Reservoir Design and Operation with Variable Lake Hydrology

Transcription

1 Reservor Desgn and Operaton wth Varable Lake Hydrology Hugo A. Loácga, P.E., M.ASCE 1 Abstract: Ths research quantfes the mpact of lake evaporaton and ranfall on optmal reservor capacty and water yeld. A reservor desgn and operaton model was developed and appled to the Santa Ynez Rver basn of central Calforna, whch endures large evapotranspraton and extreme clmatc varablty. Reservor desgn and average annual water yeld were obtaned n two cases. Frst, lake evaporaton and ranfall fluxes were taken nto account n the water balance of the reservor system. Second, those same fluxes were gnored. The optmzaton-model results ndcate that n-lake hydrology plays a consderable role on estmates of optmal reservor capacty and yeld. Furthermore, results ndcate that the lack of proper consderaton of n-lake hydrology leads us to err on the sde of greater rsk. Specfcally, reservor capacty and average water release are under and overestmated, respectvely. The optmzaton model s partcularly well-suted for modelng reservor systems wth actve n-lake hydrologc fluxes and allows a varety of objectve functons to be consdered, thus, provdng flexblty n the optmzaton of reservor desgn and operaton. DOI: / ASCE :6 399 CE Database keywords: Reservor operaton; Reservor desgn; Evaporaton; Droughts; Optmzaton; Lnear programmng; Streamflow. Introducton The Santa Ynez Rver basn of Santa Barbara County, Calf. s subject to dverse, and at tmes, conflctng water uses. The Santa Ynez Rver s the man source of water for muncpaltes, agrculture, and fsheres. Water s dverted va gravty tunnels from the Santa Ynez watershed to the south coast regon of Santa Barbara County to serve the towns of Carpntera, Goleta, Isla Vsta, Montecto, Santa Barbara, and Summerland. Wthn ts watershed, the Santa Ynez Rver supports urban water use n the towns of Buellton, Lompoc, Santa Ynez, and Solvang, as well as largescale agrculture n the Lompoc Valley. Fg. 1 shows a map of the Santa Ynez Rver basn and water-resources nfrastructure wthn t. The man water mpoundng structure s Bradbury Dam at Cachuma Reservor, whch s supplemented by two small upstream dverson dams.e., Gbraltar and Juncal dams. Water s transferred from Cachuma Reservor to Santa Barbara County s south coast va the Tecolote Tunnel shown n Fg. 1. The Msson and Doulton tunnels also shown n Fg. 1 dvert water from the Santa Ynez Rver at Bradbury and Juncal dams, respectvely. Fsheres have been affected by the regulaton of streamflow caused by Cachuma Reservor. Specfcally, the southern steelhead trout Oncorhynchus mykss has been declared an endangered speces n the Santa Ynez Rver due to declnng fshpopulaton levels durng the last 4 decades. 1 Professor, Dept. of Geography, Unv. of Calforna, Santa Barbara, CA E-mal: hugo@geog.ucsb.edu Note. Dscusson open untl Aprl 1, Separate dscussons must be submtted for ndvdual papers. To extend the closng date by one month, a wrtten request must be fled wth the ASCE Managng Edtor. The manuscrpt for ths paper was submtted for revew and possble publcaton on June 5, 2001; approved on October 1, Ths paper s part of the Journal of Water Resources Plannng and Management, Vol. 128, No. 6, November 1, ASCE, ISSN /2002/ /$8.00 $.50 per page. Reservors and dverson nfrastructure n the Santa Ynez Rver were bult to regulate ts hghly varable stream flow and to convey ts water to urban areas n whch water use has rsen rapdly snce the late 1800s Eckman 1967; Loácga and Renehan Relable water supply n the presence of recurrent drought has been an elusve goal for those who depend on Santa Ynez Rver water Lawrence et al Ths paper presents a reservor desgn and operaton model for the Santa Ynez Rver system. The model takes nto account monthly water dversons, fshery flow requrements, and, n partcular, ncludes an nnovatve technque for calculatng reservor evaporaton and precptaton. A key objectve of ths paper s to show the effect that evaporaton has on reservor desgn and operaton n regons of hgh-potental evapotranspraton such as central Calforna. In addton, ths paper addresses the role of ntal reservor storage on annual water yeld and optmal reservor capacty. Our reservor desgn and operaton model s bult wth the am of flexble mplementaton and a parsmonous structure. Examples of the model s applcablty complement ts theoretcal underpnnngs. Streamflow Varablty Fg. 2 shows a tme seres of unregulated Santa Ynez Rver annual streamflow nto Cachuma Reservor from water years through The mean and standard devatons of annual streamflow are m 3 and m 3, respectvely. Shown n Fg. 2 are the droughts that occurred n that perod. Hydrologcal droughts are defned heren as perods wth four or more consecutve years of below-average annual streamflow Loácga et al. 1992, 1993; Loácga and Lepnk There were sx such events n the perod, or about one drought every 14 years. Fg. 2 shows sharp nterannual fluctuatons n streamflow. The persstence of dry runs s a matter of concern n the Santa Ynez Rver from a water-resources standpont Loácga and Renehan It s n ths context of uncer- JOURNAL OF WATER RESOURCES PLANNING AND MANAGEMENT / NOVEMBER/DECEMBER 2002 / 399

2 Fg. 1. The Santa Ynez Rver basn and major water resources nfrastructure tan water sources and growng water use that we have developed a mathematcal programmng model for the optmzaton of reservor operaton and desgn n the Santa Ynez Rver basn. Storage Dynamcs n Santa Ynez Rver Reservor System Conceptual Representaton of Reservor System Consder the schematc representaton of the reservor system n the Santa Ynez Rver shown n Fg. 3. Cachuma Reservor and Fg. 2. Santa Ynez Rver stream flow nto Cachuma Reservor, through Source: Santa Barbara County Water Agency two small dverson dams Gbraltar and Juncal are depcted along wth the varous fluxes that determne the water balance above Bradbury Dam at Cachuma Reservor. For comparson, the trbutary dranage areas above Bradbury, Gbraltar, and Juncal dams are 1106, 553, and 36 km 2, respectvely. The dversons shown n Fg. 3 represent nterbasn water transfers from the Santa Ynez Rver watershed to Santa Barbara County s south coast, across the Santa Ynez Mountans Fg. 1. For all practcal purposes, the storages behnd Gbraltar and Juncal dams are neglgble. Therefore, these two dams are treated strctly as waterdverson nodes along the Santa Ynez Rver. Lake Evaporaton and Ranfall Fg. 4 shows the annual evaporaton and the ranfall n Cachuma Reservor from through It s seen n Fg. 4 that ranfall s more varable and smaller than evaporaton. The mean annual Cachuma evaporaton and ranfall are 179 and 46 cm/year, respectvely. For comparson, the reference crop evapotranspraton n the study regon has been estmated at 126 cm/ year Davdoff et al Therefore, evaporaton depletes reservor storage at an average rate of 133 cm/year. The precptaton nto Cachuma Reservor s measured n a standard Natonal Weather Servce ran gauge located at Bradbury Dam. The reservor evaporaton equals pan evaporaton multpled by a lake coeffcent. Pan evaporaton s measured by means of a standard Natonal Weather Servce evaporaton pan stuated at Bradbury Dam. Monthly lake coeffcents for Cachuma Reservor have been calculated by the Santa Barbara County Water Agency Alroth 400 / JOURNAL OF WATER RESOURCES PLANNING AND MANAGEMENT / NOVEMBER/DECEMBER 2002

3 Fg. 3. Schematc of Santa Ynez Rver reservor system. P and E denote reservor ranfall and evaporaton, respectvely. Drawng not at scale. and Naftal For the purpose of water balance calculatons n Cachuma Reservor, the ranfall and evaporatve fluxes n volume per unt tme are calculated by multplyng ther rates n length per unt tme tmes the reservor area n length squared. The area A versus storage S functon for Cachuma Reservor s gven by the followng equaton n whch the area and storage are gven n 10 3 m 2 and 10 3 m 3 respectvely MNS Engneerngs, Inc : A 10 3 m S r (1) n whch r 2 s the correlaton coeffcent of the regresson between area and storage. Eq. 1 plays a central role n Cachuma Reservor s water balance. Monthly Water Balance n Cachuma Reservor Let us denote the Cachuma Reservor storage at the end of perod by S and the reservor storage capacty by C. Consder a tme horzon of n perods one perod 1 month n the applcaton, and let 0,1,2,...,n be the tme ndex as shown n Fg. 5. The total number of months equals the number of water years tmes 12. The perod of analyss n ths work encompasses the water years through , whch mples that n In Fg. 5, E, P, r, w, and D denote reservor evaporaton, reservor ranfall, streamflow accreton, reservor release, and water dversons durng the th month, respectvely. The Fg. 4. Measured evaporaton and ranfall n Cachuma Reservor Source: Santa Barbara County Water Agency. water-balance equaton for Cachuma Reservor, gven that the startng or ntal reservor storage s S 0 g C, where g s a fracton between 0 and 1, s as follows: S S 1 r P D 1, D 2, D 3, w E 1,2,...,n (2) n whch D 1, s the dverson from Cachuma Reservor; D 2, and D 3, are the dversons at Gbraltar and Juncal dams, respectvely; r s the streamflow nto Cachuma Reservor produced by runoff from ts entre upstream dranage area.e., equal to 1,106 km 2. Other terms n Eq. 2 were defned above. In Eq. 2, the precptaton P and evaporaton E durng the th month are calculated based on the average reservor area (A 1 A )/2, expressed n 10 3 m 2, tmes the measured ran p n meters and evaporatve rate e n meters durng the th month, respectvely. The reservor area s related to ts storage by Eq. 1. All fluxes and storages that appear n Eq. 2 are n thousands of cubc meters (10 3 m 3 ). Bradbury Dam at Lake Cachuma does not serve flood-control purposes. The rparan and flood zone downstream from and n the vcnty of Bradbury Dam s essentally grazng rangeland. Therefore, the reservor releases that keep the reservor capacty from overtoppng maxmum-storage constrants are mposed n the reservor model presented below ncorporate an mplct flood-control rule curve, whch s to avod excessve storage. Bradbury Dam, through a combnaton of spllway and sluce gate dscharges s well-equpped to control reservor level wthout concerns for downstream damages. The ssue of flood control, n any event, s more relevant for hourly real-tme operaton durng hgh-nflow events typcally assocated wth ntense ranfall that falls durng 1 to 2 days n strong El Nño years rather than for the monthly operaton tme frame adopted n ths paper. Another aspect of reservor water balance, that concerns seepage losses, s shrouded n uncertanty. The best hydrologc nformaton suggests that the net seepage losses.e., groundwater accretons mnus groundwater losses to reservor storage are neglgble compared to streamflow, precptaton, and evaporaton Alroth and Naftal Therefore, seepage losses are not consdered further n ths paper. The water-balance Eq. 2 s of a recursve nature,.e., the end-of-perod storage depends on the begnnng-of-perod storage. Takng advantage of that property and substtutng the area versus storage Eq. 1 nto Eq. 2, the Cachuma Reservor stor- JOURNAL OF WATER RESOURCES PLANNING AND MANAGEMENT / NOVEMBER/DECEMBER 2002 / 401

4 Fg. 5. Tme dagram for the reservor-analyss problem. S reservor storage at end of perod ; P ranfall nto lake durng perod ; r lake streamflow n ; D,E, w water dverson, lake evaporaton, and reservor release n month, respectvely. See text for defnton of unts. age at the end of the th month may be wrtten n terms of the ntal storage S 0 g C: S g C A m 1 C m r m D 1m D 2,m D 3m w m B n whch B m 1 R m C m T m 1,2,...,n (3) A m 1 K m (4) q m 1 q m 1 1 b 2 p v e v K v 1 b 2 p v e v K q wth K q 1 (5) q 1 K q wth q 1 K q 1 (6) R m a p m e m 1 b 2 p m e m 1 T m 1 b 2 p m e m where v m or q (7) (8) (9) In Eqs. 7 through 9, p v and e v where v m or q are the ranfall and pan evaporaton n the vth perod respectvely n meters ; a and b are the ntercept and slope coeffcents n the area versus storage Eq. 1 respectvely. The storage Eq. 3 expresses the storage at tme n terms of the ntal storage (g C) and all other reservor fluxes. In partcular, t accounts for the effect that a changng reservor area has on the evaporatve losses of lake storage. Ths has an mportant effect on reservor capacty and water yeld. Notce that the ntermedate storage values S, 1,2,...,n have been elmnated from the water balance Eq. 3, leavng n t as unknowns the storage capacty C and the reservor releases w ( 1,2,...,n). Optmzaton Model for Santa Ynez Rver System The optmzaton model can be used for any of the followng purposes these are called modelng scenaros : 1. To obtan the optmal reservor capacty gven specfed water demands and reservor releases. In ths case, the decson varable s the reservor capacty C. The cost of buldng 10 3 m 3 of reservor capacty equals K $ Loácga and Renehan 1997 ; 2. To obtan optmal reservor releases gven the storage capacty and water demands. In ths case, the decson varables are the reservor releases w, 1,2,...,n; 3. To obtan optmal reservor capacty and reservor releases gven desgn and operatonal objectves. The decson varables are C and w, 1,2,...,n. The thrd, most general, modelng scenaro was chosen n ths work. The correspondng objectve functon n ths case s to mnmze the net cost of buldng and operatng the reservor system. The net cost equals the cost of buldng reservor capacty mnus the revenue assocated wth water producton. Specfcally, n mn K C 1 G w (10) n whch G value of one unt of water release G $ per 10 3 m 3 Loácga and Renehan The objectve functon s subject to several constrants. Eq. 3 s used n the constrants that nvolve Cachuma Reservor storage. Maxmum storage, S C, 1,2,...,n: g C A m 1 C m r m D 1m D 2,m D 3m w m B C (11) Mnmum storage, S 0, 1,2,...,n: g C A m 1 C m r m D 1m D 2,m D 3m w m B 0 (12) 402 / JOURNAL OF WATER RESOURCES PLANNING AND MANAGEMENT / NOVEMBER/DECEMBER 2002

5 Table 1. Mnmum Cachuma Fsheres Releases F n 10 3 m 3 Per Month October November December January February March Aprl May June July August September Reservor releases must equal or exceed fsheres-flow requrements: w F 1,2,...,n (13) The fsheres-flow requrements F are presented n Table 1 Entrx, Inc Dversons from Cachuma Reservor D 1 and at Gbraltar and Juncal dams, D 2 and D 3, respectvely, are gven n Table 2. All decson varables.e., C and w, 1,2,...,n) are nonnegatve n the lnear optmzaton model defned by Eqs Results Reservor Capacty The optmzaton problem defned by Eqs was coded n a Mcrosoft Redmond, Wash. Excel Spreadsheet and solved wth ts mathematcal lnear programmng package Solver. Fg. 6 depcts the calculated relatonshp between the optmal Cachuma storage capacty C and the ntal storage fracton g. Two cases are presented n Fg. 6: In the frst case, lake evaporaton and ranfall effects on reservor capacty were consdered labeled wth lake hydrology n Fg. 6 ; n the second case, lake evaporaton and ranfall were neglected labeled wthout lake hydrology. It s seen n Fg. 6 that n the frst case the optmal reservor capacty ncreases wth decreasng ntal storage when the ntal storage s less than 0.4 C. In the second case, the optmal reservor capacty ncreases wth decreasng ntal storage when the ntal storage s less than 0.3 C. Otherwse, the optmal capacty s ndependent of the ntal storage n both cases. In the wth lake hydrology case, the optmal Cachuma capacty was found to be m 3 whenever the ntal storage exceeds the threshold 0.4 C. If reservor evaporaton and ranfall fluxes were gnored, the optmal capacty would be m 3. The actual capacty of Cachuma Lake s m 3. The need for a larger reservor sze assocated wth a smaller ntal storage mght be ntutve. Low ntal storage may lead to volatons of mnmum release constrants durng dry years followng the begnnng of operaton. To avod nonfeasblty of the constrant set, the model results call for nordnately large reservor capacty when the ntal storage s close to zero. Ths drastc drought-hedgng effect vanshes when the ntal storage ncreases, as shown n Fg. 6. Clearly, the start tme of reservor operaton and the specfc hydrology to whch the reservor system s subject to are crucal n the model predcton of optmal reservor capacty. Gven that most water-supply reservors do not start operaton untl they have reached at least half ther capacty, the parts of the graphs shown n Fg. 6 that are of greatest practcal nterest are those for whch the ntal storage fracton s g 0.5. In that nstance, our results ndcate a nonneglgble optmal reservor capacty underestmaton equal to m 3 attrbutable to the lack of consderaton of lake hydrology. Ths represents close to a 22% underestmaton of the optmal reservor capacty n a practcal context. It may be concluded that proper consderaton of evaporatve lake losses calls for a more conservatve reservor-desgn approach, whereby the optmal reservor capacty s larger than what would be nferred f such losses were not accounted for. Average Annual Water Fg. 7 shows the average annual release from Cachuma Reservor. Ths water has envronmental and economc value because t supports downstream steelhead habtat and meets urban and agrcultural demands. The average annual release s seen to ether reman constant or to ncrease wth ncreasng ntal storage n the case n whch evaporatve and ranfall fluxes are consdered.e., wth lake hydrology n Fg. 7 as well as that n whch they are not wthout lake hydrology curve n Fg. 7. In the former case, the average annual release s constant when the ntal storage s less than 0.36 C, whle t s constant when the ntal storage s less than 0.2 C n the latter case. Releases are kept at ther mnmums at low ntal storage and t s not feasble to decrease the total cost Eq. 10 by larger releases wthout offsettng that cost mprovement wth large ncreases n reservor capacty and thus, wth larger reservor cost. Another nterestng feature evdent n Fg. 7 s that the average annual release calculated when evaporaton and ranfall are gnored exceeds that assocated wth the case n whch they are consdered. The average annual water yeld overestmaton s 37 and 11% when the ntal storage s 50 and 100% of the reservor capacty, respectvely. It was argued above that the range g 0.5 s that of greatest practcal nterest. The magntude of the water-yeld overestmaton poses potental rsks to water management because the analyss wthout the consderaton of evaporatve losses would suggest a reservor yeld that s, n fact, unattanable. Fg. 8 summarzes the relatonshp among optmal average annual water, optmal reservor capacty, and the ntal storage represented by the fracton g of reservor capacty. It s seen n Fg. 8 that n the case n whch lake hydrology s consdered, the average annual release ncreases whle the optmal reservor storage s kept at a constant level of m 3 whenever the ntal Table 2. Water-Supply Dversons D j, j 1,2,3n10m 3 Per Month October November December January February March Aprl May June July August September Cac. a Gb. b Jun. c a Cac.: Cachuma Reservor. b Gb.: Gbraltar Dam. c Jun.: Juncal Dam. JOURNAL OF WATER RESOURCES PLANNING AND MANAGEMENT / NOVEMBER/DECEMBER 2002 / 403

6 Fg. 6. Optmal Cachuma storage as a functon of ntal storage and lake-hydrology scenaros storage s larger than 40% of the reservor capacty. The mplcaton n that nstance s that the total cost s mnmzed by keepng the reservor sze at an optmal mnmum whle ncreasng the water releases as much as reservor sze and hydrologc condtons permt t. The same pattern s observed n Fg. 8 for the case n whch lake hydrology s not consdered and whenever the ntal storage s greater than 30% of the reservor capacty, except that n that case the optmal reservor capacty s m 3, lower than that assocated wth the wth lake hydrology case. In ether case,.e., wth or wthout lake hydrology consdered, t s seen n Fg. 8 that when the ntal storage approaches zero, the reservor capacty rses rapdly to satfy mnmum release constrants whle the average annual yeld s kept at a constant mnmum level. The upward dsplacement of the wthout lake hydrology graph relatve to the wth lake hydrology one s caused n ths nstance by the overestmaton of the average annual yeld n the latter case as ponted out above. Concluson The results of ths work provde concrete evdence about the mpact that lake evaporaton has n our estmates of optmal reservor capacty and reservor yeld. A reservor desgn and operaton model was developed and mplemented to the Santa Ynez Rver basn of central Calforna, stuated n a regon of large potental evaporaton and extreme clmatc varablty. In addton, the Santa Ynez Rver s largest reservor exhbts large surface-area Fg. 8. Optmal average annual release versus optmal reservor capacty. The ntal storage fracton g s n ncrements of 0.1 begnnng wth 0.1 and ncreasng clockwse to 1.0 for the wth lake hydrology and wthout lake hydrology graphs. changes wth changng storage. The results of ths work ndcate that n-lake hydrology plays a consderable role on estmates of optmal reservor capacty and average annual water release. Furthermore, ths work s results ndcate that the lack of proper consderaton of n-lake hydrology leads us to err on the sde of greater rsk. Concretely, reservor capacty and average water release are under and overestmated, respectvely. It s worth notng that there are alternatve methods for szng reservors. The masscurve method, for example, s a classcal one Lnsley and Franzn That method s hndered by the fact that reservor precptaton and evaporaton cannot be accurately accounted for n the analyss. Nor are reservor constrants easly ncorporated n the mass-curve method ether. The optmzaton approach presented n ths paper accounts for all factors, natural or man-made, that affect reservor storage, and t s partcularly well-suted for modelng reservor system wth actve n-lake hydrologc fluxes e.g., evaporaton. It also allows a varety of objectve functons to be consdered n the analyss, and thus provdes convenent flexblty n the optmzaton of reservor desgn and operaton. Further work s needed to test the type of reservor relatonshps dentfed n ths paper n reservor systems ether larger or smaller than the Santa Ynez Rver system. Acknowledgment Ths work was supported n part by grant HQ-96-GR from the U.S. Geologcal Survey. The methods and results of ths paper are the sole responsblty of the wrter and do not mply endorsement by the fundng agency, nor does the fundng agency endorse any commercal products cted heren. References Fg. 7. Average annual water releases from Cachuma Reservor under two lake-hydrology cases Alroth, J. H., and Naftal, M Santa Ynez Rver hydrology model manual, Santa Barbara County Water Agency, Santa Barbara, Calf. Davdoff, B., Jones, D. W., Snyder, R. L., Echng, S., and Gonzales- MacPherson, H Reference evapotranspraton map of Calforna, Calforna Dept. of Water Resources, Sacramento, Calf. Eckman, J The search for water n the south coastal area of Santa Barbara County. Notcas, 13 3, Entrx, Inc Lower Santa Ynez Rver fsh management plan. 404 / JOURNAL OF WATER RESOURCES PLANNING AND MANAGEMENT / NOVEMBER/DECEMBER 2002

7 Walnut Creek, Calf. Lawrence, C. H., Stubchauer, J. M., and Alroth, J. H Changng condtons and water electons. J. Water Resour. Plan. Manage., 120 4, Lnsley, R. K., and Franzn, J. B Water resources engneerng, McGraw-Hll, New York. Loácga, H. A., Mchaelsen, J., Garver, S., Haston, L., and Lepnk, R. B Droughts n rver basns of the western Unted States. Geophys. Res. Lett., 19 20, Loácga, H. A., Haston, L., and Mchalsen, J Dendrology and long-term hydrologc phenomena. Rev. Geophys., 31 2, Loácga, H. A., and Lepnk, R. B A stochastc renewal model of low flows. Stochastc Hydrol. Hydr., 10, Loácga, H. A., and Renehan, S Muncpal water use and waterrates drven by severe drought: A case study. J. Am. Water Resour. Assoc., 33 6, MNS Engneerng, Inc Gbraltar Lake bathymetrc study. Rep. WO 11711, Santa Barbara, Calf. JOURNAL OF WATER RESOURCES PLANNING AND MANAGEMENT / NOVEMBER/DECEMBER 2002 / 405