Water Supply and Demand Sensitivities of Linear Programming Solutions to a Water Allocation Problem

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1 Appled Mathematcs, 22, 3, Publshed Onlne October 22 ( Water Supply and Demand Senstvtes of Lnear Programmng Solutons to a Water Allocaton Problem Konstantne P. Georgaaos,2 Hydrologc Research Center, San Dego, USA 2 Scrpps Insttuton of Oceanography, UCSD, La Jolla, USA Emal: KGeorgaaos@hrc-lab.org Receved June 7, 22; revsed July 7, 22; accepted July 25, 22 ABSTRACT Ths wor formulates and mplements a mathematcal optmzaton program to assst water managers wth water allocaton and banng decsons to meet demands. Lnear programmng s used to formulate the constrants and objectve functon of the problem and tests of the developed program are performed wth data from the Castac Lae Water Agency (CLWA) n Southern Calforna. The problem s formulated as a determnstc programmng problem over a fve year plannng horzon wth annual resoluton. The program accepts annual water allocatons from the State Water Project (SWP) n Calforna. It then determnes the least-cost feasble allocaton of ths water toward meetng annual demands n the fve-year plannng horzon. Local water sources, ncludng water recyclng, and water banng programs wth ther constrants and costs are consdered to determne the optmal water allocaton polcy wthn the plannng horzon. Although there s not enough nformaton to fully account for the uncertanty n future allocatons and demands as part of the decson problem soluton for CLWA, uncertanty n the SWP allocaton s consdered n the tests, and senstvty analyses s performed wth respect to demand ncreases to derve nferences regardng the behavor of the medan mnmum-cost solutons and of the rs of falure to meet demand. Keywords: Water Allocaton; Water Banng; Lnear Programmng. Introducton The goal of the wor s to formulate and test the feasblty of solutons of a mathematcal programmng problem that wll be sutable for annual operaton and whch wll assst Castac Lae Water Agency (CLWA) wth decsons pertanng to meetng demand over a mult-year decson horzon. CLWA was formed n 962 for the purpose of contractng wth the Calforna Department of Water Resources (DWR) to provde a supplemental supply of mported water to four retal water purveyors n the Santa Clarta Valley. CLWA serves an area of 95 square mles n the Los Angeles and Ventura Countes. CLWA and ts retal purveyors meet demand through local groundwater pumpng, State Water Project water mports, water stored n water bans, and recycled water. The mported water component of the water supply s based on relablty studes conducted and publshed by the DWR. There s uncertanty that enters the development of polcy over the plannng horzon of say N years for CLWA. Ths uncertanty arses due to future State Water Project (SWP) allocatons and due to future demands n the CLWA regon. Allocatons and demands are nfluenced by the weather and clmate n Northern Calforna and the regonal and local weather and clmate n the CLWA regon and the pattern of regonal development. The statstcal parameters of ths uncertanty are largely unnown at ths tme to allow a relable formulaton of a stochastc programmng problem. Therefore, a determnstc mathematcal programmng problem formulaton s mplemented at ths tme, that s flexble enough to allow for each year the examnaton of several possble scenaros of SWP allocaton and demand (e.g. examnaton of solutons for a sequence of expected normal years, examnaton of solutons for possble sequences of dry years, and examnaton of soluton for expected sequences of possble future development). Although the solutons for water to be placed n storage and for water transfers among varous storages wll be provded for the entre plannng horzon, only the present year soluton wll be mplemented as t contans the lowest level of uncertanty. Even ths low uncertanty may be consdered n the decson process, as CLWA decson maers can examne the spectrum of possble solutons obtaned wth partcular emphass n any dfferences for the current year pror to arrvng at a decson. An annual tme step s used n the formulaton. The next secton presents the sources of water supply Copyrght 22 ScRes.

2 286 K. P. GEORGAKAKOS for the CLWA. Secton 3 formulates a lnear programmng program that ncludes the constrants pertanng to water supply sources for CLWA and mnmzes total cost whle meetng demand, Secton places the present formulaton wthn the context of the pertnent lterature of mathematcal programmng and applcaton to water resources problems, whle Secton 5 presents results of applcaton and of senstvty analyss. Concludng remars are n Secton Sources of Water Supply for CLWA ). Wholesale water mported from SWP: Maxmum allocaton 95,2 af n any one year. Actual delvery vares n a partcular year based on hydrology, amount n SWP storage at the begnnng of water year, restrctons n delvery (technologcal and legal), and total demand requested by SWP contractors. The State Water Project Delvery Relablty Report ssued n 27 (referred to as DWR7 hereafter) shows an average delvery of 66% - 69% of maxmum allocaton amount under future condtons (set for year 227). The same report shows an average delvery of 63% of maxmum allocaton amount under year 27 condtons (two scenaros of regulatory restrctons n delvery). It also shows average and dry perod delveres under 27 condtons and future condtons (for year 227), and the dstrbuton of the percent delveres dscussed above by future years. For nstance and for fve year ntervals, average percent delveres (maxmum allocaton amount) are: Year % (Ave) Wth respect to the cost for wholesale SWP water, SWP ncome from all contractors n 22 was approxmately $6 mllon for approxmately.3 mllon acrefeet (af) of allocated water. CLWA estmated current costs are approxmately $/af per year. In ths and subsequent monetary fgures we use 27 US dollars. 2). Flexble storage account n Castac Lae (Water Supply Contract DWR): CLWA has access to up to 68 af of the storage n Castac Lae. Ths amount must be replaced wthn 5 years of ts wthdrawal. CLWA polcy s to eep the account full n normal and wet years and then delver the stored amount or porton durng dry years. The account s reflled durng the next year that adequate SWP supples exst to do so. Cost of borrowng conssts of the power cost to delver out of Castac Lae, whle the cost to replace the borrowed amount of water s CLWA s per af cost plus varable power costs. For ths wor we assume an annual cost of $2/af per year for depost and no cost for wthdrawal and for storage. 3). Flexble storage account (Ventura County agences): Access to another 376 af of storage n Castac Lae for years begnnng n 26. Ths amount too must be replaced wthn 5 years of ts wthdrawal. Costs for depost and wthdrawal are as n tem 2 above plus $5/af per year for storage. ). Artcle-2 water: Water avalable on an unscheduled and nterruptble bass for average to wet years, generally only for a lmted tme. The DWR7 ndcates average delvery under 27 condtons of 9 af and under future condtons (227) of 3 af. Maxmum amounts for 27 condtons are 59 af and for future condtons are - 2 af. Mnmum amounts are. It s unlely that CLWA wll utlze ths water type n the future. 5). The Turnbac Pool: Maes a small amount of water avalable n all types of hydrologc years (less water n dry years). It depends on urban contractor demand. It s unlely that CLWA wll utlze ths water type n the future. 6). DWR Dry Year Water Purchase Programs: DWR purchases from wllng sellers n areas of adequate supply and the DWR sells bac to those wllng to purchase t. Ths program s not operated n all years and s somewhat uncertan. It s unlely that CLWA wll utlze ths water type n the future. 7). Groundwater Alluvum: Governed by hydrologc condtons n eastern Santa Clara Rver watershed. Pumpng ranges 3, -, afy durng normal and abovenormal ranfall years. For locally dry years pumpng s reduced 3, - 35, afy (, 2 or 3 dry year sequences). Annual wthdrawal costs are estmated to be $5/af. 8). Groundwater Saugus Formaton: Ths s lned to the avalablty of other water supples n a gven year, partcularly from the SWP. Durng average-year condtons wthn the SWP system, Saugus pumpng ranges 75-5, afy. Dry year pumpng ranges between 5, - 25, afy durng a SWP drought year (reduced SWP delveres). If SWP delveres are reduced for 2 consecutve years, pumpng can ncrease to 2, - 25, afy. If SWP delveres are reduced for 3 consecutve years, pumpng can ncrease to 2, - 35, afy. Such hgh pumpng s to be followed by perods of reduced (average-year) pumpng rates to recover water levels and groundwater storage. Annual wthdrawal costs are estmated to be $5/af. 9). Core transfer (Buena Vsta Water Storage Dstrct and Rosedale-Ro Bravo Water Storage Dstrct): Ths s a transfer to purchase a defned quantty of water every year,, afy. Cost s estmated (25 Urban Water Management Plan UWMP) to be $6 - $/afy. In ths wor, ths source of water s used to reduce the demand n all years and t s not part of the optmzaton problem. ). Southern water banng programs: Water banng and exchange programs that wll provde addtonal Copyrght 22 ScRes.

3 K. P. GEORGAKAKOS 287, af of storage capacty and 69 af per year of pumpbac capacty durng a sx-month emergency outage. Annual costs are assumed to be $/af for depost and $/af for wthdrawal. The storage cost s assumed to be $5/af. ). Semtropc Water Banng: There s currently a total of 5,87 af stored water, recoverable through 23 to meet CLWA demands as needed. Current operatonal plannng ncludes use of the water stored n Semtropc for dry-year supply. Wthdrawal costs are $25/af. 2). Rosedale Ro Bravo Water Storage Dstrct Water Banng: A water banng and exchange program wth up to, af storage capacty and pumpbac capacty of 2, afy. Ths s to support dry-year supply (not planned growth). Annual storage, depost and wthdrawal costs are $3/af. 3). Recycled water: Table 3- of the 25 UWMP shows 7 afy. In ths wor, ths source of water s used to reduce the demand n all years and t s not part of the optmzaton problem. 3. Mathematcal Program Formulaton The purpose of the mathematcal program formulated n ths secton s to provde feasble solutons to the water allocaton problem for CLWA that are optmal under the set of parameter values used. The mathematcal program conssts of a set of constrants that defne the feasble solutons, and an objectve functon that facltates the selecton of the best feasble soluton. For ths programmng problem, the objectve functon s an expresson of annual costs assocated wth a certan allocaton polcy and therefore, n ths context the best or optmal soluton s one that mnmzes the objectve functon. We formulate and nterpret the set of constrants and the objectve functon n the followng. It s noted that, for the purposes of ths formulaton, we adopt an annual tme step and a plannng horzon of N years. Even though solutons are obtaned for all the years n the plannng horzon, only the frst year soluton for the allocaton polcy s mplemented. 3.. Constrants that Defne Feasble Solutons The frst set of constrants concerns the State Water Project (SWP) annual allocaton to CLWA based on the exstng contract between SWP and CLWA. For each year wthn the plannng horzon the constrants assure water volume conservaton n the process of delvery, carryover storage and CLWA allocatons. The set of constrants s expressed as: C D N (),, ;,, where represents the SWP carryover storage at the end of tme step (one year n ths case), C s the SWP annual water allocaton to CLWA for tme step + (taen as the perod from the end of tme step to the end of tme step + ), and D s the water used by CLWA durng step + (defned prevously) to meet demand at that or future tme steps. Due to the fact that carryover storage of a certan year must be used n the next year, the followng constrant s used:, D ;,, N (2) The quantty C, s program nput from actual and/or forecasted SWP allocaton (a fracton of the maxmum allocaton for CLWA, whle carryover storage, and water use by CLWA D are program soluton output. As such, nonnegatvty constrants are mposed on the latter two quanttes: ;,, N (3) D ;,, N () whle the ntal condton on carryover storage specfed from: s (5) wth nown at the begnnng of the plannng horzon. In addton to the nonnegatvty constrants t s also possble to nclude upper bound constrants for carryover storage for all I, but at ths tme no such constrants are mposed by the CLWA operatonal polcy. A second set of constrants concerns wthdrawals from local groundwater resources (Alluvum and Saugus formaton). These constrants are n the form of upper and lower bounds on wthdrawals, wth the bounds dependng on the occurrence of a dry or a medum/wet year. Ths set of constrants s represented by: G G G ;,, N (6), wet dry, lo wet dry,up G GG G ;,, N (7), wet day 2, lo 2 wet day 2,up where G s the annual wthdrawal from the Alluvum aqufer for perod + (ndcated as the perod from the end of perod to the end of perod + ), G 2 s the analogous annual wthdrawal from the Saugus formaton, wet dry G,lo sgnfes the lower bound on the Alluvum aqufer wthdrawal for a wet or a dry year n the plannng horzon, wet dry G,up sgnfes the upper bound on the Alluvum aqufer wthdrawal for a wet or a dry year n the plannng horzon, and wth analogous defntons for the Saugus formaton bounds. The values of the bounds are nonnegatve and consdered nown once a certan year n the plannng horzon s characterzed as wet or dry. Ths elmnates the need for nonnegatvty constrants for the local groundwater wthdrawals. The wthdrawal amounts G, and G 2 for all are obtaned from the soluton of the programmng program beng formulated. Copyrght 22 ScRes.

4 288 K. P. GEORGAKAKOS There are three sets of constrants assocated wth water banng programs that CLWA s currently nvolved n. The frst pertans to the Semtropc ban where CLWA has transferred a sgnfcant amount of water already, the second s assocated wth a groundwater banng program wth gven constrants of maxmum amount stored and maxmum annual wthdrawal, and the thrd s assocated wth the Rosedale Ro Bravo banng program that has ts own constrants pertanng to maxmum stored amount and maxmum annual wthdrawal. These constrants are dscussed next for each banng program. The Semtropc water volume conservaton on a year to year bass may be wrtten as: S N (8), ;,, where sgnfes the water mount n Semtropc sto, rage at the end of year, and S sgnfes the annual wthdrawal from Semtropc storage for year + (same notatonal conventon as prevously defned for transfer amounts). It s noted that Equaton (8) does not nclude a supply term for Semtropc as t s assumed that CLWA does not ntend to ncrease the water stored n Semtropc. Should ths polcy change, an addtonal postve term wll be added to the rght hand sde of Equaton (8) to sgnfy annual water amount added to Semtropc storage. Nonnegatvty constrants are assocated wth the Semtropc ban transfers: ;,, N S N (9) ;,, () and an ntal condton s specfed for the amount n Semtropc storage at the begnnng of the plannng horzon: () wth beng a nown quantty. Lastly, there s a requrement that the amount n storage n Semtropc must be wthdrawn by the end of a nown year n the future (call t n), and ths s enforced by: ; n,, N (2) The southern water banng programs may be represented by the followng constrants (water amount conservaton, and bounds on amount stored and on annual amount wthdrawn from ths ban): DG B N,, 5 5 ;,, max ;,, N 5 5 (3) (), max B B ;,, N (5), DG ;,, N (6) where 5 sgnfes the water amount n ban storage at the end of year, B sgnfes the annual wthdrawal from ban storage for year +, and DG sgnfes the annual transfer amount to ban storage by CLWA for max year +. The upper bounds for ban storage, 5, and annual wthdrawal amount, B max, are nown quanttes, and the unnown quanttes whose values result from the soluton of the programmng problem are the storage amounts ( 5 ; = 2,, N), and the annual transfer amounts n and out of the southern water ban ( DG, B ; =,, N ). The ntal storage n the ban at the begnnng of the plannng horzon s consdered nown and equal to : (7) The analogous constrants for the Rosedale-Ro Bravo banng program are shown below: DR R, ;,, N (8), 6 6 max,, N (9) 6 6 ;, max R R ;,,N (2), DR ;,, N (2) 6 (22) where 6 sgnfes the water amount n Rosedale-Ro Bravo (RRB) ban storage at the end of year, R sgnfes the annual wthdrawal from RRB-ban storage for year +, and DR sgnfes the annual transfer amount to RRB-ban storage by CLWA for year +. max The upper bounds for ban storage, 6, and annual wthdrawal amount, R max, are nown quanttes, and the unnown quanttes whose values result from the soluton of the programmng problem are the storage amounts ( 6 ; = 2,, N), and the annual transfer amounts n, and out of the RRB ban (, DR, R ; =,, N ). In ths case, too, the ntal storage n the RRB ban at the begnnng of the plannng horzon s consdered nown and equal to 6. The two Castac Lae flexble storage accounts mentoned n the water sources secton can also be treated as water bans for the purpose of ths mathematcal formulaton. Ther separate consderaton s necessary because one of the accounts has a -year actve perod whle the other does not carry a termnaton date. The feasblty constrants for the two accounts may be wrtten as: 6,, N ; 7, 8, D, F, F ; ;,, N; 7, 8 max F F ;,, N ; 7,8 max DF ; (23) (2) (25),, N ; 7,8 (26) Copyrght 22 ScRes.

5 K. P. GEORGAKAKOS 289 ; 7, 8 (27) where dentfes the Castac Lae flexble storage accounts ( 7 and 8 at the end of year ), DF s the depost to the th flexble account for year +,, F s the wthdrawal from the th flexble account for year +, and sgnfes the ntal storage n the th flexble account. Because of the 5-year term refllng condtons prescrbed for the flexble accounts addtonal constrants pertanng to the repayment are mposed: * max ; 7,8 (28) where * sgnfes the end of the year durng whch refllng of the accounts must be made. A fnal set of constrants pertans to meetng demand for each year of the plannng horzon: G G S B R F,,,,,, 2 7 F DD T N,,, 8 ;,, (29) where T s the annual demand, consdered nown. T s the resdual resultng from subtractng the fxed core transfers from Buena Vsta Water Storage Dstrct (, af) and recycled water (7 af) from the nown regonal demand for CLWA. The amount DD s the amount of the SWP allocaton to CLWA used drectly to satsfy demand, and s constraned by:,, DR DG DF DF DD D,, N and the nonnegatvty constrant:,,,, 7 8 ; DD N (3) ;,, (3) Ths amount ( DD ) s obtaned from the soluton of the programmng problem formulated. Equatons ()-(3) defne the set of feasble solutons over all the years (N) of the plannng horzon. Ths set of solutons depends very much on the values of the parameters specfed and t s possble that for certan values of the parameters there are no feasble solutons that satsfy all the constrants of the problem. For nstance, when the avalable amount n ntal storage n the varous banng programs, plus the maxmum wthdrawal from the local groundwater formaton, plus the amount expected to be transferred from SWP over the plannng horzon results n an amount lower than the total demand amount expected over the plannng horzon, then the constrant set does not have a feasble soluton. In such stuatons (rather emergency stuatons) t s assumed that CLWA wll resort to emergency measures to reduce demand and to acqure addtonal amounts of water for the perod n need. Thus, n ths stuaton there s no gan from usng the mathematcal program formulated. The constrant that may be used to trgger ths stuaton s (ths s lely to happen n very dry perods): G G C, 5 6 dry, up dry 2, up, T (32) where the summaton ( ) s for all years ( =,, N ), and all the symbols have been defned prevously. It s also noted that the set of constrants for the mathematcal programmng problem formulated s lnear n the unnown quanttes thus smplfyng substantally the soluton method for the mathematcal program. Before we dscuss soluton methods we formulate the objectve functon next The Objectve Functon The objectve functon of the mathematcal program s used to select the optmal soluton from the set of feasble solutons allowed by the constrants formulated n the prevous secton. For the purposes of the CLWA operatons, the objectve functon s defned as the total cost of the CLWA operaton due to annual water transfers and due to storage of water over all the years of the plannng horzon. An optmal soluton for ths case, then, s the one that mnmzes the objectve functon. In general terms, the objectve functon may be stated as: J KY (33) wth the summaton extendng over all the unnown varables Y j and wth K j representng annual cost assocated wth water storage (f the varable represents storage) and wth annual water transfer (f the varable represents annual water transfer amount). The unnown varables Y j have been dentfed n the dscusson of the problem constrants of the prevous secton. It remans to dentfy the costs K j (n unts of constant dollars over the plannng horzon per acre feet $/af). The relablty of the cost numbers s mportant as t s through the relatve magntude of such costs that the mathematcal program wll relably allocate water among the varous storages whle meetng demand (assumng that the nfeasblty constrant of Equaton (32) among parameters and future allocatons does not hold). It s then approprate to examne the senstvty of the soluton of the mathematcal program formulated wth respect to the relatve value of the costs K j.. Lterature Revew of Pertnent Optmzaton Problems and Ther Solutons The formulated mathematcal programmng problem s lnear n both the constrants and the objectve functon. Standardzed methods exst (e.g. []) to convert any gven set of lnear equalty and nequalty constrants and a lnear objectve functon to the standard form, whch may be wrtten as: Copyrght 22 ScRes.

6 29 K. P. GEORGAKAKOS subject to J y (3) Ay b (35) y (36) where bold face letters sgnfy column vectors for lower case and matrces for upper case letters, y sgnfes the n-dmensonal vector of ndependent varables, A s an mxn-dmensonal vector of parameters, b s an m-dmensonal vector of parameters, and s an xn-dmensonal vector representng the vector transpose of, and represents the n-dmensonal vector wth all elements equal to zero. The objectve functon of Equaton (3) s maxmzed for the standard form. There are several computatonally effcent algorthms that have been developed to solve such lnear programmng problems n standard form. The orgnal algorthm was developed by George B. Dantzg [2] and t s called the smplex algorthm or the smplex technque. It s based on the premse that the optmal soluton for a lnear programmng problem that maxmzes J of Equaton (3) s at the vertces of the shape formed by the ntersecton of the lnear equalty constrants of Equaton (35) that defne the feasble regon. The smplex algorthm then dentfes sequentally non nferor vertex ponts (vertex ponts that have an objectve value that s hgher than prevously found) untl t fnds the maxmum soluton (optmal soluton for the standard form) (for detals see [3,]). It s a computatonally effcent algorthm because t taes approxmately between m and 2m teratons to arrve at the optmum. Varants of the algorthm sutable for large problems are n use today for mplementaton n modern day dgtal computers that are even more effcent (e.g. [5,6]). For the class of lnear programmng problems and for each lnear programmng problem (called the prmal n ths context), t s possble to formulate ts dual programmng problem wth an optmal soluton that matches n value exactly the value of the objectve functon of the prmal at the optmum. The sgnfcance of ths dual problem s that the optmal soluton values of the dual-problem ndependent varables represent the rates of change of the objectve functon J wth respect to the constrant parameters (elements of vector b). They are typcally called shadow prces or margnal values and n several cases have an economc nterpretaton [7]. For the case of equalty constrants, the dual varables are the so-called Lagrange multplers nvolved n the soluton of constrant optmzaton problems. Lnear programmng formulatons and algorthms have been used wdely to gude the soluton of water allocaton and water resources plannng problems (e.g. [8]). Ther wdespread use s a result of the avalablty of very effcent solvers (as dscussed prevously) that allow the planner to focus on the problem formulaton rather than on the mechancs of the soluton algorthm. Some recent examples are mentoned below as llustratons of potental applcatons. In [9], a seres of lnear programs are used to solve a water supply problem n Antelope Valley, Calforna, where volumes of water was njected and extracted each year n an unconfned aqufer. In [], lnear programmng was used to montor pollutants and maxmze the potable water n a storage reservor that s used for water delvery va a number of pumpng statons. Groundwater management n a saltwater ntruson envronment n a coastal arstc aqufer was studed n [] usng lnear programmng and physcal groundwater modelng. In [2] lnear programmng was used to study conjunctve water use and banng n southern Calforna. In addton, methods for ncorporatng uncertanty n problem parameters have been developed that retan the effcency of soluton of the lnear programmng problems. The bass of these methods s the development of chance constrants where the constrant expresses the probablty threshold mposed for the volaton of the constrant. Under ths constrant formulaton and assumng nowledge of the probablty dstrbuton of the uncertan parameter, such constrants may be readly converted to ordnary determnstc constrants (e.g. [8,3]). As an alternatve to lnear programmng problem optmzaton methods, drect smulaton s often used for complex problems. These approaches do not optmze the objectve functon, but serve as tools to explore varous propertes of the objectve functon wthn the feasble regon specfed by the constrants. If optmzaton s necessary for certan system components then smulaton s combned wth optmzaton. An example of ths approach s the CalSm model used by the US Bureau of Reclamaton for reservor management n Calforna, whereby smulaton s coupled wth mathematcal programmng to dstrbute water volumes throughout the conveyance networ of the Central Valley Project of Calforna []. Stochastc smulatons methods (also referred to as Monte Carlo smulaton methods) have also been developed and used to ncorporate uncertanty n problem nput or problem parameters durng the exploraton of the propertes of the objectve functon. For nstance, a very recent applcaton of Monte Carlo smulaton appled stochastc generaton technques to develop rs-based strateges for tradng dscharge permts n rvers [5]. Also, methods that combne stochastc smulaton wth systematc optmum search methods have been recently appled to water management problems that nvolve sequental decsons n the operatonal management envronment (e.g. [6]). The stochastc smulaton methods Copyrght 22 ScRes.

7 K. P. GEORGAKAKOS 29 are approprate when the uncertanty n model parameters can be well characterzed through probablty dstrbutons (jont probablty dstrbutons may be necessary). However, they are computatonally expensve and good ntal solutons (perhaps obtaned by the soluton of lnear programmng problems usng the effcent smplex algorthm) are necessary for effectve use. Before we close ths short excurson n the lterature of mathematcal programmng for water resources we refer to two addtonal types of optmzaton. The frst s a recent evoluton of mathematcal programmng called Postve Mathematcal Programmng [7] and the second s the substantally older and well nown class of dynamc programmng algorthms [8]. Postve Mathematcal Programmng s an enhanced mathematcal program that ncludes constrants pertanng to the hstorcal baselne operaton of the system to be optmzed to nclude constrants that assure solutons that are smlar to system past behavor, and a nonlnear objectve functon to be optmzed for best performance. Ths type of programmng s sutable for analyzng envronmental polcy when substantal data s avalable for the calbraton process and requres substantal optmzaton expertse for ts soluton, as no effcent solvers are currently commercalzed. In addton no sgnfcant experence exsts n ntroducng uncertanty n model parameters or exstng observed data. Dynamc programmng s a powerful optmzaton method sutable for very large problems that nvolve decsons n stages, wth nonlnear objectves and constrants. Well establshed algorthms exst for stochastc dynamc programmng problems wth non-neglgble uncertanty. There s a long hstory of the use of dynamc programmng n water management wth recent applcatons n partcpatory decson mang for operatonal water resources management (e.g. [9]). For the applcaton at hand, however, the mathematcal program s not large and the lnear programmng formulaton s very effcent so there s no need to ntroduce dynamc programmng as a soluton method. 5. Example Programmng Problem Solutons To llustrate the type of solutons that may be obtaned from the formulated mathematcal program and to demonstrate the feasblty of the formulaton, we have constructed a MATLAB sample code that mplements a partcular verson of the formulaton usng a number of assumed parameters and we perform a small scale senstvty analyss wth respect to demand to llustrate the benefts of such a mathematcal programmng approach. The mathematcal program formulaton s solved wth a varant of the smplex algorthm mplemented n MATLAB. It s noted that the numercal experments conducted allow for the generaton of many possble scenaros of SWP allocatons and determne the ensemble of mn- mum cost solutons as well as certan quantles of the mnmum cost soluton ensemble. In all cases, the experments assume ntal condton at the begnnng of year 29, and the plannng horzon for the problem s set through year 23 (5 years). 5.. Nomnal Input and Demand Table lsts the assumed mean annual SWP allocaton for each of the years of the plannng horzon together wth the assocated standard devaton of the annual allocaton. It s noted that the frst year n the plannng horzon (Y29) carres very small uncertanty, because t s assumed that the allocaton s already determned when the model s used. The followng years carry consderable uncertanty. It s also assumed that dry years wll occur n the mean. Table 2 shows the nput parameters used n the numercal experments. They reflect the nformaton n prevous sectons and are consdered nomnal for these numercal experments. Fgure shows the hstogram of the 5-year-average mnmum cost from a run that generated ensemble members (possble SWP allocatons). The dstrbuton of mnmum average annual costs s approxmately Gaussan wth a mean of approxmately 5. mllon dollars over the plannng horzon of 5 years. The mnmum costs range from about 5.5 to 5.7 mllon dollars. No generated sequence of SWP allocatons was assocated wth an nfeasble soluton. Table. Statstcs of generated SWP allocatons n acre-feet (AF) for CLWA. (a) Mean annual SWP allocatons Y29 Y2 Y2 Y22 Y23 5, 5 5, 5 (b) St dev of annual SWP allocatons Y29 Y2 Y2 Y22 Y , 2 Table 2. Values of ntal condtons and nput parameters for the 5-year plannng horzon. (a) Intal storages (af) CrOv Semtrp SBn RRB-Bn CLaFlex CLaFlex (b) C lae flexble accounts maxmal volumes (c) C lae flexble accounts remanng refll years.. Copyrght 22 ScRes.

8 292 K. P. GEORGAKAKOS (d) Annual demand (af) nclusve of recycle contrb (e) Annual recycle contrbuton (af) (f) Annual local-gw wthdrawal lmts (af) Alluv mn: Alluv max: Saugus mn: Saugus max: (g) Annual maxmum storage volumes (af) Carry-over: Semtropc: S. ban:..... RRB ban:..... CL flex : CL flex 2: (h) Annual maxmum wthdrawals from bans (af) South ban: 69. RRB ban: 2. CL flex : 68. CL flex 2: 376. () Annual storage costs ($/af) Carry-over: Semtropc: So ban: RRB ban: CL flex : CL flex 2: (j) Annual transfer costs ($/af) Semtropc wthdr: So ban wthdr:..... RRB ban wthdr: So ban depst:..... RRB ban depst: SWP wthdr:..... Alluvum wthdr: Saugus wthdr: CL flex dep: CL flex wthdr:..... CL flex 2 dep: CL flex 2 wthdr:..... Fgure. Hstogram of ensemble members of the 5-year average mnmum cost. The cost s n mllons of $ per year. Copyrght 22 ScRes.

9 K. P. GEORGAKAKOS 293 Although the range of 5-year-average mnmum costs s consderable, the frst year cost for the optmal polces that lead to the mnmum 5-year-cost s substantally less dspersed among ensemble members. The reader s remnded that although the soluton refers to the optmal polcy over the plannng horzon of 5 years, only the frst year soluton may be mplemented, usng an adaptve prncple. Fgure 2 shows the hstogram of the frst year costs assocated wth the mnmum 5-year-average cost solutons of Fgure. Very few ensemble members have values dfferent from the bul of values near 3.6 mllon per year. In fact, the medan and the 95th percentle frst year cost are dentcal for ths experment. Table 3 shows the medan frst year cost together wth the frst year soluton for the numercal experment. The allocaton of 2,7 af of recycled water s assumed gven and t s not consdered for the optmzaton experment, leavng a total of 66, af of frst-year demand to be satsfed from varous ban and groundwater storages (the frst year allocaton from SWP s assumed to be af for ths experment). It s shown that the frst year demand, assumed to be 78,8 af, s satsfed wth a 23,6 af wthdrawal from Semtropc ban, a 35, af wthdrawal from Alluvum groundwater storage, and a 75 af wthdrawal from Saugus groundwater storage. The medan frst year cost s estmated to be $ The optmal polcy for the entre plannng horzon for the medan frst year soluton s gven n Table. Table also shows the generated SWP allocatons for ths medan soluton ensemble member. Wthdrawals from Semtropc, Alluvum and Saugus formatons satsfy the resdual annual demands after the recycled water allocaton s subtracted from the total annual demand of Table 2 for each year. It s noted that the Semtropc storage s depleted by 23 as requred, and no carryover SWP storage s generated due to the low SWP allocatons expected for ths experment. No actvty s projected for ths soluton pertanng to the southern banng storage, the RRB and the Castac Flexble storages for mnmum Table 3. Medan of frst-year cost solutons. Carryover storage:. So ban wthdrawal:. Semtropc storage: 587. Alluvum wthdrawal: 35. South ban storage:. Saugus wthdrawal: 75. R-RB storage:. CL flex depost:. CL flex storage: 68. CL flex wthdrawal:. CL flex 2 storage: 376. CL flex 2 depost:. SWP wthdrawal:. CL flex 2 wthdrawal:. SWP use for demand:. R-RB ban depost:. Semtropc wthdrawal: 236. RRB ban wthdrawal:. So ban depost:. Fgure 2. As n Fgure but for frst year costs. Copyrght 22 ScRes.

10 29 K. P. GEORGAKAKOS Table. Optmal 5-year polcy for medan of frst-year cost solutons. (a) Generated annual SWP allocatons (af) (b) Solutons for CLWA lnear program Caryovr store: Semtropc: 5,87 27,27 3, South ban: RRB store: CL flex store: CL flex 2 store: SWP wth:., SWP use:., Semtropc Wth 23, So ban dep:..... So ban wth:..... RRB ban dep:..... RRB ban wth:..... Alluvum wth: 35, 35, 35, 35, 35, Saugus wth: , CL flex dep:..... CL flex wth:..... CL flex 2 dep:..... CL flex 2 wth:..... cost over the 5-year horzon gven the SWP allocatons generated. The frst year total cost s $ , whle the 5-year average annual cost (nclusve of ntal storage) s Increasng Demand In ths secton we explore the dependence of the optmal solutons to ncreases n demand. The SWP allocaton statstcs are as shown n Table, and the rest of the nput parameters are as shown n Table 2. At frst, a 5% ncrease of demand s mposed for all the years of the 5-year plannng horzon. Fgure 3 shows the hstogram of the 5-year-average mnmum costs for realzatons of SWP allocatons wth the statstcs of Table. The Fgure shows that the hstogram n ths case devates sgnfcantly from the Gaussan dstrbuton and t has a medan cost that s about 6 mllon dollars per year; a substantal ncrease from the 5. mllon dollars per year assocated wth the nomnal soluton. That s a 5% ncrease n demand generates more than a % ncrease n the medan 5-year-average mnmum cost. The shape of the dstrbuton s truncated for hgh values of cost due to the fact that there are now realzatons of SWP allocatons that are nfeasble (cannot meet demand due to the volaton of the constrant n Equaton (32)). In fact,.5% of the realzatons generated do not have feasble solutons. There s a.5% chance that gven the statstcs of the SWP allocatons of Table, a 5% ncrease n demand wll not be met. Fgure 3. As n Fgure but for a 5% ncrease n annual demand for the entre plannng horzon. Copyrght 22 ScRes.

11 K. P. GEORGAKAKOS 295 Fgure shows the frst year cost hstogram for the feasble solutons (analogous to Fgure 2). Sgnfcant ncrease of the medan frst-year cost s mpled by the long tal of the hstogram. The medan fst year cost and assocated soluton are shown n Table 5. The dfference n frst year medan cost from that of the nomnal run s due to the ncreased wthdrawal from the Semtropc ban and from the Saugus groundwater formaton. The frst-year medan cost ncrease s about 5% wth respect to the run wth nomnal demand. Ths s equal to the 5% ncrease n demand. In varance wth the nomnal case, n ths case of ncreased demand the 95th-percentle frst-year cost s dfferent (hgher) from the medan such cost. Table 5 also shows the 95th-percentle cost and assocated soluton. There s an ncrease of more than % of frst year cost wth respect to the medan frst year cost. The demand ncrease resulted n an ncrease of frst year cost assocated wth a hgh probablty of meetng demand (rs reducton s n ths case assocated wth substantally ncreased cost). The cost ncrease s due to the ncrease of the wthdrawal from the Saugus groundwater storage and a decrease of the Semtropc wthdrawal. The full 5-year solutons assocated wth the medan and the 95th-percentle frst-year costs dscussed above are shown n Tables 6 and 7, respectvely. The frst year costs are $ and $567.5, respectvely. The 5-year average annual costs are $5965. and $ , respectvely. Dfferent strateges are exhbted over the Table 5. Medan and 95th-percentle of frst-year cost for a demand ncrease of 5% for all years. SWP carryover storage:.. Semtropc storage: South ban storage:.. R-RB storage:.. CL flex storage: CL flex2 storage: SWP wthdrawal:.. SWP use for demand:.. Semtropc wthdrawal: So ban depost:.. So ban wthdrawal:.. R-RB ban depost:.. R-RB ban wthdrawal:.. Alluvum wthdrawal: Saugus Wthdrawal: CL flex depost:.. CL flex wthdrawal:.. CL flex 2 depost:.. CL flex 2 wthdrawal:.. Frst-year ($/YR): Medan 95th-percentle. Fgure. As n Fgure 2 but for a 5% ncrease n annual demand for the entre plannng horzon. Copyrght 22 ScRes.

12 296 K. P. GEORGAKAKOS Table 6. Optmal 5-year polcy for medan of frst-year cost solutons for a 5% ncrease n demand (a) Generated annual SWP allocatons (af) (b) Solutons for CLWA lnear program Semtropc: 5, South ban: RRB store: CL flex sto: CL flex 2 sto: SWP wth: SWP use: Semtrop wth ,38 So ban dep:..... So ban wth:..... RRB ban dep:..... RRB ban wth:..... Alluvum wth: 35, 35, 35, 35, 35, Saugus wth: , 25, , CL flex dep:..... CL flex wth:..... CL flex 2 dep:..... CL flex 2 wth:..... Table 7. Optmal 5-year polcy for 95th percentle of frst-year cost solutons for a 5% ncrease n demand. (a) Generated annual SWP allocatons (af) (b) Solutons for CLWA lnear program Semtropc: South ban: RRB store: CL flex sto: CL flex 2 sto: SWP wth: SWP use: Semtrop wth So ban dep:..... So ban wth:..... RRB ban dep:..... RRB ban wth:..... Alluvum wth: 35, 35, 35, 35, 35, Saugus wth: ,96 25, 25, 25, CL flex dep:..... CL flex wth:..... CL flex 2 dep:..... CL flex 2 wth:..... plannng horzon. The SWP allocatons are dfferent for the medan and the 95 th -percentle solutons and ths s reflected n the optmal polcy for storage and wthdrawal. In all cases, the SWP carryover storage remaned at zero, and no actvty was noted for the southern and RRB bans and the flexble Castac lae storage. An addtonal senstvty experment was conducted to determne the varaton of the frst year cost and of the percent nfeasble solutons for a % ncrease of demand for each year of the 5-year plannng horzon. The medan frst-year cost n ths case was $ and the percent of runs wth nfeasble solutons rose to 3.%. It s apparent that a % ncrease of demand under the dry condtons for SWP allocaton depcted n the statstcs of Table yelds a hgh chance of not beng able to meet the demand over the plannng horzon. Among the feasble solutons, there s an ncrease of the medan frst-year cost of about 5% wth respect to the nomnal demand run. The senstvty of the soluton costs to demand changes may be seen n Fgure 5. The medan frst-year cost of the optmal solutons s shown by the bars for ncreases n demand that span the range from to % for each year of the plannng horzon. Read on the rght vertcal axs and ndcated by the blac heavy lne s the correspondng percent of the nfeasble solutons (that also descrbes the rs of falure to meet demand). It s apparent that for the set of assumptons regardng the statstcs of the SWP allocaton of Table and for the parameters of Table 2 (apart from the demand values), the senstvty of the medan frst-year soluton wth respect to the demand ncrease as well as the senstvty of the rs of falure to meet demand ncrease substantally for percent ncreases n demand hgher than 5%. Up to that value of demand ncrease, the CLWA system as defned heren exhbts robustness wth respect to medan frst-year cost and rs of falure to meet demand. 6. Concludng Remars A lnear programmng formulaton and applcaton s demonstrated for a surface- and ground-water management problem that nvolves a varety of water supply sources and the ablty for water banng operatons. The formulated mathematcal optmzaton problem s desgned to mnmze the cost of water management operatons due to water allocatons and water storage over the management horzon of N years wth the objectve to meet annual water demands. The mathematcal solutons of the resultant mnmum cost problem are then used to gude water supply allocaton and to study the senstvty of the mnmum cost solutons to water supply uncertanty and to demand ncreases. The results allow quanttatve evaluaton of the possble strateges of the water management agency for Copyrght 22 ScRes.

13 K. P. GEORGAKAKOS 297 the frst year allocatons under the cases examned. Notable s the sgnfcant ncrease n cost and nfeasble solutons wth even a moderate ncrease n water demand. Relable water demand estmaton s then a prerequste for useful applcaton of the mathematcal programmng formulatons presented to decson mang. REFERENCES [] D. A. Perre, Optmzaton Theory wth Applcatons. Chapter 5, Dover Publcatons, Inc., New Yor, 986, pp [2] G. B. Dantzg, Maxmzaton of a Lnear Functon of Varables Subject to Lnear Inequaltes, In: T. C. Koopmans, Ed., Actvty Analyss of Producton and Allocaton, Cowles Commsson Monograph, No. 3, Wley, New Yor, 95. [3] G. B. Dantzg, A. Orden and P. Wolfe, The Generalzed Smplex Method for Mnmzng a Lnear form under Lnear Inequalty Constrants, Pacfc Journal of Mathematcs, Vol. 5, No. 2, 955, pp [] G. B. Dantzg, Lnear Programmng and Extensons, Prnceton Unversty Press, Prnceton, 963. [5] S. Mehrorta, On the Implementaton of a Prmal-Dual Interor Pont Method, SI Journal on Optmzaton, Vol. 2, No., 992, pp do:.37/8228 [6] Y. Zhang, Solvng Large-Scale Lnear Programs by Interor-Pont Methods under the MATLAB Envronment, Techncal Report TR96-, Department of Mathematcs and Statstcs, Unversty of Maryland, Baltmore, 995. [7] D. M. Smmons, Nonlnear Programmng for Operatons Research. Chapter 2, Prentce-Hall, Inc., Englewood Clffs, 975, pp [8] D. P. Loucs, J. R. Stednger and D. A. Hath, Water Resource Systems Plannng and Analyss, Prentce-Hall, Inc., Englewood Clffs, 98. [9] D. P. Ahlfeld and G. Baro-Montes, Solvng Unconfned Groundwater Flow Management Problems wth Successve Lnear Programmng, Journal of Water Resources Plannng and Management, Vol. 3, No. 5, 28, pp. -2. do:.6/(asce) (28)3:5() [] I. Ioslovch and P.-O. Gutman, Optmal Montorng and Management of a Water Storage, Envronmental Mon- torng and Assessment, Vol. 38, No. -2, 28, pp do:.7/s [] S. M. Kasteras, G. P. Karatzas, I. K. Nolos and M. P. Papadopoulou, Applcaton of Lnear Programmng and Dfferental Evolutonary Optmzaton Methodologes for the Soluton of Coastal Subsurface Water Management Problems Subject to Envronmental Crtera, Journal of Hydrology, Vol. 32, No. 3-, 27, pp [2] M. Puldo-Velazquez, M. W. Jenns and J. R. Lund Economc Values for Conjunctve Use and Water Banng n Southern Calforna, Water Resources Research, Vol., 2, pp. -5. do:.29/23wr2626 [3] K. P. Georgaaos and N. E. Graham, Potental Benefts of Seasonal Inflow Predcton Uncertanty for Reservor Release Decsons, Journal of Appled Meteorology and Clmatology, Vol. 7, No. 5, 28, pp do:.75/27jc67. [] A. J. Draper, A. Munévar, S. K. Arora, E. Reyes, N. L. Parer, F. L. Chung and L. E. Peterson, CalSm: Generalzed Model for Reservor System Analyss, Journal of Water Resources Plannng and Management, Vol. 3, No. 6, 2, pp do:.6/(asce) (2)3:6(8) [5] S. M. Mesbah, R. Kerachan and M. R. Noo, Developng Real Tme Operatng Rules for Tradng Dscharge Permts n Rvers: Applcaton of Bayesan Networs, Envronmental Modelng and Software, Vol. 2, No. 2, 29, pp do:.6/j.envsoft [6] N. E. Graham, K. P. Georgaaos, C. Vargas and M. Echevers, Smulatng the Value of El Nno Forecasts for the Panama Canal, Advances n Water Resources, Vol. 29, No., 26, pp do:.6/j.advwatres [7] R. E. Howtt, Postve Mathematcal Programmng, Amercan Journal of Agrcultural Economcs, Vol. 77, No. 2, 995, pp do:.237/2353 [8] R. Bellman, Dynamc Programmng, Dover Edton 23, Dover Publcatons Inc., Mneola, New Yor, 972. [9] H. Yao and A. Georgaaos, Assessment of Folsom Lae Response to Hstorcal and Future Clmate Scenaros: 2. Reservor Management, Journal of Hydrology, Vol. 29, No. -, 2, pp do:.6/s22-69()8-8 Copyrght 22 ScRes.