A Simulation Based Optimal Control System For Water Resources

Transcription

1 Cy Unversy of New York (CUNY) CUNY Academc Works Inernaonal Conference on Hydronformacs 8--4 A Smulaon Based Opmal Conrol Sysem For Waer Resources Aser acasa Maro Morales-Hernández Plar Brufau Plar García-Navarro Follow hs and addonal works a: hp://academcworks.cuny.edu/cc_conf_hc Par of he Waer Resource Managemen Commons Recommended Caon acasa, Aser; Morales-Hernández, Maro; Brufau, Plar; and García-Navarro, Plar, "A Smulaon Based Opmal Conrol Sysem For Waer Resources" (4). CUNY Academc Works. hp://academcworks.cuny.edu/cc_conf_hc/34 hs Presenaon s brough o you for free and open access by CUNY Academc Works. I has been acceped for ncluson n Inernaonal Conference on Hydronformacs by an auhorzed admnsraor of CUNY Academc Works. For more nformaon, please conac AcademcWorks@cuny.edu.

2 h Inernaonal Conference on Hydronformacs HIC 4, New York Cy, USA A SIMUAION BASED OPIMA CONRO SYSEM FOR WAER RESOURCES ASIER ACASA (), MARIO MORAES-HERNÁNDEZ (), PIAR BRUFAU (), PIAR GARCÍA- NAVARRO () (): IFEC-CSIC, Unversy Zaragoza, C/ Mara de una 3, Zaragoza, Span Nowadays, he ncreasng demand of waer resources has become one of he mos relevan ssues when waer supply mus be guaraneed as n he case of rrgaon comunes. I becomes dffcul o provde he predcve opmal behavor of regulaon srucures when he regme of he flow s changng n me and, moreover, when hs regulaon modfes he hydraulc condons of he srucure. One of he mos wdely used soluon for conrol s PID. he dsadvanages of PID conrol are he dffcules n he presence of non-lneares as well as he PID parameers unng. akng no accoun prevous works abou sluce gae formulaon and s hydraulcs, an equvalen formulaon s gong o be appled o oban he conrol sgnal for he gae. hs wll make possble o buld a sysem ha allows he opmal conrol of he gae n order o oban, for nsance, a desred volume of waer n he leas me possble wh he mnmum waer loss. In hs work, a reformulaon of he problem s analyzed and hen, an equvalen he conrol s appled o sluce gae flow o oban a smulaon based conrol sysem and a near real-me soluon s explored usng adjon varable formulaon. INRODUCION he dea of regulaon appled o PDE's s he use of he mahemacal model as he sysem whch delvers he resuls of he physcal varables whch are used o apply he regulaon. In oher words, he mahemacal model s used as he represenaon of he realy when he conrol s appled. Fnally, he man queson s he nverse, ha s how o deermne he conrol o opmze a gven funconal. PDE-consraned opmzaon and he adjon mehod for solvng he opmal conrol problem appears o be an neresng opon when he complexy of he problem does no allow o oban he easy way o know he opmal soluon. Followng he Algorhm () and usng he echnque provded n hs work, s possble o fnd he opmal conrol. In he fuure hs conrol wll be ransformed n a gae openng sgnal. he man deaas has been obaned from he work [] and [] usng he numercal resoluon for he physcal model descrbed n [3].

3 ONE DIMENSIONA SCAAR PROBEM WIH SOURCE ERMS In order o esablsh he bass n a smple example, he echnque s nroduced by s applcaon o he D scalar equaon wh consan velocy and source erm denfed such as a subracon or njecon of some quany. I s modelled as C( x, ) C( x, ) A f ( x, ) C(, ) C( x,), x () beng A a consan propagaon velocy. he role of f ( x, ) for our conrol wll be he puncual locaon of he njecon n order o produce some profle n $$ of he conrollable quany. hs means ha, f ( x, ) wll be placed a x x (, ) beng n any oher locaon n order o sasfy some condon n anoher locaon x he funconal ha covers he quadrac error may be wren, such as ( x, ; h, q) dxd ( (, ) ( )) C x Cobj d J () he conrol appled n () consders a physcal locaon over he enre me doman. Adon Formulaon In order o esablsh some relaon beween he error and he conrollable source f ( xs, ) s necessary o mulply () by ( x, ) and hen, negrae over x(, ) and (, ), akng no accoun ha sasfes s

4 C C Q A f dxd x (3) he new varable ( x, ) s also known as a agrange Mulpler and s connuous and dfferenable (once a leas). Inegrang (3) by pars and akng he frs varaon wh respec o C and f follows ha Q [ C ] dx C dxd [ A C] d A C dxd fdxd x (4) I s also possble o ake he frs varaon of J n order o oban J as J ( C C ) obj Cdxd (5) Addng boh (4) and (5) o oban for C and placng he followng conrans on JJ Q applyng boundary and nal condons C and C(, ), C(, ), C( x,), (, ), (, ), ( x, ) (6) he varaon of he objecve funcon can be rewren as J C( A ( C C )) obj dxd fdxd x (7) hen, he graden of J can be expressed as J J ( x, ) (8) f ( x, ) If A ( C C obj ) x (9) For our purposes and akng no accoun he puncual source locaon n x s he regulaon wll be appled by means of he perurbaon n he value of f ( xs, ) usng he dscree verson of () for every mes (, ) such J J( xs, ) ( xs, ) () f ( x, ) s

5 Dscrezaon he numercal resoluon of equaon () s performed usng an upwnd scheme for he spaal negraon and Forward Euler for he me negraon, makng possble o formulae he updaed value a cell and me sep C n, n C ( C C ) C A f f A n n n ( ) n x ( x) n n n ( C C ) ( ) C A f f A x he same scheme can be appled n order o solve he adjon varable as () n bu akng care wh he forward negraon by means of choosng negave me lenghs and updang from he me n o n n n n ( ) A( f) f A x ( x) ( ) A( f) f A x n n n n () Beng necessary o sasfy he CF condon for he elecon of he me sep c A x (3) Applcaon In order o show he applcably, he usng of () nsde Algorhm () wh he prevous schemes are gong o appled o he nex case. he condons for he physcal problem are,, x., CF, x.45, x.65 (4) Where C ( x, ) s defned as obj s f.5 (.5) f.5.6 Cobj ( x, ) f.6.7 (.7 ) f.7.8 f.8 (5) Usng he nex parameers for he graden mehod ol max 7, Ier, 6. (6) max

6 I s possble o oban exponenal convergence o he soluon, reachng he olerance n 37 eraons. I s dsplayed n Fgure () Fgure Convergence o he objecve funcon. Evoluon of C( x, ) (op), ( xs, ) he graden and convergence (boom, lef) and J (boom, rgh) ONE DIMENSIONA SHAOW WAER EQUAIONS used n he D Shallow Waer Equaons derve from he deph-averaged equaons of mass conservaon and of momenum. hey form a x hyperbolc sysem of equaons: U ( x, ) d ( x, ) F U H ( x, U ) (7) dx Where (8) In hs sudy, we wll consder consan secon of wdh $B=$ n order o make easer he analyss. hs allow o rewre he erms I (/ ) Bh and I Moreover, he objecve funcon s nroduced n a general form as J ( x, ; h, q) dxd (9)

7 hs funcon ncludes he aspec of he flow o be characerzed or regulaed. Adjon Formulaon Nex, he formulaon of he adjon equaon s obaned by means of mulplyng he adjon varable ( x, ) by he connuy equaon and, he momenum equaon by he adjon varable ( x, ).he sum of hese wo producs s negraed n me and space. h q q q gh f Q gh( S S ) dxd () x x h hs expresson can be negraed by pars as before. hen, varaons are esmaed by akng ncremens respec o h and q. he frs varaon of he funconal becomes Q Q( h, q) () Incremens are also aken n he funconal J, leadng o ( J h q ) dxd h q () Addng boh erms and groupng () and (), follows ha whenever he followng s sasfed q S f gh g ( S S f ) gh h x h h S q f gh x h x q q (3) hen J can be rewren as q q J [ h q] dx [ q ( q ( gh))] d (4) h h ( a) ( b) he expresson of (4) esablshes he relaon beween he error and he quanes ha may be regulaed (h.q). hs wll allow us o evaluae he graden whch wll be nroduced n he opmzaon mehod (Algorhm ). Dscrezaon Boh he shallow waer sysem and (3) are hyperbolc. hey nclude a Jacoban marx n he formulaon and share he same se of egenvalues and egenvecors. he upwnd fne volume mehod appled o boh sysems reles on he sgn of he egenvalues. An upwnd Remann

8 solver [4] for he locally lnearzed problem s formulaed for boh problems. For a gven cell wh edges -/ and +/ dfferences he updang follows he scheme [5]: U n n m m U ( ) / ( ) / x e e m m (5) Where and are he egenvalues and e are he egenvecors. he coeffcens, defned as: n m / m / Are dfferen for every sysem of equaons. he common defnon of (5) usually ncludes he evaluaon of he nex value n+ as a funcon of he prevous one n. In hs case, he negraon s beng backward n me and hen, he expresson s reversed. Obvously, n he Adjon problem he condons are usually esablshed n formng a Fnal Value Problem. Applcaon he adjon varables are very useful o fnd he graden of J o changes n nflow or ouflow. When conrollng one of hese parameers, he value of J may be nroduced n a Ierave mehod o fnd he soluon, (he opmal conrollng parameers). In hs case, he developmen has been made o conrol he nflow. he frs sep s o choose nal condons for boh SWE and Adjon Equaons, (6) ( x, ), ( x, ), x h( x,), q( x,), x akng no accoun (7), J of (4) can be smplfed as (7) J (, ) q(, ) d (8) Consderng (8) sensvy of J, perurbaons n q can be appled be means of he dscree verson of he graden J (, ) q(, ) (9) he case was proposed n [6] where here s a channel whou frcon nor bed slope wh consan wdh B=m = m and nal deph h.m and nal flow q m s he me doman s defned as (, ) wh =4 s. and he nle.67 /. boundary condon s q q h. he funconal s orened o o ( ) sec (.3( )) regulae he waer deph, consderng h ( x, ).5m for (, ) wh x.5. he evoluon of he waer deph as well as ( x, ) s dsplayed n fgure () obj

9 Fgure Evoluon of waer deph (op) and he adjon varable ( x, ) for dfferen mes In hs case, ( x, ) evaluaed n x= only provdes nformaon relave o he sensvy of J respec o q bu he evaluaon of (, ) can be used o regulae he nle condon forbddng such dscharges ha may produce overflow n he channel. ACKNOWEDGMENS hs research s funded by he Mnsry of Economy and Compeveness of Span under he Research Projec BIA-39-C-. REFERENCES [] Murllo J., García-Navarro P. Weak soluons for paral dfferenal equaons wh source erms: Applcaon o he shallow waer equaons, J. Compu. Phys. 9. () pp [] BF Sanders, ND Kaopodes. Adjon sensvy analyss for shallow-waer wave conrol Journal of Engneerng Mechancs 6 (9), [3] BF Sanders, ND Kaopodes. Conrol of canal flow by adjon sensvy mehod Journal of rrgaon and dranage engneerng 5 (5), [4] Randall J. eveque Numercal Mehods for Conservaon aws ecures n Mahemacs, EH-Zurch [5] Morales-Hernández, M., García-Navarro, P., Burguee, J., Brufau, P., A conservave sraegy o couple D and D models for shallow waer flow smulaon, Compuers & Fluds, 8, pp.6-44, 3 [6] Sanders B.F. Conrol of shallow-waer flow usng he adjon sensvy mehod PH.D hess. Dep of Cv. And Envr. Engrg. Unversy of Mchgan, Ann Arbor, Mch.