DESIGN OF AN IRRIGATION WATER DELIVERY SYSTEM UNDER THE RESTRICTION OF CANAL CAPACITY

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1 DESIGN OF AN IRRIGATION WATER DELIVERY SYSTEM UNDER THE RESTRICTION OF CANAL CAPACITY Dmtra Alexou 1, Constantnos Tsouros 2 1.Department of Spatal Plannng and Development Engneerng, A.U.Th., Greece 2. Department of Mathematcal, Physcal and Computer Scences, A.U.Th., Greece ABSTRACT The water supply to an rrgaton system s drectly related to agrcultural productvty. An mportant factor that nfluences the effectveness of an rrgaton system s the desgn of the canal network that meets the water requrement at dsparate locatons. The amount of water conveyed down a canal s ts capacty. Canal capacty n dverse areas depends on varous factors, e.g., land structure, topography and constructon cost. Here a canal water dstrbuton system s determned, whch receves an unlmted quantty of water from a man source, and dstrbutes the maxmum possble amount of water to dscrete locatons at the mnmum possble cost havng taken nto account the canal capacty restrctons and the correspondng canal settlement cost. The problem s dealt wth n the context of graph theory and a correspondng algorthm has been developed. KEYWORDS: Irrgaton system, Bottleneck, Graph theory # 1,! "! 2 1.,,... 2.,,..!" #! "#" $ " "$" % %%. &" % ' "! # "" "! "! # # % " "! (". " ) )" $ "% $!.! # "! $! % %! %!!, %, "" ". *! "! ' $!! "#"! # ) " " # %! $ " "% $ (" $% "! " # ) + "! ( " " ". ) ' " " ( % #"" " % (. $ : #"!",,! $% ", %. 359

2 Τομέας Υδραυλικής και Τεχνικής Περιβάλλοντος, Τμήμα Πολιτικών Μηχανικών, Πολυτεχνική Σχολή Α.Π.Θ. 1. INTRODUCTION. The water supply to an rrgaton system s drectly related to agrcultural productvty. An mportant factor that nfluences the effectveness of an rrgaton system s the desgn of the canal network that meets the water requrement at dsparate locatons. The amount of water conveyed down a canal s ts capacty. Canal capacty n dverse areas depends on varous factors, e.g., land structure, topography [2] and constructon cost. Here a canal water dstrbuton system s determned, whch receves an unlmted quantty of water from a man source [4], and dstrbutes the maxmum possble amount of water to dscrete locatons at the mnmum possble cost, havng taken nto account canal capacty restrctons and the correspondng canal settlement cost. The problem s dealt wth n the context of graph theory, namely We consder a weghted connected drected graph G = ( V,E,C) where the nodes of V = { v 0, v 1,v 2,...,v n } represent the water demand locatons and the couples of E = { (v,v j),v,v j V) express the canddate canals to be settled. The elements of C = { c j } are the capacty of canal (v,v j) E. 2. BOTTLENECK PROCEDURE. In order to desgn a network water rrgaton system that comples wth the stated problem, we start by applyng a procedure so as to detect the maxmum amount of water that can be conveyed to each demand locaton by only one canal path, that s the bottleneck problem [1, 3, 5]. We remnd that the maxmum flow that a path P j can convey from a node v to a node v j s the mnmum capacty of an arc n P j. Thus, the bottleneck problem conssts of determnng the maxmum flow from v to v j by a path P j. We apply procedure B usng a labelng method, [1] on graph G as formulated above. In procedure B: A label L( v ) s assocated to every node v V. a label L( v ) s a numercal number whch at a certan nstant of the procedure s characterzed as temporary or permanent. A permanent label L( v ) ndcates the maxmum flow that can be conveyed by one canal path from source v 0 to locaton v. Set Q contans the nodes of V, the labels of whch are temporary at a current nstant of B, the algorthm termnates when the labels of all nodes n V become permanent, namely when Q =, varable p denotes the most recent node, the label of whch became permanent. We say that two nodes v, v j V are neghbors f (v,v j) E. Procedure B Set Q V - v0 v 1, Set L( v 0 ), p v 0. For each node v Q Set L (v) 0. Do whle Q s not empty For each neghbor v of p, v {V Q} Set L ( v ) max { L ( v ), mn { L ( p ), end for L(x) max{ L ( y ), y { V Q }}. c p v } } 360

3 Τιμητικός τόμος για τον ομότιμο καθηγητή Δημήτρη Τολίκα Set p x and Q Q x. end do In order to facltate the presentaton n the followng we also express the nodes v by the natural number. The demand locatons and the canddate canals to be settled are represented here by the n nodes graph G = ( V,E,C) of fgure 1 whch s smlar to the Latn square model. The capactes c j C of arc ( v, v j)e were produced n the range [ 10, 20 ] wth the use of a unform random generator functon. The square number n bold talc assocated to each node v s the value L( v ) obtaned after the applcaton of procedure B / / / Let P0k v0, y1, y2,..., yk1, vk Fgure 1. be a path that leads to value L( v k ). We say that P 0k s a bottleneck path from v 0 to v k, e.g., value L (14) = 15 s reached by 3 alternatve bottleneck paths, namely, P 0, 14 = { }, P 0, 14 ={ }, P 0, 14 ={ }. Table 1 shows the number of alternatve bottleneck paths produced to the nodes of the network of fgure

4 Τομέας Υδραυλικής και Τεχνικής Περιβάλλοντος, Τμήμα Πολιτικών Μηχανικών, Πολυτεχνική Σχολή Α.Π.Θ. Table 1 Clearly, numerous rrgaton desgn systems can be generated due to the exstence of alternatve bottleneck paths. The objectve here s to determne the correspondng system at the mnmum possble cost. Wthout loss of generalty, we consder that the constructon cost of each canal s numercally the same as ts capacty. 3. GENERATION OF THE FAMILY OF BOTTLENECK PATHS. A bottleneck path P 0k = { v 0 = 1, 2,, k = T} from source s to locaton T must verfy the condton ( L( j ) 7 L( T ) and c j 1, j 7 L( T ) for all j = k, k-1, 1. ( 1 ) The followng procedure BP s a depth frst search method that generates n a backward manner all the paths from a source s to a gven node T. Frstly, the explanaton of the notatons used n BP are gven. GM: A two dmenson array for whch every row contans n an ordered way the ng() }, where predecessors of node, that s GM(, j) = { v, v 1 2,, v ng( ) ( v, j), ( v 1 2, j ),, ( v ng( ),j ) E. Number nm() expresses the poston of a node n row of GM, e.g., nm() = 3 means the. thrd node n GM, namely node GM(, nm() ) = v 3 Procedure BP 8 T, s 8 0, k8 1, P (k) 8 T Do nm( ) 8 nm() Branch 9 f nm( ) : ng () then 8 GM (,nm() ), k 8 k + 1, P ( k ) 8 f = s then Wrte P( j ), j = 1,2,,k 9 a path s to T 9 Else Cycle "goto loop" end f " = s" end f nm( ) 8 0, k 8 k 1 9 Backtrack 9 loop f k < 0 then end 8 P ( k ) 362

5 Τιμητικός τόμος για τον ομότιμο καθηγητή Δημήτρη Τολίκα 4. BOTTLENECK TREE. We developed the subsequent procedure TBP that proceeds for every water demand locaton vv, = 1,2,,n, from the closest to the farthest nodes from the source, so as to select among alternatve bottleneck paths the most sutable one that wll be contaned n the fnal soluton. The famly of the derved bottleneck paths forms a tree that we call bottleneck tree. TBP s derved from BP where the necessary actons have been ncorporated so to frstly produce the alternatve bottleneck paths S to T (satsfyng condton (1) ) and successvely select a path as follows. In a specfc nstance durng the executon of TBP let P p1 p v 2 pq v,,.., P ( p1 p2 ), ( p2 p3 ),.., ( pq1 v ) be an alternatve path where! > 1 and or v,,, B = ( p p p p j, j, ), ( j, j, ),... are the arcs already assgned to a path at a prevous stage. We form set Q B P and v P Q, by cost( ) we symbolze the total v cost of the arcs n. Clearly contans the new unassgned arcs to be added n therefore, the selected P, v q P s the one that corresponds to the mnmum cost of the arcs among v, namely, Z q = mn {cost ( ) }. (): The set of successor nodes of n network G. n: number of locatons n the network nalt: the number of alternatve paths. np(): number of nodes n the th alternatve path. Durng the executon bool (, j) = 1 or 0 respectvely, dependng on whether arc (, j) has been assgned or not as an arc n the bottleneck tree. The cost of the arcs orgnatng from the source are not calculated n the fnal cost. Statements: 2 to 7and 25 to 32 are smlar to procedure BP. 9 to 13 check for the satsfacton of condton (1). 17 to 23 store the alternatve paths and fnd the correspondng cost of the added arcs among alternatve paths. 33 to 36 detect the mnmum cost of the added arcs The bottleneck tree gven by TBP s shown n Fgure 2 wth bold lnes, the correspondng cost of whch s 384 monetary unts. The total cost of all canddate canals (wth the excepton of the ones that orgnate from the source) s 674, namely, 56.9% of the total cost. The method wth whch the frst drectly generated bottleneck path s consdered, wthout takng nto account the alternatve paths, wll result n a cost of 441 unts, namely, 65.4%. Ths means, we have a gan of 8.5% unts accordng to the total cost of the canddate canals. 363

6 Τομέας Υδραυλικής και Τεχνικής Περιβάλλοντος, Τμήμα Πολιτικών Μηχανικών, Πολυτεχνική Σχολή Α.Π.Θ. Procedure TBP Set Bool (, j) 8 1, for =0,1,2,,n, j = 0,1,2,,n. Set Bool (, (j ) ) 8 0, =1,2,,n, j = 1,2,,; () ; "ntal condton " 1. Do T = 1 (1) n 2. 8 T, s 8 0, k 8 1, pk8 T, mn 8 1E+6, nalt 8 0, Sum Do " loop 1" 4. nm( ) 8 nm( ) f nm( ): ng () then 6. 8 GM (,nm() ), k 8 k + 1, pk8 7. f = s then 8. lg = true 9. Do Whle lg = true 10. Do j = k - 1 (-1) z18 p j 1, z test C 7 L(T) and L( p z 1, z j 1) 7 L(T) If test = false then lg 8 false 14. end do " j " 15. end do " whle lg " 16. If lg = true then 17. nalt 8 nalt + 1, np(nalt ) 8 k 18. Do j = k -1 (-1) z18 p j 1, z28 p j p j, Bpath(nalt,j) 8 p j 20. f bool ( z 1, z 2 ) = 0 then 21. Sum (nalt) 8 Sum (nalt) + C z 1, z end f 23. end do " j " 24. end f " lg " 25. nm( ) 8 0, k 8 k f k < 0 then Ext " from loop 1 " 27. = p k 28. Else 29. Cycle "goto loop 1" 30. end f " = S " 31. end f " nm( ): ng () " 32. loop "1" 33. Do = 1 (1) nalt 34. If Sum() < mn then 35. mn 8 Sum(), keep end do " " 37. Set bool (z 1,z 2 ) 8 1 for every arc n Bpath 38. Wrte Bpath (keep), = 1,2,, np(keep) 39. end loop " T " 40. end. 364

7 Τιμητικός τόμος για τον ομότιμο καθηγητή Δημήτρη Τολίκα / / / Fgure 2 5. CONCLUSION. The method presented n the precedng paragraphs s a heurstc procedure that can n a feasble executon tme solve stuatons concernng large networks. Dverse components are nvolved n real-lfe rrgaton systems, such as the water flow velocty [6], the amount of water requred at certan locatons, etc. These factors may n certan cases be ncorporated n procedure TBP. REFERENCES 1. Alexou D., Tsouros K.(2008), "Elements of Graph Theory and Networks, (n Greek), Unversty notes", School of Technology, Department of Spatal Plannng and Development Engneerng, Arstotelan Unversty of Thessalonk. 2. Lynn E. Johnson (2008), "Gs for water-supply and Irrgaton systems", Geographc Informaton Systems n water resources engneerng, p ,ebook, CRC press Renje et all (2012), "Identfy the Bottleneck of Water Network by usng Graph Theory", Advanced Materal Research Vols (2012),pp , Trans Tech Publcatons, Swtzerland,