Development of a fuzzy logic based software for automation of a single pool irrigation canal

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1 Developen of a fuzzy logc based sofware for auoaon of a sngle pool rrgaon canal Dr. R. Gopakuar, Reena Nar, Vnuraj R. 2, Sony Davs 3, Bjeesh V. 4, Junl Jacob 5 Governen Engneerng College, Sreekrshnapura, Palakkad, Kerala , Inda. E-al: gopak2@yahoo.co.n Absrac A fuzzy logc based sofware for auoaon of a sngle pool rrgaon canal s presened. Purpose of he sofware s o conrol downsrea dscharge and waer level of he canal, by adjusng dscharge release fro he upsrea end and upsrea gae sengs. The sofware s developed on a fuzzy conrol algorh proposed by he frs auhor durng hs docoral research work and publshed n leraure. Deals of he algorh are gven. The algorh was orgnally developed usng fuzzy logc ool box of MATLAB, whch s propreary sofware no avalable freely and hence canno be adoped for general use. Presen sudy descrbes developen of a canal auoaon sofware based on hs algorh usng open source ools, whch are freely avalable. The sofware s ransparen and nuve, whch can be easly appled by feld engneers. The effor requred n unng he fuzzy odel has been reduced by ncludng an opzaon echnque. Also, a new procedure has been nroduced for fuzzy nference based on he Madan Iplcaon ehod. The sofware s esed by applyng o waer level conrol proble n a canal wh a sngle pool, as repored n leraure, and sasfacory resuls are obaned. Keywords- Irrgaon canals; canal auoaon sofware; open source ools; Fuzzy syses I. INTRODUCTION Irrgaon s he larges user of waer, whch coes ore han 8 % of world waer consupon. Hence s essenal ha rrgaon canals are o be operaed o her axu effcency by nzng wasage of waer and axzng flexbly n operaons. Also, any rrgaon canals are under dlapdaed condons and her rehablaon can be econocally acheved by operang he effecvely (and no by rebuldng he as orgnally desgned, whch wll be hghly expensve). Effecve operaon of he canals n accordance wh hese prncples can be acheved hrough canal auoaon, whch eans applcaon of auoac devses or logc o asss n he operaon of he canal []. In he presen conex, canal auoaon ples developen and applcaon of a fuzzy logc based sofware o conrol he downsrea dscharge and waer level of a sngle pool canal (Fg. ). Here he er pool ples he srech of he canal beween wo check gaes. The canal s under known seady flow condon nally wh waer level a he downsrea Reservor Upsrea gae Canal bed Pool offake Reservor Fgure. Scheac dagra of he sngle-pool canal end a s specfed arge value. Perurbaons n he syse are due o changes n he dscharges hrough he offake, whch s locaed a he downsrea end of he pool (Fg. ). Under he effec of hese perurbaons, he waer level a he downsrea end sars devang fro s arge value. Purpose of he fuzzy conrol sofware s o deerne he nflow dscharges and he gae sengs a he upsrea end, a every regulaon e sep, so ha sable conrol a he downsrea end s acheved n a nu of e. Dfferen ypes of canal auoaon algorhs have been proposed n he pas as descrbed n [2]. Soe of he are developed on conrol syses heory. Exaples are classcal (e.g., Proporonal plus Inegral, PI), predcve (e.g., Model Predcve Conrol, MPC) and opal (e.g., Lnear Quadrac Regulaor, LQR) conrollers. Dffcules assocaed wh applcaon of hese conrollers have been dscussed n leraure. Tunng of PI conroller, o deerne he conroller paraeers, s a edous ask [3]. Also, boh LQR and MPC assue ha he syse can be descrbed by lnear dfferenal equaons, where as he underlyng unseady flow equaons are non-lnear. The opzaon nvolved n MPC s a dffcul ask and consue sgnfcan aoun of copung power [4]. Also, for LQR conrollers, he relaonshp beween he easured waer levels and he conrol acons s aheacally coplex and hence he conroller appears as a black box [5]. Such black box ype conrollers are less lkely o be acceped by he canal operaors han ones ha are nuve [6]. There s anoher class of canal auoaon algorhs, whch ake use of nverson of he followng dynac wave equaons (San-Venan equaons) [7], as he desgn echnque: Connuy equaon A Q + = x () Inernaonal Journal of Copuer & Councaon Technology (IJCCT), ISSN (PRINT): , Volue-3, Issue-4, 24 38

2 Moenu equaon Q + 2 Q h + ga gas x A x + gas f = where A s flow cross seconal area, h s flow deph, Q s dscharge, S s bed slope, S f s frcon slope, g s acceleraon due o gravy, s e varable and x s space varable. Gae Srokng [8] and CLIS [9] are exaples for he algorhs cong under hs class. The Gae Srokng s a feed forward conrol algorh, suable for scheduled offake dscharge changes, where he offake dscharge changes are known a pror. I s based on he ehod of characerscs, whch s ore coplex. The nheren dffcules and praccaly of Gae Srokng have been deonsraed n leraure [, ]. Fne dfference alernaves for he Gae Srokng ehod were also proposed [2, 3]. These algorhs nvolve nverson of he governng equaons () and (2) n boh e and space. Bu he nforaon fro nal condons and ransen boundary condons degrades wh e due o frcon and hence backward copuaon n e can cause nsably [4]. However, he proble of nsably s no appled o nverse copuaons n space and a feed back conrol algorh of hs ype, suable for unscheduled offake dscharge changes, was repored laer [5]. CLIS [9] s a odfed verson of hs algorh, ade suable for scheduled flow changes also. For all hese algorhs, he recovery e, whch s he e perod n whch he conrolled varables are expeced o aan her arge values, s aken as he upsrea wave ravelng e, whch s uch ore han he physcally correc downsrea wave ravelng e. Addonally, CLIS akes use of lnearzed verson of he San-Venan equaons for he nverse odelng where as he underlyng syse dynacs are nonlnear. Reference [6] descrbes developen of a fuzzy logc based dynac wave odel nverson algorh for canal regulaon, whch was capable of overcong he aforeenoned drawbacks of exsng algorhs. Tunng of he algorh was easy and copuaons were easy o pleen. No lnearzaons of he governng equaons were nvolved and he recovery e was rghly aken as he downsrea wave ravelng e. The fuzzy conrol algorh was ransparen and nuve and can be expeced o be ore accepable for canal operaors as copared o black box ype conrollers. However, he algorh was developed usng he fuzzy logc ool box of MATLAB, whch s propreary sofware no avalable freely and hence canno be adoped for general use. Presen sudy descrbes developen of a canal auoaon sofware based on hs algorh, for a sngle pool canal, where n freely avalable coponens are used. II. THEORETICAL ASPECTS (2) funcons and fuzzy rules can be found n leraure [8, 9]. Fuzzy rule based odels are suable for nonlnear npuoupu appng [8]. In fuzzy rule based odelng, a gven npu daa undergoes he processes of fuzzfcaon, applcaon of fuzzy operaors, plcaon, aggregaon and defuzzfcaon, before fnally forng no oupu fro he odel [2]. Varous seps nvolved n he desgn of fuzzy logc conrollers have been dscussed n leraure [8]. As enoned prevously, presen sudy descrbes developen of a canal auoaon sofware based on he fuzzy conrol algorh proposed n [6]. The fuzzy algorh nvolves replacng (2) by a fuzzy rule based odel whle reanng () n s coplee for. The fuzzy rule based odel has been developed on fuzzfcaon of a new aheacal odel for dynac wave velocy, obaned as: ( Dsconnuy n graden of Q ) ( Dsconnuy n graden of A) ( ΔQ) ( ΔA) ( Vw ) = = (3) where, ( V ) w s he wave velocy a pon along he canal, a e. Dsconnuy n graden of a flow paraeer (Q or A), a pon and a e, has been deerned as he dfference beween s graden correspondng o he fnal seady condon (whch he flow wll aan under he effec of he perurbaon) and s graden correspondng o he condon a e, boh gradens beng evaluaed a pon. The fuzzy rule based odel s developed on fuzzfcaon of (3). Whle copung he dynac wave velocy (V w ) a any node (,j+) n he copuaonal doan (Fg. 2), he correspondng values of A and Q are deerned approxaely by adopng he followng dscrezaon: ΔA ΔQ j + = j + = F.S F.S j+ j+ [ A + - A ] -[ A + - A ] (4) F.S F.S j+ j+ [ Q - Q ] -[ Q - Q ] (5) + + In (4) and (5), he noaons wh superscrp F.S ndcae he values of he flow paraeers correspondng o he fnal seady sae. Equaon () has been dscrezed a he node (,j), n he x- copuaonal doan (Fg. 2), usng he Vaslev schee [2], whch s an uncondonally sable plc fne dfference schee, as follows: j+ j j+ j+ [ A A ] + [ Q Q ] + - = Δ 2Δx where = node nuber n he x drecon and j = node nuber n he drecon. (6) The concep of fuzzy logc was frs nroduced by Zadeh [7]. Deals of fuzzy ses, fuzzy nubers, ebershp Inernaonal Journal of Copuer & Councaon Technology (IJCCT), ISSN (PRINT): , Volue-3, Issue-4, 24 39

3 Te ( ) -,,j+,,j+,,j Δ +,,j+ 2 N- Dsance (x) Fgure 2. Copuaonal doan for he fuzzy conrol algorh: N = nuber of nodes n he x drecon, = node nuber n he x drecon, j = node nuber n he drecon. Here, s assued ha he conrol s acheved by rapd oveen of check gaes, whch wll resul n he foraon of gravy waves. If he wave velocy (V w ) correspondng o a ceran nflow dscharge s assued o rean consan over he enre pool under consderaon hen a se of crsp rules can be wren, based on (3) and applcable everywhere n he pool, as IF Q s low THEN A s low, IF Q s edu THEN A s edu, IF Q s hgh THEN A s hgh ec. where low, edu, hgh ec. are quanave varables and Q and A ndcae crsp nuercal values. If he above assupon on V w holds good only approxaely hen he rules becoe fuzzy, low, edu, hgh ec. becoe lngusc varables and Q and A ndcae fuzzy subses. The fuzzy rule based odel s developed on hs prncple. A ajor advanage of he above fuzzy forulaon s ha s developed on fuzzfcaon of a aheacal odel (3) relang he npu and oupu varables. Hence he rules lnkng hese varables can be accuraely specfed. Such a fuzzy odel works n he bes way and also he resuls are sgnfcanly beer han he case where he odel s developed usng saple daa [22]. The error, assocaed wh he assupon on V w reanng approxaely consan n a pool, reduces as he wave aplude reduces because as he aplude reduces he wave resebles ore closely a lnear gravy wave whch propagaes downsrea wh nu dsoron, aenuaon and change n velocy [23, 24, 25]. Hence can be expeced ha he algorhc error wll be a nu and consequenly sable conrol a he downsrea end wll be acheved early n hose cases where he fnal seady profle confor o unfor flow. Such a case s very dffcul o conrol by oher exsng algorhs [26]. Sofware n he presen conex has been developed for hs parcular case. III. DEVELOPMENT OF THE SOFTWARE The sofware consss of hree odules; he frs one for unng, he second one for fuzzy nference and he hrd one for copung he requred nflow rae, a he upsrea end, Δx N usng he frs and second odules. The Graphcal User Inerface of he sofware s developed usng Q and he copuaonal odules are developed usng 'C' language. For he applcaons specfed n he algorh, only one o one correspondence s used. Therefore he fuzzy lbrares and ool boxes gven n [27, 28, 29] are no requred as hey are desgned for a wde varey of applcaons. A. Tunng Tunng of he fuzzy conrol algorh nvolves deernaon of he ebershp funcon paraeers (core, suppor and boundary) of each fuzzy subse and consrucon of he fuzzy rule base. I s done by consderng a fcous case where he flow correspondng o axu desgn dscharge n he canal s unfor a he upsrea end. For a base flow correspondng o he axu desgn dscharge and for varous nflow dscharges, respecve values of Q and A are obaned by solvng (3) along wh he basc equaon for gravy wave velocy [3] gven below, whch has been derved fro oenu equaon: V ga = V2 + (A Y A2 Y ) (7) A (A A ) w In (7), he subscrp ndcaes he upsrea end fro where he wave s produced and he subscrp 2 ndcaes a neghborng secon where he effec of he wave has no ye reached, V s flow velocy and Y s deph of cenrod of he secon below he free surface. The soluon procedure s as follows. For a ceran nflow dscharge, Q = Q Q 2 s a known value. Also, as flow s unfor a he upsrea end, A can be replaced by A -A 2 n (3). Thus n (3) and (7) he only unknowns are A and Y. Bu Y s a funcon of A. So for a ral value of A correspondng value of Y s deerned and appled n (7) o deerne V w. Sae value of V w should be obaned when he above value of A s used n (3). If no, he above procedure s repeaed wh oher ral values of A unl V w obaned fro he wo equaons are equal. An opzaon echnque has been adoped o reduce he nuber of rals requred n unng. I nvolves coparng he dfference beween V w values, resulng fro (3) and (7), durng successve rals and he copuaon wll proceed n ha drecon where he above dfference wll keep reducng. Ths s connued unl he dfference les whn soe pered olerance. Once he correc value of A s obaned, he value of A= A -A 2 s calculaed. Thus values of Q and A are obaned for he specfed nflow dscharge. Ths procedure s repeaed for oher nflow dscharges also and respecve values of Q and A are deerned. These values are sored n wo arrays. Each value n he npu array ( Q) has a correspondng value n he oupu array ( A) also, boh havng degree of ebershp equal o one (core values of he fuzzy ses). Then by adopng he ehod of paronng [8], a se of rangular ebershp funcons and fuzzy rules are derved relang Q and A. Inernaonal Journal of Copuer & Councaon Technology (IJCCT), ISSN (PRINT): , Volue-3, Issue-4, 24 4

4 B. Fuzzy Inference The sofware nroduces a new procedure o copue he oupu ( A) for a gven npu ( Q), based on he Madan Iplcaon ehod of nference [3] and he Cenrod Mehod of defuzzfcaon [8], as descrbed below. Suppose x n s he gven npu daa ( Q), wh and 2 as he correspondng degrees of ebershp n he neghborng ebershp funcons shown n Fg. 3. Snce hese ebershp funcons have been developed by ehod of paronng, hey wll for sosceles rangles. µ(x ) Fgure 3. Fuzzfcaon of npu daa The value of x 2 neares o x n s found fro he array of Q. The values x, x 3 and x 4 are also obaned fro he sae array. Then he values of and 2 are copued as follows: anθ = / (x 2 - x ). = (x 3 -x n ) anθ. 2 = (x n -x 2 ) anθ There s one o one correspondence beween he npu and oupu fuzzy ses sored n he wo arrays. Hence and 2 can be easly apped o he correspondng oupu ebershp funcons, as shown n Fg. 4. µ(y ) 2 2 x (a, b ) θ (a 2, b 2 ) x 2 x 3 x 4 x n θ θ 2 Fgure 4. Copung he oupu ebershp funcon by plcaon. The values of a 2, a 3, a 4 and a 5 shall be obaned as follows: Calculaon of a 2 an θ = /(y 2 a ). Suppose o = a 2 a. Then, an θ = b 2 /o. Bu b 2 =. Therefore, o = /an θ. Hence, a 2 = a + o. Calculaon of a 3 an θ 2 = /(y 3 y 2 ). Suppose o 2 = y 3 a 3. Then, an θ 2 = b 3 /o 2.. Bu b 3 = Therefore, o 2 = /an θ 2. Hence, a 3 = y 3 o 2. Calculaon of a 4 Suppose o 3 = y 3 a 4. Then, an θ 2 = b 4 /o 3. θ (a 3, b 3 ) (a 4, b 4 ) (a 5, b 5 ) y 2 y 3 (a 6, b 6 ) θ ΔQ ΔA Bu b 4 = 2 Therefore, o 3 = 2 /an θ 2. Hence, a 4 = y 3 o 3 Calculaon of a 5 s slar o ha of a 4 The above calculaons are for he case when > 2. Calculaons for he case where < 2 s slar. For he resulng polygon wh coordnaes as shown n he Fg. 5, he x-coordnae of he cenrod, ΔA, whch s he requred oupu, s calculaed usng followng equaons [32]. 6 P + + = = / 2 ( a b ) ( a b ) (8) 6 ΔA = /(6P ) ( a + a (9) = + ) ( ab+ a+ b ) Ths odule perfors only aheacal calculaons. The copuaonal cos s ndependen of any characerscs of he canal and hence he e and space coplexy s consan. C. Seps n Copuaon. Recovery e (Δ) s obaned as he downsrea wave ravelng e. 2. The canal reach s dscrezed no (N-) subreaches usng N nodes, as shown n Fg. 2; node beng a he upsrea end and node N a he downsrea end. 3. Copuaon sars a he las node N, assung ha Q and h a hs node wll be equal o her arge values a he end of he e sep (Δ). 4. Q and h a he nodes N-, N-2,2, are obaned successvely, by applcaon of (6) and he fuzzy rule based odel. Noe ha A s a funcon of h dependng on geoery of he canal. 5. The upsrea gae s adjused o release he dscharge copued a node for he recovery e perod (Δ). 6. If he flow deph (h) a node N a he end of Δ devae fro s arge value hen repea he above seps for he nex Δ. 7. Oherwse sop. The copuaon odule perfors a se of operaons for each node n he pool. Hence he e coplexy s lnear wh he nuber of nodes n he pool. Bu he nuber of nodes s no a paraeer ha akes very hgh values. Therefore he copuaonal e of hs odule wll no vary uch wh he nuber of nodes. IV. SOFTWARE TESTING The sofware s esed by coparng s oupus wh ha obaned usng MATLAB fuzzy logc ool box [6]. The sofware has been appled o waer level conrol proble n a fcous canal wh a sngle pool [5]. Scheac dagra of he canal s shown n Fg. 5. The canal lengh (L) s 5, boo wdh 5. and sde slope.. Is boo slope s Inernaonal Journal of Copuer & Councaon Technology (IJCCT), ISSN (PRINT): , Volue-3, Issue-4, 24 4

5 .3. Wdh of upsrea and downsrea recangular gaes s 5. wh dscharge coeffcen.75. There s a consan head reservor a upsrea end and anoher one a downsrea end. Ther respecve waer levels are 6.3 and 2.8. Also boo elevaon of he pool a he upsrea end s.5 and a he downsrea end s.. An offake canal s provded jus upsrea of he downsrea gae. Targe waer deph a downsrea end of he canal s 3.46, whch corresponds o noral deph a a peak flow rae of 4 3 /s. Fgure 5. Scheac dagra of he sngle-pool canal used for esng he canal auoaon sofware. For a base flow correspondng o he peak flow rae of 4 3 /s, and for varous nflow dscharges rangng fro 35 3 /s o 45 3 /s, correspondng values of Q and A are obaned and ebershp funcons are developed by followng he procedure explaned under he secon Tunng. They are shown n Fg. 6. In he fgure L ples low values and H ples hgh values. Hgher he coeffcen of L and H hgher wll be her gradaon owards low and hgh values respecvely. The fuzzy rules, connecng he aneceden Q and he consequen A, are derved n he for IF Q s a THEN A s a where a ake values n ascendng order fro 9L o 9H (Fg. 6). Hence a oal of egheen rules are ncluded n he fuzzy rule base. (a) Mebershp degree Mebershp degree.5 9L 8L 7L 6L 5L 4L 3L 2L L H 2H 3H 4H 5H 6H 7H 8H 9H Q ( 3 /s) (b) Flow Bed slope =.3 5 Offake L 8L 7L 6L 5L 4L 3L 2L L H 2H 3H 4H 5H 6H 7H 8H 9H A ( 2 ) Fgure 6. Mebershp funcons of he fuzzy odel for he sngle-pool canal: (a) npu ebershp funcons, (b) oupu ebershp funcons. Inally he flow s seady n he canal wh a dscharge of 2 3 /s. There s no flow n he offake canal and he enre flow s dvered o he downsrea reservor. Flow n he offake canal s hen ncreased fro o 2 3 /s n 2 nues. Wh such a dsurbance, he response of he syse s suded. The copuaonal nodes are placed a a dsance nerval (Δx) of each. The recovery e (Δ) s aken as downsrea wave ravelng e of 797 s [5]. A every recovery e sep, he upsrea gae openng s adjused, n accordance wh he gae equaon gven n [5] so as o anan he consan gae dscharge oupu fro he fuzzy conrol algorh. The sulaon resuls nclude varaon of he upsrea gae openng wh e (Fg. 7), developen of waer surface profle along he canal wh e (Fg. 8), varaon of he dscharge wh e a boh upsrea and downsrea ends (Fg. 9) and varaon of he change n waer level wh e a boh hese locaons (Fg. ). These resuls are coparable o hose obaned usng MATLAB fuzzy logc ool box [6] and hence can be concluded ha he sofware s gvng accurae oupus. Gae openng () Waer surface elevaon () Te (nues) Fgure 7. Varaon of he upsrea gae openng of he sngle-pool canal wh e durng conrol Targe profle 45 s 25 s Canal bed 2 s Dsance () Fgure 8. Developen of waer surface profles wh e n he sngle-pool canal durng conrol Inernaonal Journal of Copuer & Councaon Technology (IJCCT), ISSN (PRINT): , Volue-3, Issue-4, 24 42

6 Change n waer level () Dscharge ( 3 /s) Upsrea end Downsrea end Te (nues) Fgure 9. Varaon of dscharge wh e, a upsrea and downsrea ends of he sngle-pool canal durng conrol Upsrea end Downsrea end Te (nues) Fgure. Varaon of change n waer level wh e, a upsrea and downsrea ends of he sngle-pool canal durng conrol V. CONCLUSIONS A sofware for auoaon of a sngle-pool canal, based on a fuzzy conrol algorh, s presened n hs sudy. The fuzzy algorh was orgnally developed usng MATLAB fuzzy logc ool box, whch s no freely avalable. Thus he orgnal work was unsuable for coon feld applcaons. Ths proble has been overcoe n he presen sudy usng free and open source packages. Addonally, exsng canal conrol sofware are based on eher conrol syses heory or dynac wave odel nverson, boh requre srong heorecal background for applcaon. Ths dffculy does no arse for he presen sofware. I s ransparen and nuve and can be easly appled by feld engneers. Nuber of rals requred for unng he fuzzy odel has been effecvely reduced by ncludng an opzaon echnque. Also, a new procedure has been nroduced for fuzzy nference based on he Madan Iplcaon ehod. The echncal deals of copuaon are absraced fro he end user and herefore he sofware s exreely user frendly. As he fron end was desgned usng Q and he copuaonal odules uses 'C' language, only a nu nuber of freely avalable packages are requred o nsall he sofware. The sofware has been esed by applyng o a canal conrol proble repored n leraure. The resuls obaned are coparable o hose obaned usng MATLAB fuzzy logc ool box, whch proves ha he sofware s gvng accurae oupus. ACKNOWLEDGMENT The fundng provded by CERD (Cenre for Engneerng Research and Developen), Kerala, n carryng ou hs research s graefully acknowledged. REFERENCES [] A.J. Cleens. Edoral, Journal of Irrgaon and Dranage Engneerng, ASCE, vol. 24(), 998, pp. -2. [2] P-O. Malaerre, D.C. Rogers and J. Schuurans. Classfcaon of canal conrol algorhs, Journal of Irrgaon and Dranage Engneerng, ASCE, vol. 24(), 998, pp. 3-. [3] B.T. Wahln and A.J. Cleens. Perforance of hsorc downsrea canal conrol algorhs on ASCE Tes Canal, Journal of Irrgaon and Dranage Engneerng, ASCE, vol. 28(6), 22, pp [4] B.T. Wahln. Perforance of odel predcve conrol on ASCE Tes Canal, Journal of Irrgaon and Dranage Engneerng, ASCE, vol. 3(3), 24, pp [5] A.J. Cleens and J. Schuurans. Sple opal downsrea feedback canal conrollers: heory, Journal of Irrgaon and Dranage Engneerng, ASCE, vol. 3(), 24, pp [6] V.M. Ruz-Carona, A.J. Cleens and J. Schuurans. Canal conrol algorh forulaons, Journal of Irrgaon and Dranage Engneerng, ASCE, vol. 24(), 998, pp [7] V.T. Chow, D.R. Maden and L.W.Mays. Appled Hydrology, McGraw-Hll Inernaonal Edons [8] E.B. Wyle. Conrol of ransen free-surface flow, Journal of Hydraulc Dvson, ASCE, vol. 95(HY), 969, pp [9] F. Lu, J. Feyen, P-O. Malaerre, J-P Baue and P. Kosuh. Developen and evaluaon of canal auoaon algorh CLIS, Journal of Irrgaon and Dranage Engneerng, ASCE, vol. 24(), 998, pp [] E. Bausa, T.S. Srelkoff, and A.J. Cleens. General characerscs of soluons o he open-channel flow, feed forward conrol proble, Journal of Irrgaon and Dranage Engneerng, ASCE, vol. 29(2), 23, pp [] E. Bausa and A.J. Cleens. Volue copensaon ehod for roung rrgaon canal deand changes, Journal of Irrgaon and Dranage Engneerng, ASCE, vol. 3(6), 25, pp [2] G.Chevereau. Conrbuon a l éude de la régulaon dans les sysèes hydraulques a surface lbre, Ph.D. hess, Insu Naonal Polyechnque de Grenoble, Grenoble, France. 99. [3] F. Lu, J. Feyen and J. Berlaon. Copuaon ehod for regulang unseady flow n open channels, Journal of Irrgaon and Dranage Engneerng, ASCE, vol. 8(), 992, pp Inernaonal Journal of Copuer & Councaon Technology (IJCCT), ISSN (PRINT): , Volue-3, Issue-4, 24 43

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