Dynamic optimal groundwater management considering fixed and operation costs for an unconfined aquifer

Transcription

1 Dynamc optmal groundwater management consderng fxed and operaton costs for an unconfned aqufer MING- SHENG. YEH, YU-WEN. CHEN 2, CHIN-TSAI. HSIAO 3, LIANG-CHENG. CHANG 5. Graduate Student of Department of Cvl Engneerng, Natonal Chao Tung Unversty, Hsnchu, Tawan 2. Graduate Student of Department of Cvl Engneerng, Natonal Chao Tung Unversty, Hsnchu, Tawan 3. Assstant Professor, Department of Informaton Management, Chung Chou Insttute of Technology, Changhwa, Tawan 4. Professor, Department of Cvl Engneerng, Natonal Chao Tung Unversty, Hsnchu, Tawan Abstract The smulaton and plannng of unconfned aqufers are much more complex than those of confned aqufers because the former are nonlnear. However, unconfned aqufers are mportant n groundwater management because they are the upper aqufers of a groundwater system. Computng an optmal strategy for the management of a groundwater system s a nonlnear, dscrete and dynamc optmzaton problem f the total costs nclude both fxed and operatng. The nonlnearty s caused by the objectve functon and the smulaton of the groundwater flow of an unconfned aqufer. The computatonal dffculty makes conventonal groundwater management models unable to solve the problem wthout smplfcaton. Therefore, ths nvestgaton proposes a groundwater management model that can determne the optmal network and pumpng rates of the pumpng wells by smultaneously consderng fxed and operatng costs. The proposed model nvolves a novel hybrd algorthm that combnes the Genetc Algorthm (GA) and Constraned Dfferental Dynamc Programmng (CDDP), a knd of the optmal control theorem. The man part of the hybrd algorthm s the GA, n whch each chromosome represents a potental network desgn of the pumpng wells. The CDDP s used to compute the optmal pumpng rates and operatng costs assocated wth each pumpng network. A two dmensonal fnte element model s embedded n the CDDP to smulate the groundwater flow. Hypothetcal cases are consdered to demonstrate the computatonal capablty. The results of the smulaton reveal that the proposed algorthm can solve the complex problem usng affordable computaton resources. Comparng the management strategy obtaned by the proposed algorthm wth that computed usng an algorthm that consders only the operatng cost ndcates that the proposed algorthm sgnfcantly reduces the costs of mplementng management strategy. Accordngly, the proposed model can be employed to obtan a cost-effectve strategy, ncludng network desgn and pumpng polcy, for managng a groundwater system.

2 Key Words: Dynamc Optmal Control, Genetc Algorthm, Groundwater Management, Unconfned Aqufer. Introducton Groundwater s an mportant resource, so numerous scholars have nvestgated the topc of groundwater management to ncrease the effcency of groundwater use. The goal s to meet demand by pumpng groundwater usng the most economc desgn (wth optmal pumpng well locatons and pumpng rates). Ensurng the sustanablty the use of groundwater has been extensvely studed (Gorelck, 983[]; Yeh, W. W-G, 992[2]; Ln and Yang, 99[3]; Pezeshk et al., 994[4]; Takahash and Peralta, 995[5]. Varous optmzaton technques have been employed n the plannng stages of groundwater management, ncludng lnear programmng (Aquado and Remson, 974 [6]; Molz and Bell, 977 [7]), nonlnear programmng (Murtagh and Saunders, 982 [8]; Ahlfeld et al. 988 a,b [9][0]; Gorelck et al. 984 []), mxed-nteger programmng (Rosenwald and Green, 974 [2]), genetc algorthms (GAs) (McKnney and Ln, 994 [3]; Wang and Zheng, 998 [4]) and dfferental dynamc programmng (DDP) (Jones et al., 987 [5]). Of these methods, DDP markedly reduces the dffcultes of dmensonalty assocated wth nonlnear dynamc groundwater management problems (Jones et al. 987 [5], Chang et al. 992 [6]; Culver and Shoemaker 992 [7]). Unconfned aqufers are mportant n groundwater management as they are the upper aqufers n a groundwater system and groundwater can be pumped from the layers at low cost usng a short well and small pumpng lft. Therefore, unconfned aqufers are always the most exploted aqufer layers. Not only the property of tme-varyng pumpng rates but also the smulaton of the unconfned aqufer affects the dffculty of solvng the problem of groundwater management. In the optmal desgn of the groundwater management problem for unconfned aqufers, the governng equaton of groundwater flow s the necessary system response equaton. Therefore, the smulaton and plannng of unconfned aqufers are much more complex than those of confned aqufers, because the former are nonlnear. In prevous nvestgatons, the nonlnearty of the unconfned aqufers have generally been smplfed, and the nexacttude of the desgn used for ground water management has ncreased. In past studes of appled DDP, only Jones et al. (987) [5] and Mansfeld and Shoemaker (999) [8] addressed the problem of groundwater management for an unconfned aqufer. Solvng the remedaton problem of unconfned-aqufer, Mansfeld and Shoemaker (999) [8] appled the nonlnear optmzaton control algorthm (SALQR) to determne the tme-varyng pumpng rate of each well. However, they neglected dscrete varables, such as the fxed cost of well nstallaton. The calculaton lmts and optmal methods used n earler studes of groundwater 2

3 management desgn for unconfned aqufers have prevented the smultaneous consderaton of fxed cost and operaton costs. The authors are unaware of any nvestgaton that smultaneously consders the fxed costs of nstallng the wells and the operatng costs of tme-vary pumpng rates for an unconfned aqufer. Effectvely managng unconfned aqufers has rarely been addressed. Based on the above dscusson, n desgnng the optmal groundwater pumpng system, the decson varables are classfed nto two types - dscontnuous varables, lke the number of pumpng wells and the locatons of the pumpng wells, and contnuous varables lke the pumpng rates of each well n each tme step. In groundwater management, the total cost covers well nstallaton (fxed costs) and pumpage (operatng costs). The fxed cost functon s dscontnuous, so fxed costs are commonly neglected n the applcaton of gradent-based optmzaton algorthms. Chang et al. 992 [6] appled an optmal control method, called the Successve Approxmaton Lnear Quadratc Regulator (SALQR), to desgn a pumpng system for the remedaton of contamnated aqufers. Culver and Shoemaker (993)[9] used a quas-newtonan method wth SALQR, called Quas Newtonan Dfferental Dynamc Programmng, (QNDDP), to mprove the effcency of convergence of the optmal control method. Mansfeld et al. (998)[20] exploted the sparseness of the structure to reduce the huge computatonal requrement of the dynamc optmal control model (SALQR). All of these studes offered pumpng polces for a tme-varyng stuaton. However, they dd not explore the dscrete varables, such as the numbers and locatons of the well system nstallaton. Determnng the numbers and locatons of pumpng wells and the tme-varyng pumpng rates of each well are essental to groundwater management. However, conventonal groundwater management models cannot smultaneously consder dscontnuous and contnuous varables and so cannot consder both fxed and operatng costs. Rogers and Dowla (994)[2] used an artfcal neural network (ANN) and a genetc algorthm (GA) to determne the number of optmal wells and the locatons of stes n the groundwater remedaton problem. However, n ther work, the pumpng rates had to be determned before the artfcal neural network (ANN) was traned. McKnney and Ln (995)[22] appled mxed-nteger nonlnear programmng (MINLP) and nonlnear programmng (NLP) to consder smultaneously both dscrete varables and contnuous varables n the remedaton problems. However, n ther nvestgaton, the pumpng rates of the wells remaned constant throughout the perod, even as the plume kept movng away. Zheng and Wang (999) [23] ntegrated the tabu search method and lnear programmng to desgn the remedaton system. However, the pumpng rates of the wells dd not vary wth the locaton of the pollutant plume. Wang and Zheng (998) [4] successfully assocated the genetc algorthm and smulated annealng wth MODFLOW to optmze the well ste and the tme-varyng 3

4 pumpng rate to solve the groundwater management problem. However, n dynamc problems, the number of decson varables ncreases wth the operatng perod, and the computatonal requrements ncrease, too. In ther study, the maxmum number of operatng perods s four and the optmal pumpng polcy does not satsfy the requrements of the dynamc problem. Consequently, n the most general a complex groundwater management problem, fxed cost and operaton costs of an unconfned aqufer must be consdered. Hsao and Chang (2002a)[24] appled GA and CDDP to solve the groundwater supply problem for a confned aqufer, and appled the model to solve the remedaton problem (Chang and Hsao, 2002b[25]). They smultaneously consdered both the fxed costs of well nstallaton and the operatng costs assocated wth the tme-varyng pumpng rate. However, n ther nvestgatons, the model could not solve the groundwater management problem or the remedaton problem for an unconfned aqufer. Ths study develops a novel management model that combnes GA and CDDP to optmze groundwater basn development and management. The groundwater flow numerc model, ISOQUA, was modfed for an unconfned aqufer by the Pcard teraton method. The proposed model solves a groundwater supply problem for an unconfned aqufer that smultaneously consders both the fxed costs of well nstallaton and the operatng costs of tme-varyng pumpng by explotng the advantages of both methods, GA and CDDP, and embeddng the modfed groundwater flow numerc model. 2. Groundwater flow numeral model of an unconfned aqufer Ths nvestgaton addresses a two-dmensonal unconfned aqufer groundwater system. Its governng equaton s as follows. h h K xx h + K yy h + x x y y where j Ω ( x, y ) h u jδ j j = S...() y t h, groundwater head K xx, prncpal components of conductvty algned along the x axs; K yy, prncpal components of conductvty algned along the y axs; Ω, set of pumpng wells n the aqufer; S y, storage yeld ; u j, pumpng rate at ( x j, y j ) ; δ x j, y ), delta functon evaluated at x j, y ). ( j ( j Numercal modelng of ths study was based on Eq. (), and derved by modfyng the ISOQUA for the confned aqufer (Pnder, 978[26]). Applyng Galerkn s fnte element method and mplct fnte dfferent method yelded a system of nonlnear equatons from Eq. 4

5 (): [ A( h )] where, [ B ] [ B ] h t h F L t + + t t + = t + h + t...(2) t u [ A ], the global conductance matrx of the smulaton regon, combnes the conductance matrces of all elements. Each conductance matrx of each element s expressed as, ω ω x x ω ω y y e j j [ A( ht )] = j K xx h K yy h dxdy +...(3) + [ B] s the global capactance matrx of the smulaton area. As a rule, each capactance matrx for each element s expressed as, e [ ] j ( j e B = S ω ω dxdy)...(4) F, the boundary condton n the form of a vector. The boundary vector of each element s gven by { F} e e nd ω j ω j ω fa dxdy ω K xx h nx K yy h ny hj dl j x y Γ...(5) = = Equaton (2) can be rewrtten as E ( h ) h = E h + L u F t + t+ 2 t h t +...(6) Equaton (6) s nonlnear, and s the system-transform functon for optmal groundwater management. In Eqs. (2) and (6), A( h ) t+, [B], E ( h t + ) and E2 are L h s the poston of the pumpng well, and are F s an n-dmensonal vector. n n matrces. n n matrces. n s the number of nodes, excludng the Drchlet Boundary Condtons node. m s the number of pumpng wells. ω s the base functon of node. ω j s the base functon of node j. n v x n v y f a s the normal vector n the x drecton. s the normal vector n the y drecton. s the flux through the boundary. ht s the ground water head at tme step t, and s an n x vector. ut s the pumpng rate at tme step t, and s an m x vector. A further descrpton of each coeffcent matrx n Eqs. (2) to (6) can be found n Chang et al. (992)[6]. Whereas the thckness of a confned aqufer remans constant n tme, that of an 5 e

6 unconfned aqufer vares wth the groundwater table. Accordngly, ths nvestgaton modfes the ISOQUA (Pnder, 978[26]) by the Pcard method. ISOQUA was orgnally developed for a confned aqufer. The modfcaton focuses on updatng the thckness of the aqufer accordng to the groundwater table, as follows. Step: At the begnnng of each tme step, the aqufer thckness, Thck (L), of each element s assumed to be the mean head (water table) of the element n the prevous tme step. Step2: The aqufer s assumed to be a confned aqufer wth thckness Thck (L); then the head (water table) of each element of the aqufer s computed. The updated thckness, Thck (L), s then the mean of the new head of each element. Step3: Calculate the dfference between Thck (L) and Thck (L). Step4: If the dfference between Thck (L) and Thck (L) does not satsfy the convergence crteron, the value of Thck (L) s assgned to Thck (L). Return to Step2. Otherwse, the computaton of the tme step s complete and f ths step s not the fnal tme step, return to Step for the subsequent tme step. 3. Formulatng the groundwater management model The am of the management model s to mnmze the total cost of a groundwater management system for an unconfned aqufer. The system costs nclude the fxed costs of well nstallaton and the operatng costs assocated wth the tme-varyng pumpng rates. The optmal model, consderng both fxed costs and tme-varyng operatng costs, can be formulated as, and Objectve functon mn I Ω ut, I, t=,..., T subject to E J ( I) = I { c y ( I) + ( h ) h = E h + L u F t + t+ 2 t h t + T t= c u ( I)[ L ( I) h 2 t * t+ ( I)]}...(7), t =,2,..., T...(8) h t + h mn, t =,2,...,T...(9) ut dt, t =,2,...,T...(0) I umn ut I) ( u, t =,2,...,T, I Ω...() max where Ω s an ndex set that defnes all of the canddate well locatons n the aqufer, I s a subset of Ω and s a possble alternatve network desgn. The upper ndex refers to a well n the network desgn. Equaton (7) represents the total cost assocated wth the alternatve network desgn I. The frst term n the objectve functon specfes the fxed costs of the well network, n whch ($/m) represents the nstallaton cost per unt length of well; y (I) are the depths of the wells and are the dstances between the ground surface and the lower datum of the aqufer for each well. c 6

7 The second term n the objectve functon represents the operatonal costs, where ($/m) s the cost coeffcent of pumpage. The (I ) are the pumpng rates at tme step and the dmensons are altered wth the number of wells n I. L *( I) s the level of the ground surface of each well. The expresson L I) h ( ) s the pumpng lft at pumpng well. u t *( t+ I Equaton (8) s the transton equaton of the management model and s assocated wth the unconfned smulator, modfed ISOQUAL. Equaton (9) defnes the lowest lmt on the hydraulc head to prevent damage by over-pumpng, ncludng land subsdence. Equaton (0) specfes the total demand for water to be satsfed. Equaton () represents the desgn lmtaton of the well capacty, whereas the lowest lmt prevents well pumpng at unrealstcally small rates. The groundwater management model defned by Eqs. (7) to () has three key characterstcs. Frst, the objectve functon s a mxed-nteger nonlnear functon. Restated, the decson varables nclude dscontnuous varables (well locatons) and contnuous varables (tme-varyng pumpng rates). The characterstcs of dscrete varable are such that c 2 t general gradent-based algorthms, such as nonlnear programmng, cannot be used alone to solve the problem. The search for optmal network alternatves s a dscrete combnatoral optmzaton problem, so general gradent-based algorthms cannot be appled. Second, the decson varable of pumpng rate vares wth tme, and the management system s dynamc and may cause an excessve computatonal load when a dscrete-based algorthm, such as nteger programmng or dscrete dynamc programmng, s appled. Thrd, the constranng transform functon of an unconfned aqufer causes the soluton of the proposal groundwater management model to be a non-convex set, rasng the non-convex problem. Before the developed algorthm can be used to solve the defned problem, ths problem must be redfned. Accordng to Eqs. (7) to (), when a network alternatve s selected, the dscreteness and nseparablty can then be rewrtten as follows. Prmary problem mn z = cl* + J I { I Ω} * ( I ) of the problem are elmnated and the optmzaton model...(2) Slave problem (for each network alternatve desgn, I) J * T ( I ) = c u ( mn L h )...(3) ut I t= 2 t t+ Subject to Eqs. (8), (9), (0) and ()...(4) 7

8 Accordngly, the orgnal problem, specfed by Eqs. (7) to (), s a dscrete, combnatoral, dynamc, nonlnear optmzaton problem. However, restructurng the problem reveals clearly that the prmary problem s the dscrete combnatoral property, and the slave problem becomes a dynamc, nonlnear optmzaton problem. The slave problem (for each network alternatve desgn) s to mnmum the operaton costs, whch are a contnuous functon of the state and control varables. Then, the control varables are separable for each staget. Hence, the slave problem can be solved usng CDDP, and the beneft of usng CDDP s the avodance of curse of dmensonalty because of the use of the conventonal dscrete DP. The computatonal effcency of CDDP exceeds that of conventonal DP and mathematcal programmng algorthms (Murray and Yakowtz, 979[27]; Jones et al., 987[5]). The CDDP algorthm does not requre the dscretzaton of the state and control vectors. Therefore, CDDP overcomes the curse of dmensonalty, a serous lmtaton on conventonal DP (Bellman and Dreyfus, 962[28]). CDDP substantally reduces the workng dmensonalty of the algorthm below that of mathematcal programmng algorthms, by takng advantage of the dynamc nature of groundwater hydraulc or water qualty optmzaton problems through stage-wse decomposton. In contrast, mathematcal programmng algorthms do not explot the sequental structure of these problems (Jones et al., 987[5]). When the CDDP s appled to solve the slave problem, the frst and second order dervatve term of the objectve functon and the transform functon (Eq. (2)) must be derved. The nonlnearty of the governng equaton of an unconfned aqufer (Eq. (2)) s such that the frst order dervatve term ncludes a three-dmensonal matrx. Therefore, the second order dfferental term of the transform functon (Eq. (2)) becomes more complex, and the computatonal tme ncreases. Ths study neglects the second order dervatve term of the transform functon (Eq. (2)), whch wll not affect the results (Chang et al. (992). The frst order dervatves of the transform functon (Eq. (2)) are as follows. [ B] [ B] ht + = [ G] t+ + [ A] t+ +...(5) h t t t [ B] ht + = [ G] t+ + [ A] t+ + [ Lh ]...(6) ut t where [ ] n Eqs. (5) and (6) s [ G] t+ where ht G t+ [ A] n t+ T = ht + el...(7) l= ht +, l [ A] ( e) = 4 K ω ω j x x + K xx yy +, l y j ω ω j...(8) y 8

9 Therefore, the dfferentaton calculaton above s smlar to that n the nvestgaton of Mansfeld and Shoemaker (999) [8]. The operaton cost and the constrants for network alternatve desgns are always consdered n the slave problem and the mnmum operaton cost s determned by CDDP. Hence, the prmary problem s an optmzaton problem of combnaton, and does not nvolve constrants. Therefore, the Genetc Algorthm, a near global optmzaton algorthm, can locate the optmal well stes whle CDDP s employed to calculate the optmal tme-varyng pumpng rates for each well. When the locatons of the pumpng wells are predetermned, the CDDP algorthm s an effcent tool for determnng the optmal pumpng rates assocated wth each well. However, the optmzaton of pumpng rates does not consttute complete optmzaton n the management model because the number of wells and ther locatons are pre-specfed. CDDP cannot easly solve prmary problems wth fxed costs, where the number of wells and ther locatons are consdered as decson varables, because the fxed cost functon s dscontnuous. 4. Soluton procedure: Integraton of GA and CDDP Ths study) combnes GA and CDDP to develop a groundwater management model of an unconfned aqufer defned by the prmary problem (Eq. (2)) and the slave problem (Eqs. (3) and (4)) as presented n Fg.. The algorthm s a smple GA wth CDDP embedded n the total cost evaluaton. It has two man features. Frst, the GA accommodates the dscreteness of the search for alternatve optmal well locatons of the canddate well stes. Second, the CDDP algorthm s used to calculate optmal operatng costs for tme-varyng pumpng assocated wth each network alternatve (chromosome). These features are clarfed by the followng sequental explanaton of the algorthm. Step 0: Intalzaton Encode the network alternatves as chromosomes and randomly generate an ntal populaton. The GA s extensvely known to use bnary codng to represent a varable. Ths study uses a bnary ndcator to represent the status of the well nstallaton at a canddate ste. Accordngly, a chromosome, represented by a bnary strng, defnes a network alternatve. Each bt n a chromosome s assocated wth a canddate well, and the length of the chromosome equals the total number of canddate wells avalable for nstallaton. If the value of a bt equals one, then a well wll be nstalled at the assocated canddate ste; otherwse, the value of a bt s zero and no well wll be nstalled at the assocated canddate ste. The selecton of wells s bnary, so the encodng and decodng of the chromosome are straghtforward. For nstance, the chromosome n Fg. 2 represents a network desgn that selects only three wells; the well numbers are 3, 0 and 30. Step : Evaluate the total cost and ftness of each chromosome 9

10 The chromosome descrbed n step 0 can be represented mathematcally by I k = x, x2,..., x M where I k represents a chromosome (as a network alternatve) n the populaton and M s the number of total canddate wells. Each element has a bnary value of or 0, so the number of wells n ths chromosome can be calculated as follows; well M = l= l ( xl I k N k x )...(9) where k refers to the k-th chromosome n the populaton When the number and locatons of the pumpng wells have been determned, the fxed costs are easly calculated and the problem then nvolves optmzng the operatng costs for the network desgn. In the prmary problem, I c L * s a constant that represents the fxed costs and does not nfluences the operatng costs. Therefore, the CDDP can be used to evaluate the optmal operatng costs assocated wth the selected network desgn. The CDDP employed heren was that presented by Murray and Yakowtz (979)[27] and s a successve approxmaton approach to solvng optmal control problems, teratvely determnng, whch teratvely determnes the optmal soluton to the stated slave problem (Eqs. (3)~ (4)). The CDDP algorthm depends on a quadratc approxmaton of the orgnal problem. Equaton (8) s substtuted nto Eq. (3) to make the objectve functon a functon of the control and state varables wth dentcal tme ndex ( t ).Murray and Yakowtz (979) [27], Chang (986)[29], Jones et al.(987)[5], and Chang et al.(992)[6] all dscussed the CDDP algorthm n detal. Step 2: Reproduce the best strngs The best stngs were reproduced heren usng the roulette wheel approach. In roulette wheel reproducton, each chromosome has a probablty (I) of beng selected. f j ( I) p j ( I) = pop...(20) f ( I) j= j where pop s the sze of the populaton. Ths operaton smulates natural selecton, and a hgher ftness value of a chromosome corresponds to a hgher probablty of the survval of that chromosome. The, the algorthm can converge to a set of chromosomes wth hgh ftness values. Step 3: Perform crossover Crossover nvolves the random couplng of newly reproduced strngs; each par of strngs partally exchange nformaton. Crossover seeks to exchange gene nformaton to generate new offsprng strngs that preserve the best materal from the two parent strngs. Crossover normally occurs at a partcular probablty ( ) and so s performed on a majorty of the populaton. In ths study, one pont crossover s selected, as dsplayed n Fg. 3, p j p cross x l 0

11 where p cross ranges from 0.5 to.0. Step 4: Implement the mutaton Mutaton restores lost or unexplored genetc materal to the populaton to prevent the GA from prematurely convergng to a local mnmum. A mutaton probablty ( specfed so that random mutatons can be appled to ndvdual genes. DeJong (975)[30] orgnally suggested that a probablty of mutaton that was nversely proportonal to the populaton sze would prevent the search from lockng onto a local optmum. Ths study mplements DeJong s suggeston. A random number s generated wth a unform dstrbuton before a mutaton. If ths number s less than the probablty of mutaton, then the mutaton occurs; otherwse, t does not. Such a mutaton changes a specfc gene (0 or 0) n the offsprng strngs produced by the crossover operaton. Step 5: Termnate p mutat When steps to 4 have been completed a new populaton s formed. The new populaton requres evaluatng the total cost as n step, whch s employed to evaluate the stoppng crteron. The stoppng crteron s based on the change n ether the value of the objectve functon (total cost) or the optmal parameters. If the user-defned stoppng crteron s satsfed or the maxmum allowed number of generatons has been reached, the procedure termnates; otherwse, t returns to step to perform another cycle. The success and performance of GA depend on varous parameters - populaton sze, number of generatons and the probabltes of crossover and mutaton (Mcknney and Ln, 994[3]). Goldberg (989)[3] has asserted that good GA performance depends on the choce of hgh-crossover and low-mutaton probabltes and a moderate populaton sze. Therefore, solutons obtaned usng a GA cannot be guaranteed to be optmal. However, GAs s robust and easly hybrdzed wth other optmzaton methods or smulaton models. ) s

12 Start Parameter encodng Generate ntal populaton of strngs Fgure 2. Chromosome representaton ( I ) Total cost evaluaton for each chromosome Obtanconfguraton of well setup ( I ) Calculate fxed costs Determne nomnal trajectory for CDDP Calculate operatoncosts usng CDDP Reproducton Crossover Mutaton Calculate ftnesses for chromosome Fgure 3. Crossover operator Is stoppng crteron satsfed? No Yes End Fgure. Flowchart of groundwater management model 5. Numercal results A hypothetc problem, whch s a modfcaton of the example from Chang et al. (992)[6], s used to confrm the effectveness of the methodology. Fgure 4 depcts the aqufer. The aqufer s assumed to be homogeneous and sotropc. The 3,000 5,000 m ste was descrbed by 77 fnte element nodes and 35 potental well locatons. Constant-head and no-flow boundares crcumvent the flow doman. Intal condtons of hydraulc head dstrbuton before pumpng are assumed to correspond to a steady state. Table lsts the propertes of the unconfned aqufer. In the management model, the plannng horzon s dvded nto 36 stages over 9 years. Three examples are examned. CASE :The ntal condtons on hydraulc head are h = h =80m, and the level of ground surface s L * =20m. CASE 2:The ntal condtons on hydraulc head are h = h =40m, and the level of ground surface s L * =80m. CASE 3:The ntal condtons on hydraulc head are h =80m, h =70m, and the level of ground surface s L * =20m. The performance of examples depends on the proper settng of the crossover probablty ( p ), populaton sze and mutaton probablty ( p ). Numercal experments wth a unt cross m a a a b b b 2

13 b fxed cost ( c )of $20 m are conducted to analyze the senstvty of converge to the GA s parameters wth p n the range of , populaton sze from 60 to 90, and p = /populaton (adapted from DeJong s suggeston (975)[3]). The senstvty analyss demonstrates that the parameters mnmally affect the optmal values. Therefore, the solutons of the followng examples are obtaned wth = 0.8 and a populaton sze of 70 chromosomes. cross p cross mutat y=3000m no-flow boundary ground surface ground datum h a hb L lower between to Dstance surface ha h Aqufer Thckness (x,y)=(0,0) no-flow boundary x=5000m Impervous Fgure 4. Aqufer n water supply examples Table. Aqufer propertes n example applcaton Parameter Value Hydraulc conductvty Specfc yeld L * the level of ground surface ( ) m / s 0. 80~20m 5. Comparson of total costs n these cases Comparng the true total cost of the desgned network for all cases consdered demonstrates the advantages of consderng both the fxed and operatng costs. Ths secton consders the varous unt fxed cost n the three cases as above. Table 2 summarzes the total cost n these cases, gven a range of unt fxed costs. The results n Table 2 ndcate that ncreasng the unt fxed cost of drllng wells ncreases the operatng and total costs, but reduces the total number of wells. When the ntal condtons on the hydraulc head are h =80m, =70m and L =20m (Case 3), the optmal number of well nstallatons s always fve. In Table 3, Case was used to analyze the effect of total cost to save, wth and wthout the consderaton of fxed cost. In Table 3, when the optmal groundwater management model dd not consder the fxed cost, the optmal well desgn nvolved all 35 of the possble well locatons. The dfference between the total cost, consderng fxed cost, and that not consderng fxed cost, s 8% when the unt fxed cost equals $00.0 a h b *, provng that the fxed cost may markedly affect a groundwater management system and that these fxed costs should be explctly ncorporated nto a groundwater management model. 3 m

14 Table 2. Comparson of costs wth varous values of coeffcent c Coeffcent c $20.0 m $60.0 m $80.0 m $00.0 Case No. of wells Fxed cost($) Operatng cost($) Total cost ($) Case 2 No. of wells Fxed cost($) Operatng cost($) Total cost ($) Case 3 No. of wells Fxed cost($) Operatng cost($) Total cost ($) m Table 3. Comparson of total cost wth and wthout consderaton of fxed costs n Optmal desgn consderng fxed cost Optmal desgn consderng no fxed cost The dfferent of total cost wth and wthout consderaton of fxed costs optmzaton model Coeffcent C $20.0 m $60.0 m $80.0 m $00.0 m No. of wells Fxed cost($) Operatng cost($) Total cost ($) No. of wells Fxed cost($) Operatng cost($) Total cost ($) % 0.6% 4.3% 8.0% 5.2 Comparson of dfferent aqufer thcknesses A comparson was made to determne how the thckness of the unconfned aqufer affects the optmal groundwater management desgn and the total cost. The thck and thn aqufer cases (CASE and CASE2) were consderng wth the same pump lft. In CASE, the thckness of the aqufer (L) s 20m, and the ntal condtons on the hydraulc head are h = h =80m, so the pumpng lft (L-h) s 40m. In CASE 2, the thckness of the aqufer (L) s 80m, and the ntal condtons on the hydraulc head are h = h =40m, so the pumpng lft (L-h) s 40m, the same as that n CASE. Table 4 consders the varous unt fxed costs (Coeffcent ), and compares the total costs between the thck aqufer and the thn aqufer. Table 4 demonstrates that the total cost assocated wth the thn aqufer exceeds that of the thck aqufer by 6%. Therefore, for a gven water demand, ncreasng the thckness of the unconfned aqufer reduces the total cost. Fgure 5 presents the optmal head dstrbuton at the last tme step, the number of wells and the locaton of the wells, assumng a unt fxed cost of $20. a m b c a b 4

15 Table 4. Comparson of total costs n Cases and 2 Coeffcent c $20.0 m $60.0 m $80.0 m $00.0 m Thck aqufer (case ) Thn aqufer (case 2) Percentage of dfference 6.26% 6.7% 6.% 6.3% pumpng well pumpng well 3000 no-flow boundary 3000 no-flow boundary h a =80m ha =40m h b =80m 000 h b =40m no-flow boundary no-flow boundary Fgure 5. Head dstrbuton, optmal number and locaton of wells assocated wth the thck aqufer and the thn aqufer when the unt fxed cost s $20.0 m 5.3 Comparson of varous head lmtatons Theoretcally, the head lmt can prevent over pumpng whch would cause land subsdence, and ensure that groundwater resources are sustanable. Table 5, whch consders dfferent unt fxed costs (Coeffcent ), compares the total cost under varous head lmts wth the thck aqufer and the thn aqufer. Table 5 reveals that, regardless of the unt fxed cost (Coeffcent ), a hgher hydraulc head lmtaton corresponds to a greater total cost. The fxed cost s ncreases because the hgher hydraulc level of the head lmtaton ncreases the number of pumpng wells. c c CASE CASE2 Table 5. Unt fxed costs (Coeffcent c ) wth varous head lmts c $20 $80 Head lmt No head lmt 76.95m 77.0m No head lmt 76.7m 77.0m (76.9m) (76.9m) No. of wells Fxed cost($) Operatng cost($) Total cost ($) c $20 $80 head lmt No head lmt 33.75m 33.8m No head lmt 33.5m 33.8m (76.9m) (76.9m) No. of wells Fxed cost($) Operatng cost($) Total cost ($)

16 The results of ths study demonstrate that the proposed model helps decson makers n desgnng a cost-effectve groundwater supply system. 6. Concluson The proposed GCDDP algorthm determnes the mnmum total cost whle smultaneously consderng the fxed and tme-varyng operatng costs of managng water resources of an unconfned aqufer. The decson varables nclude the number and locaton of wells as well as the tme-varyng pumpng rates. The number and locaton of the wells together form a dscrete optmal combnatoral problem, and the tme-varyng pumpng rates ncrease the computatonal complexty. The proposed model can ncorporate bnary varables smply nto the optmzaton problem. A numercal nvestgaton based on a homogeneous, sotropc unconfned aqufer ndcates that the fxed cost markedly affects the number and locatons of the wells n the optmal desgn. Hence, the GCDDP algorthm s a feasble groundwater management plannng method. In summary, the novel GCDDP algorthm used n the desgn process can consder the fxed cost, whch s a sgnfcant factor n groundwater plannng, to yeld a realstc soluton. Although ths study addresses only groundwater supply, the proposed algorthm can be further extended to groundwater remedaton plannng. The computatonal loadng requred to solve groundwater remedaton models ncreases wth the complexty of the problem. Addtonally, parallel mplementatons of the GA and the smulaton model are lkely to be requred. Acknowledgement The authors would lke to thank the Natonal Scence Councl of the Republc of Chna for fnancally supportng ths research under Contract No. NSC E References. Gorelck, S. M., A revew of dstrbuted parameter groundwater management modelng methods, Water Resour. Res., 9(2), , (983). 2. Yeh, W. W-G., System Analyss n ground-water plannng and management, J. Water Resour. Plng. and Mgmt., ASCE, 8(3), , (992). 3. Ln, X. and Yang, Y., The optmzaton of groundwater supply system n SHI JIAZHUANG cty, Chna, Water Scence and Technology, 24(), 7-76, (99). 4. Pezeshk, S., Helweg, O. J. and Olver, K. E., Optmal operaton of ground-water supply dstrbuton systems, J. Water Resour. Plng. and Mgmt., ASCE, 20(5), , (994). 5. Takahash, S. and Peralta, R. C., Optmal perennal yeld plannng for complex nonlnear aqufers: Methods and examples, Advances n Water Resources, 8(), 49-62, (995). 6. Aquado, E., and Remson. I., Groundwater hydraulcs n aqufer management, J. hydr. Dv., ASCE, 6

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