THE EFFECTS OF RISK AND RELIABILITY ON OPTIMAL RESERVOIR DESIGN'

Transcription

1 VOL. 0, NO. WATER RESOURCES BULLETIN AMERICAN WATER RESOURCES ASSOCIATION JUNE 98 THE EFFECTS OF RISK AND RELIABILITY ON OPTIMAL RESERVOIR DESIGN' Hasan Yazcgl and Mark H. Houck ABSTRACT: A chanceconsraned lnear programmng model, whch ulzes mulple lnear decson rules and s useful for rver basn plannng, s used o evaluae he effecs of rsk and relably on opmal reservor desgn. Sreamflow forecass or predcons can be explcly ncluded n he lnear program. The rsk assocaed wh he predcons s ncluded n he model hrough he use of cumulave dsrbuon funcons (CDF) of sreamflows whch are condoned on he pre dcons. A mulplepurpose reservor on he Gunpowder Rver n Maryland s used o llusrae he effecveness of he model. In order o provde he decson makers wh complee and useful nformaon, rade-off curves relang mnmum reservor capacy (a surrogae for dam coss), waer supply and flood conrol arges, and he relably of achevng he arges are developed. The rade-off curves may enhance he decson maker's ably o selec he bes dam capacy, consderng echnologcal and fnancal consrans as well as he rade-offs beween arges, rsks, and coss. (KEY TERMS: reservor desgn; opmzaon; rsk and relably; sreamflow forecass; mulple objecves; rade-off curves.) INTRODUCTION Inal effors o suppor and mprove he desgn and operaon of waer reservor sysems by use of sysems analyss and operaons research echnques commenced some wo decades ago. In mos prevous work whch nvolved opmzaon modelng of reservor sysems, he varably of sreamflows has been accommodaed by wo mehods: sngle or mulple deermnsc sequences of sreamflows, and explcly so- chasc modelng of sreamflows. Thomas and Waermeyer (96), Buras (966), Young (967), Hall and Dracup (970), Becker and Yeh (97), KaramQuz and Houck (98), and ohers have demonsraed he use of deermnsc sequences of sreamflows n lnear and dynamc programmng models. Two of he mos sgnfcan drawbacks of hese models are: ypcally, hey do no provde general operang rules because all operaon of he sysem s relaed o he specfc sequence of sreamflow npus o he model; and forecasng of fuure sreamflows s assumed o be perfec because he enre sequence of sreamflows s known before he model s solved. However, he use of forecas nformaon and perodc revson of hese forecass n accordance wh he changng hydrologc condons can be exremely useful n arrvng a opmal reservor operang polces. Hosh and Burges (979) and Yazcgl, e al. (98), have shown he uly of forecas nformaon n opmal reservor operaons. Many auhors, ncludng Loucks (969), Jacoby and Loucks (97), Bucher (97), Caselon and Russell (976), and Houck and Cohon (978), have aemped o descrbe he sochasc properes of sreamflows explcly whn an opmzaon model. Usually he descrpon s a dscree, lag-one Markov model so ha hgh frequency (shor-erm) effecs may be adequaely ncluded bu low frequency (long-erm) effecs may no. These models yeld general operang rules bu forecasng s dffcul or mpossble o accommodae whn he model. Houck (979a) proposed a model whch explcly ncludes he rsk assocaed wh predcons or forecass of fuure sreamflows hrough he use of he cumulave dsrbuon funcons of sreamflows whch are condoned on he predcons, The npus o he model are he relables of he forecass: he probables ha gven sreamflows acually occur when a parcular sreamflow s forecased. The model s a lnear program wh chance consrans and employs mulple lnear decson rules. Many auhors have descrbed relaed work. Chance-consraned programmng usng lnear decson rules has been dscussed by Charnes and Cooper (96); ReVelle, e al. (969); Joeres, e al. (98); Houck and Daa (98); Sednger (98), and many ohers. Only a few (e.g., Joeres, e a.. 98; Houck, 979), however, have consdered forecass explcly n he model. The purpose of hs paper s o demonsrae he use of he model proposed by Houck (979a) wh acual daa n order o evaluae he effecs of rsk and relably on opmal reservor desgn. The model s consruced and solved for a hypohecal reservor se on he Gunpowder Rver n Maryland. Trade-off curves relang mnmum reservor capacy and he relably of achevng varous sysem objecves are developed. The remander of hs paper s dvded no four secons: he Mulple Lnear Decson Rule model s descrbed frs; he resuls from he s-duon of he model are presened nex; he use of he model n developng rade-off curves follows; and fnally, a summary of he paper s presened. 'Paper No. 807 of he Waer Resources Bullen. Respecvely, Asssan Professor, Deparmen of Earh Scences, Unversy of Peroleum and Mnerals, Dhahran Inernaonal Arpor, P.O. Box, UPM No. 77, Dhahran, Saud, Araba; and Assocae Professor, School of Cvl Engneerng, Purdue Unversy, Wes Lafayee, Indana WATER RESOURCES BULLETIN

2 Yazcgl and Houck MULTIPLE LINEAR DECISION RULES MODEL In hs secon, a bref descrpon of he Mulple Lnear Decson Rules (MLDR) model wll be presened. A sngle mulpurpose reservor desgn and managemen problem wll be used o descrbe he basc pars of he model. For hs case, a sngle dam se s avalable for consrucon and he reservor wll be used o conrol sreamflows o enhance he local waer supply, o mgae floodng damages and o ncrease recreaon and aquac lfe poenal. The desred levels of ncreased flood conrol, waer supply and recreaon have been specfed and he decsons requred are: how large o consruc he dam and how o operae he reservor so ha he waer supply, flood conrol, and recreaon demands are me and consrucon coss are mnmzed. In he absence of moneary uns for he acual consrucon coss, mnmzaon of he dam capacy (an ndcaor of nal economc coss) s chosen as he objecve of he model. The demands for flood conrol and recreaon wll be expressed as consrans on he amoun of freeboard and mnmum sorage avalable a he dam wh hgh relables. Smlarly, he waer supply demand wll be expressed as a consran on he amoun of waer released from he reservor wh hgh relably. These represen he sysem s performance crera, whch are defned n erms of maxmum and mnmum requred sorages and he mnmum permssble release. The bass of he model s a se of lnear decson rules (LDR) whch defnes reservor operaon under dfferen condons. Le X represen he release from he reservor durng season when he sreamflow level has been forecased o be n a range or nerval ndexed by n season and he sreamflow was acually n nerval j n season -. The acual sreamflow n season may or may no be n nerval bu a some pas me, a predcon of sreamflow n nerval had been made. The lnear decson rule saes ha release durng a season equals he sorage a he begnnng of he season (S ) mnus a decson parameer (b ), whch can vary from negave nfny o posve nfny: - b. Thus boh sorage and release are defned n erms of he decson parameers and sreamflows. The use of complex superscrps and subscrps s awkward bu necessary. The subscrps (e.g., -,, l) denoe he season of he year. The superscrps denoe he nervals or ranges of sreamflows whch have been forecased or have already occurred. The superscrp always ndexes he sreamflow nerval for season, and j always ndexes he sream- flow for season -. The rule for deermnng wheher and 7 are forecased sreamflows or acual sreamflows whch have already occurred s: he subcrp ndcaes presen me; any superscrps whch correspond o a prevous me are ndexes of acual prevous sreamflows; any superscrps correspondng o he piesen me are ndexes of forecased sreamflows. Thus, $ s he release n season condoned on acual sreamflow n season - whch was n nerval j and on fhe predcon ha sreamflow n season wll be n nerval. Sll (g) s he sorage a he begnnng of season +l () con- doned on pas, acual sreamflow n season (-) occurrng n nerval (j). R: (R-l) s he sreamflow n season (-) condoned on forecased or predced sreamflow for season (-) n nerval u). Fnally, b: (b:-l) s he LDR decson parameer for season (-) condoned on forecased sreamflow n nerval u) for season (-). Le a, P, and y represen he mnmal probables wh whch acual * releases, sorages, and freeboards equal or exceed XMIN, SMIN, and SFREE, respecvely, n season. Furhermore, le denoe he jon probably of he forecased sreamflow for season beng n nerval and he acual sreamflow for season - beng n nerval j; hs value can be esmaed usng hsorcal daa and a forecasng model. Usng hs se of noaon and he defnon of CAP as he reservor sorage capacy, s possble o formulae a chance consraned opmzaon model as shown n Equaons () hrough (9). Mnmze CAP () The sorage a he begnnng of season s superscrped wh he ndex j o ndcae ha s a funcon of he sreamflow whch acually occurred n season -. The connuy equaon, or mass balance equaon, assures ha sorage a he begnnng of one season equals he sorage a he begnnng of he prevous season plus any nflows mnus any ouflows: Subjec o: PIX; > XMIN] = a p j j a V,,j u () (5) s;+l = sl; + < - Subsung he LDR (Equaon ) n he connuy equaon (Equaon ) yelds S+l = R + b or S: = RPl + d-l. Sub-... sung n he LDR (Equaon () yelds Xy = RPl b- v (7) 8 WATER RESOURCES BULLETIN

3 The Effecs of Rsk and Relably on Opmal Reservor Desgn V (9) Consrans, 6, and 8 defne he probables of par cular release and sorage evens. For example, 0 s defned n Equaon () as he probably ha he release equals or exceeds he mnmum release n season. Smlarly, and y ; denoe he probables of sorage beng no less han he mnmum sorage for season and of freeboard (CAP- S) j beng no less han he mnmum freeboard for season. Consrans 5, 7, and 9 ensure ha he mnmal release (XMIN), sorage (SMIN), and freeboard (SFREE) are sasfed wh he desgnaed relables (or, f, T, respecvely) n each season. Consder Equaon (5). To resrc he probably of sasfyng he mnmum release (XMIN) requremen n.season, s necessary o consder all of he release rules (Xy) ha may be used and he probably of usng each n ha season. The lef-hand sde of Equaon (5) s he overall probably ha he release n season exceeds or equals XMIN; and he rgh-hand sde s he mnmal acceped value of hs probably. Consrans, 6, and 8 are no n a proper forma for npu o a lnear programmng soluon roune. The chance consran on mnmum permssble release (Equaon ) wll be used o demonsrae he procedure requred o ransform he chance consrans o deermnsc equvalens compable wh lnear programmng packages. or or P[*-l GXMIN b] = -or! V, j, () Because R-l j s a random varable, he lef-hand sde of he equaon s he cumulave dsrbuon funcon (CDF) of he sreamflows n season - condoned on he predcon (made a -) ha hs sreamflow would be occurrng n nerval j. Ths CDF, denoed by F. (.), can be wren as R:- F (XMIN -$Yl + b)= - R- or V,j, () Denong he nverse CDF by F-! (-) allows Equaon () Rl- o be wren as XMIN-b:-l+b-F-/ ( -5)=O V,j, () R- The convex poron of he nverse CDF can be pecewse lnearzed so ha he regular smplex algorhm or s common varans may be used o deermne a global opmal soluon. Houck (979b) has shown ha n he range of neres of he model, he nverse CDF s ypcally convex. The same procedure can be followed o assure a mnmum sorage resrcon (Equaon 6): The resrcon o assure a mnmum accepable freeboard (Equaon 8) s gven as CAP - bj-l - F- (+) = SFREE V j, (6) R:- The enre MLDR model comprses a lnear program wh an objecve funcon of mnmze CAP subjec o consrans 5, 7, 9,, 5, and 6. Nonnegavy resrcons apply o all varables excep he LDR decson parameers. Ths lnear program can be solved usng he regular smplex algorhm o oban a global opmum provded ha he nverse CDF s convex. SOLUTION OF THE MLDR MODEL The Mulple Lnear Decson Rule (MLDR) model descrbed prevously was consruced and solved for he Gunpowder Rver sysem n easern Maryland. Monhly sreamflow daa for he Gunpowder Rver are avalable for he years The uns are n mllon gallons per day per square mle (mgpd/sq. m.). The pernen waershed area for hs sreamflow record s 0 sq. m. correspondng o he dranage no Loch Raven Reservor. In usng he daa, all monhs are assumed o have 0. days. However, for he example problem descrbed here, a seasonal sreamflow record, whch s generaed from he monhly daa, s used n he forecasng and opmzaon modelng. Due o lmed sofware capables, only four seasons are modeled and each season s assumed o have 9.5 days. Auoregressve Movng Average, ARMA (p,q), models wh p auoregressve and q movng average parameers were fed o a poron (0 years) of he sreamflow daa by he maxmum lkelhood mehod (Box and Jenkns, 970). All of he ARMA models consruced passed sandard wheness (Anderson and Pormaneau) and perodcy (cumulave perodogram) ess. The Akake (97) creron was used o selec he bes ARMA model o use n he opmzaon model. ARMA (,O) wh he one auoregressve parameer (#I) equal o was seleced. The ARMA (,O) model was hen used o oban oneseason ahead sreamflow forecass for he remander (50 years) 9 WATER RESOURCES BULLETIN

4 Yazcgl and Houck of he sreamflow record,.e., ha poron of he record no used o calbrae he model. The acual and forecased sreamflow daa were used o generae he condonal cumulave dsrbuon funcons (Fgures hrough ). Each of he condonal CDF s was approxmaed n he model by sx lnear peces; hree peces for he 0-50 percenle poron and hree peces for he percenle poron. The number of sreamflow nervals ( or j) on whch release and sorage were condoned was wo. The sreamflow levels ha dvded he wo nervals n each season are shown n Table. The probables ha he acual sreamflows n season - were n he j-h class nerval and he forecased sreamflows for season were n he -h class nerval (.e., e) are shown n Table for all seasons. The values assgned o he relably levels for assurng he mnmum release (a), assurng he mnmum sorage volume (&), and assurng he maxmum sorage level (r) were all The mnmum sorage level (SMIN) was made equal o 0 percen of he capacy (CAP), and he mnmum seasonal release (XMIN) and mnmum freeboard (SFREE) were assgned values of 0.5 mgpd/ sq. m. (6. * 06m) and 0. mgpd/sq. m. (5.5 * 06~, respecvely Fd (6 ) = r ( mgpd Isq.m) Fgure. Condonal Cumulave Dsrbuon Funcons for Season. - F ( 6) =r ( mgpdfsq.m.) d , F I (6)= r, R, (rngpdlsq. m.) Fgure. Condonal Cumulave Dsrbuon Funcons for Season a - Fgure. Condonal Cumulave Dsrbuon Funcons for Season o - F ( 6 ) = r ( mgpd/sq.m.) R* -I TABLE. Relaonshp of Seasons o Monhs and Desgnaon of Sreamflow Inervals. Sreamflow Exceeded n Monhs Included n 50 Percen of all Season, Season Seasons JanWy-Mach.05 - mad sq. m. Aprl-June mad sq. m. July -Sepember mad sq. m. Ocober-December mkpd sq. m. Fgure. Condonal Cumulave Dsrbuon Funcons for Season. 0 WATER RESOURCES BULLETIN

5 The Effecs of Rsk and Relably on Opmal Reservor Desgn TABLE. Soluon Values of a and Coeffcen Values of P. j mnmum capacy (CAP) was obaned by solvng he lnear program. These are ploed as shown n Fgure 5. As relably levels change from 0.80 o 0.90, capacy ncreases from.57 o.985 mgpd/sq. m. (58.8 * lo6 o 07.7 * 06m). TABLE. Soluon Values ofp, d. p, TABLE. Soluon Values of b, (mgpd/sq. m.). The lnear program comprsed consrans and 6 varables, 96 of whch had fne upper bounds. Soluon me was approxmaely 5.6 CPU seconds usng he Mul-Purpose Opmzaon Sysem (MPOS) resden on a CDC 600 compuer. The oal cos per soluon was less han one dollar. However, should be noed ha he sze of he lnear program may be subsanally ncreased by ncreasng he number of seasons, he number of sreamflow nervals, and he number of sragh-lne approxmaons of he CDF s. The mnmal dam capacy requred o mee he specfed resrcons s.7 mgpd/sq. m. (8.5 * 06m). Tables,, and conan he release and sorage relables a:,, and he values of he MLDR parameers b desgnaed n he soluon of he lnear program. From examnaon of Table, s evden ha he mnmum desred release s me wh a relably exceedng or equal o 0.85 n all four seasons. Sorage levels exceed he mnmum desred sorage level wh relables equal o 0.87, 0.85, 0.89, and 0.85 n seasons,,, and, respecvely. The mnmum desred freeboard capacy s me wh relables equal o 0.9,0.85,0.87, and 0.9 for seasons,,, and, respecvely DEVELOPMENT OF TRADE-OFF CURVES I s ofen helpful o provde he decson makers wh nformaon relang performance relably levels and he requred mnmum reservor capacy n order for hem o be able o selec he bes capacy by akng no consderaon oher fnancal, polcal, and economc consrans. Several example rade-off curves were developed o llusrae her use. Frs, he relably levels (.e., a, &, r) were vared smulaneously from 0.80 o For each level, he requred..6.8.o Mnmum Requred Capacy (mgpd/sq.m.) Fgure 5. Relably Levels q, P, and T Versus Mnmum Requred Capacy. WATER RESOURCES BULLETIN

6 Yazcgl and Houck In sprng, he probably of occurrence of floods s hgher han n any oher season. Thus, a hgh relably for freeboard capacy s requred n sprng (season ). By varyng y;! from 0.85 o 0.99 and keepng y = 0.85 for seasons,, and, he rade-off curve beween he mnmum requred capacy and freeboard relably levels s obaned as shown n Fgure 6. I s clear from hs fgure ha he capacy ncreases more rapdly as 7 ncreases. Ths s an expeced resul n all real world problems of waer resources sysems. N a0 I.8.o..L.6 Mnmum Requred Capacy (rngpdisq.m.) Fgure 6. Mnmum Freeboard Relably Level for Season Versus Mnmum Requred Capacy. In summer, he demand for waer s hgher han n any oher season because of he hgh consumpon and rrgaon requremens. Thus, he reservor should be able o provde he mnmum release wh a hgh relably n summer (season ). By keepng a = 0.85 for seasons,, and, and varyng from 0.85 o 0.97, he mnmum requred capacy curve s obaned as shown n Fgure 7. The requred capacy s no sensve up o a = 0.9; hereafer, ncreases exponenally wh relably level. h v z" c.97.9 J SUMMARY AND COMMENTS A chance-consraned lnear programmng model, whch ulzes mulple lnear decson rules and whch faclaes he developmen of reservor sysem desgns and operaon polces, s used o evaluae he effecs of rsk and relably on opmal reservor desgn. The prmary aspec of he model s he explc consderaon of sreamflow forecasng. The rsk assocaed wh predcons or forecass of fuure sreamflows s ncluded n he model hrough he use of cumulave dsrbuon funcons (CDFs) of sreamflows whch are condoned on he predcons. The model was consruced and solved for a hypohecal reservor se on he Gunpowder Rver n Maryland. For hs se, a sngle dam s consdered for consrucon. I wll be used o conrol sreamflows o enhance he local waer supply, recreaon, and flood conrol. The condoned CDFs were obaned by: fng an Auoregressve Movng Average, ARMA (p,q), model o a poron of he hsorcal sreamflow record; usng he ARMA model as a forecasng ool; and comparng he forecased and acual sreamflows from he remanng poron of he hsorcal record. In order o provde he decson makers wh complee and useful nformaon, rade-off curves relang mnmum reservor capacy (drecly relaed o dam coss) and he relably of achevng waer supply and flood conrol arges are developed. The generaed rade-off curves can hen be presened o he decson maker o allow he selecon of he bes dam capacy consderng echnologcal and fnancal consrans as well as he rade-offs beween arges, rsks, and coss. The enre model developmen and generaon of resuls requred approxmaely wo person-monhs of work. A sgnfcan poron of hs me was spen on he consrucon and esng of he ARMA models. The basc MLDR model can be exended as approprae for oher reservor sysems o many oher forms by condonng he release rules and sorages on dfferen acual sreamflows and forecass made a dfferen mes. The model used heren was consruced wh releases and sorages condoned on one predcon and on one prevous season's sreamflow. I could be exended o accommodae a sequence of predcons made a dfferen mes and correspondng o varous operang horzons as more daa become avalable. The model can also be modfed o consder mulple reservor sysems. on of a Mulple-Reservor Sysem. Waer Resources Research 0(6):07-. : Box, G. E. P. and G. M. Jenkns, 970. Tme Seres Analyss: Fore o Mnmum Requred Capacy (mgpd/sq.m.) Fgure 7. Mnmum Release Relably Level for Season Versus Mnmum Requred Capacy. LITERATURE CITED Akake, H., 97. A New Look a he Sascal Model Idenfcaon. IEEE Trans. on A.C. 9(6):76-7. Becker, L. and W. W-G Yeh, 97. Opmzaon of Real Tme Opera- casng and Conrol. Holden-Day, Inc., San Francsco, Calforna. Buras, N., 966. Dynamc Programmng n Waer Resources Developmen. In: Advances n Hydroscence, V. T. Chow (Edor). Academc Press, New York, New York. WATER RESOURCES BULLETIN

7 The Effecs of Rsk and Relably on Opmal Reservor Desgn Bucher, W. S., 97. Sochasc Dynamc Programmng for Opmum Reservor Operaon. Waer Resources Bullen 7():5-. Caselon, W. F. and S. 0. Russel, 976. Long-Term Operaon of Sorage Hydro Projecs. ASCE, Journal of he Waer Resources Plannng and Managemen Dvson 0(WR):6-76. Charnes, A. and W. W. Cooper, 96. Deermnsc Equvalens for Opmzng and Sasfcng Under Chance Consrans. Operaons Research ():8-9. Hall, W. A. and J. A. Dracup, 970. Waer Resources Sysems Engneerng. McGraw-Hll Book Company, New York, New York. Hosh, K. and S. J. Burges, 979. Incorporaon of Forecased Toal Seasonal Runoff Volumes no Reservor Managemen Sraeges. In: Relably n Waer Resources Managemen, E. A. McBean, K. W. Hpel, and T. E. Unny (Edors). Waer Resources Publcaons, For Collns, Colorado, pp Houck, M. H., 979a. A Mehod o Include Rsk Explcly n Opmal Reservor Desgn. In: Relably n Waer Resources Managemen, E. A. McBean, K. W. Hpel, and T. E. Unny (Edors). Waer Re sources Publcaons, For Collns, Colorado, pp Houck, M. H., 979b. A Chance Consraned Opmzaon Model for Reservor Desgn and Operaon. Waer Resources Research 5(5): Houck, M. H. and J. L. Cohon, 978. Sequenal Explcly Sochasc Lnear Programmng Models for Desgn and Managemen of Mulpurpose Reservor Sysems. Waer Resources Research ():6-69. Houck, M. H. and B. baa, 98. Performance Evaluaon of a Sochasc Opmzaon Model for Reservor Desgn and Manag* men wh Explc Relably Crera. Waer Resources Research 7(): Jacoby, H. D. and D. P. Loucks, 97. Combned Use of Opmzaon and Smulaon Models n Rver Basn Plannng. Waer Resources Research 8(6):0-. Joeres, E. F., G. J. Seus, and H. M. Engelmann, 98. The Lnear Decson Rule (LDR) Reservor Problem Wh Correlaed Inflows;. Model Developmen. Waer Resources Research 7():8-. Karamouz, M. and M. H. Houck, 98. Annual and Monhly ReservoI Operang Rules Generaed by Deermnsc Opmzaon. Waer Resources Research 8(5): 7-. Loucks, D. P., 969. Sochasc Mehods for Analyzng Rver Basn Sysems. Cornell Unversy Waer Resources and Marne Scences Cener Techncal Repor 6, Ihaca, New York. ReVelle, C. S., E. Joeres, and W. Krby, 969. The Lnear Decson Rule n Reservor Managemen and Desgn:. Developmen of he Sochasc Model. Waer Resources Rsearch 5(): Sednger, J. R (n press). Mulple LDR Models for Screenng and Reservor Operaon. Waer Resources Research Thomas, H. A., Jr. and P. Waermeyer, 96. Mahemacal Models: A Sochasc Sequenal Approach. In: Desgn of Waer-Resource Sysems, A. Maass, e al. (Edor). Harvard Unversy Press, Cambrdge, Massachuses. Yazcgl, H., M. H. Houck, and G. H. Toebes, 98. Daly Operaon of a Mulpurpose Reservor Sysem. Waer Resources Research 9():-. Young, G. K., Jr., 967. Fndng Reservor Operang Rules. ASCE, Journal of he Hydraulc Dvson 9(HY6):97-. WATER RESOURCES BULLETIN