) ouds of each vaiales ae deteied. The iitial value of the j th paaete of the i th vecto is calculated uifol distiuted via (4). Mutatio The solutio poit that the vecto epesets oves i solutio space this opeato. This pocess ca e ealized adol selected vectos. The utatio vecto ca e otaied ; (5) F, a positive eel ue, is called scale facto.. Cossove tial vecto is otaied though the cossove of two vectos, utat vecto ad taget vecto, as follows:

7 Optiizatio of Ope Caal Coss Sectios Diffeetial Evolutio lgoith 8 C is called cossove poailit ad takes the values etwee ad 1. (6) 4. Selectio The idividuals to fo the ew geeatio ae selected at this stage. Each tial vecto otaied though cossove is copaed with the taget vecto. The vecto with the lowest ojective fuctio value suvives ito the ext geeatio g+1. This pocess is expessed as follows: (7) 5. Teiatio Citeia Sice the pocess is iteative, the teiatio citeia ae oall equied to stop the pocess. Fo this pupose, the sufficietl sall diffeece, ael eo, etwee the values otaied o a specific ue of iteatio ae used [1]. 4. PPLICTIONS The optiizatio poles set foth i Sectio fo fou diffeet shapes of caal coss sectios have ee solved diffeetial evolutio algoith ad the sectio vaiales otaied. Duig the opeatio of DE, soe use-defied paaetes ae equied. These paaetes iclude the ue of idividuals i populatio, NP, scale ate, F, ad cossove poailit C. These paaetes should e selected i such a wa that the solutio pocess is speeded up. I this stud, followig values wee used: Np=1, F=.85 ad C=.5. The esults otaied i this stud DE have ee copaed with those give i kso ad Sakaa fo the sae sectio tpes. Fist of all, classical optiu coss sectio pole whee excavatio cost does ot chage with excavatio depth, e.g. =, was solved. This pole was solved i Liteatue Lagage Multiplies (LM) ad the esults of such solutio ae take ito accout fo copaig the esults of DE as show i Tale (1) ad (). Tale 1. The esults of LM ad DE fo optiu odiesioal sectio vaiales fo = Sectio Vaiales Tiagula sectio Rectagula sectio LM DE LM DE Side slope, Botto width, Flow depth, ,

8 84 M.E. Tua ad M.. Yudusev Sectio Vaiales Tapezoidal sectio Cicula sectio LM DE LM DE Side slope, Botto width, Flow depth, Radius, s a secod execise, the pole is solved DE fo the diffeet values of ad ad sectio vaiales otaied. kso ad Sakaa solved the sae pole a ueical optiizatio (NO) techique. Fo tiagula coss sectio, the esults of oth DE ad NO have ee peseted i Figue 6 ad 7 as the esults fo othe tpes ae ot taceale i kso ad Sakaa. (a) () Figue 6. Vaiatio of (a) optiu odiesioal side slope ad () oal depth of a tiagula sectio with = 1. L

9 Optiizatio of Ope Caal Coss Sectios Diffeetial Evolutio lgoith 85 (a) () Figue 7. Vaiatio of (a) optiu odiesioal side slope ad () oal depth of a tiagula sectio with =.5 6. CONCLUSION The classical pole of optial caal coss sectio desig has ee evisited. The pole is solved Diffeetial Evolutioa lgoith fo the diffeet tpes of sectio tpes icludig cicula tiagula, ectagula ad tapezoidal shapes. The pole is cosideed the sae as that foulated i elated liteatue so as to ake oe-to-oe copaisos to see the capailit of DE esults of DE have cofied the esults otaied i the liteatue. Theefoe, it has ee cocluded that DE ca successfull e applied to the pole ad also is capale of satisfacto esults.

10 86 M.E. Tua ad M.. Yudusev 7. REFERENCES 1. Chow, V.T Ope-chael hdaulics. McGaw-Hill Book Copa, New Yok.. Baaea-Koopaei, K., Valetie, E. M., ad Swailes, D. C.. Optial desig of paaolic-ottoed tiagle caals. Joual of Iigatio ad Daiage Egieeig. 16(6), Baaea-Koopaei, K. 1. Diesioless cuves fo oal depth calculatios i caal sectios. Joual of Iigatio ad Daiage Egieeig. 17(6), Chaha, B.R. 7. Optial Desig of a special Class of Cuviliea Bottoed Chael Sectio. Joual Hdaulic Egieeig, SCE. 1(5), Swaee, P.K Optial iigatio caal sectios. Joual of Iigatio ad Daiage Egieeig. 11, Swaee, P.K., Misha, G.C., ad Chaha, B.R.. Miiu cost desig of lied caal sectios. Joual of Wate Resouces Maageet. 14, Tag H, Xue S ad Fa C. 8. Diffeetial evolutio stateg fo stuctual sste idetificatio. Coputes ad Stuctues. 86, 1-8. Redd, M.J. ad D.N. Kua, 7. Multiojective diffeetial evolutio with applicatio to esevoi sste optiizatio. J. Coput. Civil Eg. 1, Hull P. V., Tike M. L., Dozie G. 6. Evolutioa optiizatio of a geoeticall efied tuss. Stuctual ad Multidisciplia Optiizatio. 1, Sto R, Pice K. (1995). Diffeetial evolutio a siple ad efficiet adaptive schee fo gloal optiizatio ove cotiuous spaces. Techical Rep. No. TR-95-1, Iteatioal Copute Sciece Istitute, Bekle (C). 11. Sto R., Pice K., Lapie J.. 5. Diffeetial Evolutio: Pactical ppoach to Gloal Optiizatio. Gea: Spige-Velag Beli Heideleg. 1. li, M. M., Tö,., 4. Populatio Set-Based Gloal Optiizatio lgoiths: Soe Modificatios ad Nueical Studies. Copute & Opeatios Reseach. 1, kso, B., ad lta-sakaa,. B. 6. Optial lied chael desig. Caadia. J. Civ. Eg., (5),