DERIVED OPERATING RULES FOR RESERVOIRS IN SERIES

Transcription

1 DERIVED OPERATING RULES FOR RESERVOIRS IN SERIES OR IN PARALLEL By Jay R. Lud 1 ad Joel Guzma 2 ABSTRACT: Ths paper revews a varety of derved sgle-purpose operatg polces for reservors seres ad parallel for water supply, flood cotrol, hydropower, water qualty, ad recreato. Such rules are useful for real-tme operatos, coductg reservor smulato studes for real-tme, seasoal, ad log-term operatos, ad for uderstadg the workgs of multreservor systems. For reservors seres, several addtoal ew polces are derved for specal cases of optmal short-term operato for hydropower producto ad eergy storage. For reservors parallel, addtoal ew specal-case rules are derved for water qualty, water supply, ad hydropower producto. New operatg polces also are derved for reservor recreato. INTRODUCTION Despte the developmet ad growg use of optmzato models (Labade 1997), the vast majorty of reservor plag ad operato studes s based predomatly or exclusvely o smulato modelg ad thus requres tellget specfcato of operatg rules. Practcal real-tme operatos also usually requre the specfcato of reservor operatg rules. These rules determe the release ad storage decsos for each reservor at each tme-step durg the smulato ad help gude reservor operators (Bower et al. 1966; Hufschmdt ad Ferg 1966). Ths paper revews a varety of commo ad ewly derved operatg rules for sgle-purpose reservors seres ad parallel. These derved rules ca all be supported by coceptual or mathematcal deducto from prcples of egeerg optmzato for specal cases. These rules ca be cotrasted wth the may ad ofte hghly effectve emprcally based rules commo practce, such as varous poolbased rules ad balacg rules ( Reservors 1977; Wurbs 1996; Nalbats ad Koutsoyaa 1997). The rules examed here are teded mostly for seasoal ad log-term studes. Real-tme studes, wth a hourly or daly tme-step, ofte have more detaled safety, habtat, ad faclty lmtatos ot usually mportat for studes usg coarser tme-steps or loger operatg horzos. The rules preseted here offer some gudace for real-tme operato but geerally are more applcable to seasoal ad log-term operatos plag ad modelg studes. Desrable operatg rules for reservor systems wth mxed purposes mght have very dfferet forms from those preseted here. Prevous revews of reservor operatg rules clude Sheer s fe cocse revew (1986) ad Loucks ad Sgvaldaso (1982). Several U.S. Army Corps of Egeers (USACE) reports (cted below) also developed reservor operatg rules for specfc purposes. The work preseted here s draw from work doe for the USACE (Lud ad Guzma 1996). For those developg operatg polces for real reservor systems, the rules preseted are part of a bag of trcks wth some theoretcal or practcal bass to recommed them. The rules dscussed here are orgazed by reservor cofgurato, seres or parallel, ad by varous operatg purposes. Operato for multple purposes (.e., most real systems) re- 1 Prof., Dept. of Cv. ad Evr. Egrg., Uv. of Calfora, Davs, CA E-mal: jrlud@ucdavs.edu 2 Formerly, Udergrad. Studet, Dept. of Cv. ad Evr. Egrg., Uv. of Calfora, Davs, CA. Note. Dscusso ope utl November 1, To exted the closg date oe moth, a wrtte request must be fled wth the ASCE Maager of Jourals. The mauscrpt for ths paper was submtted for revew ad possble publcato o August 5, Ths paper s part of the Joural of Water Resources Plag ad Maagemet, Vol. 125, No. 3, May/ Jue, ASCE, ISSN /99/ /$8.00 $.50 per page. Paper No qures some combato of these rules or use of other operatg rule forms (Lud ad Guzma 1996). The presumpto may of these rules s that a system of reservors ca be operated to produce greater beefts tha operatg the dvdual reservors depedetly. Whle ths s ofte the case (Palmer et al. 1982), t s ot always the case (Needham 1998). Ths paper begs wth examato of rules for reservors seres for specfc sgle purposes. Varous sgle-purpose rules for reservors parallel are the revewed. A geeral storage-effectveess-based rule for recreato storage s the preseted, followed by a short geeral dscusso of specfcato of operatg rules by stochastc dyamc programmg (SDP). The coceptual rules are summarzed Tables 1 ad 2. A commetary ad coclusos ed the paper. Dervatos of selected rules appear appedces. Pseudocode for mplemetg may of these rules appear Lud ad Guzma (1996). RULES FOR RESERVOIRS IN SERIES Ths paper begs wth examato of operatg rules for reservors seres, llustrated Fg. 1, for water supply storage, flood cotrol, eergy storage, ad hydropower producto. Most rules preseted are assembled from earler cted work. Ths work s exteded some cases. The coceptual rules for reservors seres are summarzed Table 1. Water Storage Rules For reservors seres provdg water supply, a reasoable objectve s to maxmze the amout of water avalable, whch s the same as mmzg splled water. The resultg rule for sgle-purpose water supply reservors seres s smply to fll the hghest reservors frst ad the lowest oes last (Sheer 1986). The lkelhood ad severty of shortages s reduced by prevetg ay water from leavg the system as ucotrolled ad uproductve splls. For reservors seres, wth termedate flows, the probablty of spll from the system s mmzed by frst fllg the uppermost reservors ad retag storage capacty the lower reservors to capture potetally large flows ad reduce the lkelhood of splls from the system. Spllage from ay but the lowest reservor ca the be captured by a lower reservor. Durg the drawdow seaso, where system flows are less tha demads, the system should be draw dow order of the dowstream reservors frst, to provde storage to accommodate potetally excess termedate flows or a early oset of the refll seaso. For reservors seres servg water supply as a sole purpose, the above rule seems uversal. A excepto mght be where hgher reservors suffer hgher rates of water loss from evaporato ad seepage (Kelley 1986). I ths case, ay creased evaporato or seepage from hgher reservors would JOURNAL OF WATER RESOURCES PLANNING AND MANAGEMENT / MAY/JUNE 1999 / 143

2 TABLE 1. Coceptual Rules for Reservors Seres a Purpose (1) Water supply Refll (2) Fll upper reservors frst Fll upper reservors frst Fll upper reservors Seaso/Perod Drawdow (3) Empty lower reservors frst Flood cotrol Empty lower reservors frst Eergy storage Empty lower reservors frst frst Hydropower producto Maxmze storage Maxmze storage reservors wth reservors wth greatest eergy productoducto greatest eergy pro- Recreato Equalze margal recreato mprovemet of addtoal storage amog reservors a Exceptos ad refemets are dscussed text. TABLE 2. Purpose (1) Water supply Flood cotrol Eergy storage Water qualty Hydropower producto Recreato Coceptual Rules for Reservors Parallel a Refll (2) Equalze probablty of seasoal spll amog reservors Leave more storage space reservors subject to floodg Equalze EV of seasoal eergy spll amog reservors Equalze EV of margal seasoal water qualty spll amog reservors Seaso/Perod Maxmze storage reservors wth greatest eergy producto Equalze margal recreato mprovemet of addtoal storage amog reservors Note: EV = expected value. a Exceptos ad refemets are dscussed text. b Not applcable. Drawdow (3) Equalze probablty of emptyg amog reservors b For last tme-step, equalze EV of refll seaso eergy spll amog reservors For last tme-step, equalze EV of refll seaso water qualty spll amog reservors Maxmze storage reservors wth greatest eergy producto Equalze margal recreato mprovemet of addtoal storage amog reservors have to be weghed agast the creased potetal for loss due to spll from cocetratg storage at lower elevatos. Flood Cotrol Rules For reservors seres wth termedate flows ad storage servg solely for flood cotrol dowstream, t s optmal to regulate floods by fllg the upper reservors frst ad emptyg the lower reservors frst. The objectve s to mata as much cotrol over flows eterg the system above a crtcal flood-proe reach as possble. Flood storage at the reservor closest upstream from a crtcal flood cotrol reach always provdes greater flood cotrollablty tha for ay other reservor (Marë et al. 1994). These are typcally the lowest reservors the seres. Thus, for sgle-purpose flood cotrol storage a seres of reservors, t s best to fll the hgher reservors frst ad empty the lower reservors frst. A llustrato of ths rule ca be foud a large Brazla multreservor system (Kelma et al. 1989). FIG. 1. Reservors Seres A excepto to ths rule ca be where the outflow capacty of the lower reservor s restrcted. Here, t ca be better to fll the lower reservor frst to crease head o the outlet, thereby creasg release capacty from the etre system to the dowstream chael capacty ( Flood 1976). Ths allows hgher prereleases to crease total storage avalable for a comg flood evet. The operato of reservors seres for flood cotrol s farly complemetary wth water supply operatos, at least regard to the preferred locato of storage. The mateace of water supply storage stll preys o the absolute flood cotrol capablty of a system, ad vce versa. The USACE ( Flood 1976) preseted methods for allocatg flood cotrol space reservors seres wth flood cotrol locatos both dowstream of the system ad betwee reservors. I such cases, except for small flood evets, there are lkely to be heret trade-offs protecto of these stes from dfferet storage allocato ad operatos decsos. Hydropower Rules Hydropower rules for reservors seres vary betwee refll ad drawdow seasos or perods. Durg a refll perod, the problem usually s to maxmze the storage of eergy at the ed of the perod. Durg a drawdow perod, the objectve s to maxmze hydropower producto for a gve total storage amout. Dfferet rules are employed for each perod. A dffcult problem s the trasto betwee seasos or perods. Eergy Storage Rules The objectve of the eergy storage rule for reservors seres s to maxmze the total eergy stored at the ed of a refll seaso or perod. Here, the refll seaso s defed as the seaso whe system flows exceed those eeded to meet water supply or hydropower producto demads. The eergy storage rule for reservors seres s to always fll the upper reservors frst. To maxmze the eergy stored for a future tme, water storage s preferred upstream reservors. Water stored at hgher elevatos has a hgher eergy cotet (klowatt-hours/ut volume of water stored) tha water stored at lower elevatos. Ths s partcularly true for water stored reservors seres, where water evetually released from upper reservors geerates hydropower at the lower reservors as well. Ay splls from upper reservors are avalable for capture space avalable lower reservors. Kelma et al. (1989) mathematcally examed the allocato of eergy storage ad flood cotrol storage capacty complex multreservor systems. Ther results wll ofte dcate the compatblty of the desrable dstrbuto of eergy ad flood cotrol storages such twopurpose systems. Fortuately, eergy storage ad water supply storage rules for reservors seres are qute compatble for the refll sea- 144 / JOURNAL OF WATER RESOURCES PLANNING AND MANAGEMENT / MAY/JUNE 1999

3 so, at least terms of where storage s preferred the system ad ther geeral tet to accumulate the maxmum amout of water. However, wth the comg of the drawdow seaso, hydropower producto rules are requred. Hydropower Producto Rule Whe t comes tme to produce eergy, rules to maxmze hydropower producto may be employed. Upper reservors geerate hydropower by releases that cosequetly also crease dowstream power geerato by creasg heads, f stored dowstream, or by subsequet turbe releases dowstream. The steady-state hydropower producto rule attempts to maxmze hydropower geerato durg a sgle tme-step, gve a total storage target for the system, prmarly durg the drawdow seaso. Ths problem volves allocatg a gve total storage to maxmze hydropower producto. I geeral, the rule favors allocato of storage to those reservors that create a hgher head per ut volume of storage, have hgher geerato effceces, ad have hgher releases, as hydropower producto s the product of head, effcecy, ad release. A varat of ths rule s the USACE s storage effectveess dex preseted Appedx I (Egeerg 1985; Lud ad Guzma 1996). Ths rule typcally apples to a drawdow perod or seaso. The maxmum amout of power that ca be geerated a reservor system occurs whe head levels all reservors are at ther hghest. Where the total amout of water stored the system s lmted, perhaps due to other operatg purposes or varatos hydrology ad demads, the problem the becomes oe of allocatg a lmted amout of storage amog the dvdual reservors to maxmze hydropower producto. Ths hydropower maxmzg water storage allocato depeds o reservor capactes, flows, effceces of eergy producto, ad the total amot of water (or eergy) to be stored. Water ofte s stored frst smaller reservors, where head usually creases more per ut volume of addtoal storage tha most large reservors. Ths s llustrated Fg. 2, where the volume of water eeded to crease head by a amout x the lower reservor, V2, s much less tha that eeded to acheve a equvalet crease head for the upper reservor, V1. Aother cosderato s release flow rate. All else beg equal, hydropower producto s maxmzed by allocatg avalable stored water to reservors wth the greatest release rates. Thus, for reservors seres wth termedate flows, t s ofte desrable to maxmze storage the lower reservors frst. Typcally, dowstream reservors receve more drect ad drect flows tha upstream reservors. Storage dowstream reservors s therefore ofte kept at hgh levels to take advatage of creased flows (wth some creased chace of eergy splls). Fally, reservors wth hgher geerato effceces should be mataed at hgher levels of storage at the expese of reservors wth lower effceces. The combato of reservor capacty, amout of total flows, ad power geerato effceces determes the overall potetal of the reservor to produce power, assumg all reservors have adequate turbe capacty. Whe reductos storage are ecessary, they are made from reservors wth the least ablty to produce power. Coversely, f a crease storage ca be made, t should be reservors wth the greatest ablty to produce power. The teracto of these factors s examed mathematcally Appedx II. The result s to calculate the followg rato for each reservor at each smulato tme-step: j j=1 V = ae I (1) where V = creased power producto per ut crease storage; a = ut chage hydropower head per ut chage storage (the slope of the head-storage curve); e = power geerato effcecy of reservor ; ad I j = drect flows ad releases to reservor j, for all reservors upstream of reservor. Here reservor 1 s the uppermost reservor the seres of reservors. For ths steady-state hydropower producto rule, reservors are ordered terms of ther values of V ad are flled from hghest to lowest values of V utl the total water storage target s met. For drawdow perods, these rules ca be employed for hydropower producto systems of reservors parallel, seres, as well as mxed systems. Where the total storage costrat s desred to be terms of eergy storage, rather tha water storage, the more elaborate lear programmg (LP) approach preseted Appedx II s the requred. May systems of reservors seres mata ther lower, smaller reservors full for hydropower producto. Ths s the case for the Mssour Rver System ad the Columba Rver System. Ofte, matag hgh storage levels the lower reservors also ads avgato through the lower reaches of the reservor system, as the lower Columba Rver System. The hydropower producto rule for reservors seres ca be mplemeted drectly usg the values of V preseted above, or by usg the more elaborate LP formulatos of the problem preseted Appedx II. Ths rule should work best for dvdual tme-steps uder early steady-state codtos, where small chages storage are atcpated. A more dyamc ad geeral verso of ths smple rule s dscussed by Lud (upublshed paper,.d.), based o a cocept by Sheer (1986). RULES FOR RESERVOIRS IN PARALLEL The operato of reservors parallel (Fg. 3) dffers from reservors seres that dowstream reservors caot be used to capture addtoal water from uderestmated flows or beeft from the trasfer of water stored upstream f flows are overestmated. The followg sectos preset balacg rules for water supply, eergy storage, water qualty, ad flood cotrol. These rules typcally apply to the reservor system s refll seaso. Computatoal studes suggest that these rules ted to work rather well over a wde rage of codtos, perhaps because the performace respose surface s flat for such stor- FIG. 2. Chage Head wth Varyg Capactes FIG. 3. Reservors Parallel JOURNAL OF WATER RESOURCES PLANNING AND MANAGEMENT / MAY/JUNE 1999 / 145

4 age allocato decsos (Sad 1984). Several hydropower producto rules also are preseted. Coceptual rules for operatg parallel reservors are summarzed Table 2. Water Supply, Eergy Storage, ad Water Qualty Rules Several types of rules have bee developed for water supply, eergy storage, ad water qualty operatos for parallel reservors durg refll perods. For reservor systems provdg ether water supply or eergy producto, a reasoable objectve s to mmze expected shortages. The severty of shortages s reduced by avodg ay water leavg the system as ucotrolled ad uproductve splls (Sad 1984). These rules prescrbe deal release or storage levels for reservors parallel to avod the effcet codto of havg some reservors full ad spllg, whereas other reservors have uused storage capacty (Bower et al. 1966). These rules are derved Appedx III. New York Cty (NYC) Rules The NYC rules use the probablty of splls rather tha the drect amouts of physcal spll the mmzato of expected shortages. Whe the probabltes of spllg at the ed of the refll seaso are the same for each reservor, t follows that physcal spll also s mmzed (Appedx III). Water supply shortfall s cosequetly mmzed as well. The NYC rule was frst stated by Clark (1950) for the NYC water supply system, I operatg ths system a attempt s made to have the storage each of the watersheds, at all tmes, fall o the same percetage year. The draw from each reservor s adjusted to equalze the probablty of refll by the ed of the refll seaso, about Jue 1 (Clark 1956). There are three requremets for rgorous applcato of the NYC rule as follows (Sad 1984): 1. The system cotas reservors operatg parallel. 2. The system provdes for a sgle demad dowstream of all reservors. 3. Expected shortages are to be avoded or mmzed. Modfed forms of the NYC rule also ca hadle stuatos where the ut value of water vares betwee reservors but s costat ay dvdual reservor. I terms of water supply, the qualty of water mght affect ts ut value. For example, hgher total suspeded solds cocetratos correspod to greater treatmet costs ad lower desrablty as a water supply source. For eergy producto the value of water s based o the maxmum head of each reservor, so that water cotaed reservors wth greater head has proportoally greater value. Applcato of the NYC rule depeds o predcted flows. Thus greater accuracy these predctos should yeld better results (fewer splls). Because releases are recalculated at each perod, a hgh degree of accuracy predcted flows s ot crtcal the early perods of the refll seaso. Toward the ed of the refll seaso, relable flow forecasts become more mportat as the chaces of spll crease (Bower et al. 1966). Therefore, t s mportat to have eough hstorcal ad watershed data for probablstc streamflow forecasts. Optmalty also depeds o the coeffcet of varato of mea mothly flows ad the correlato betwee flows o adjacet streams (Bower et al. 1966; Sad 1984). The NYC rule has bee foud to behave optmally or ear-optmally for a wde varety of operatg codtos ad system cofguratos (Sad 1984). The geeral form of the NYC rule equates the probabltes of spll at the ed of the refll seaso adjusted by the ut value of water for each reservor h Pr[CQ K S ]=, (2) f where h = ut value of water reservor ; CQ = cumulatve flow to reservor from the ed of the curret perod to the ed of the refll seaso; K = storage capacty of reservor, assumed to be the same every perod; S f = ed-of-perod storage for the curret perod for reservor ; ad = costat across all reservors parallel. The values of h deped o water qualty ad eergy storage ssues as descrbed later. Hstorcal data are typcally used to establsh the cumulatve flows CQ. Idvdual reservor releases for the curret perod R are foud by takg the tal storage S 0 plus expected flow for the curret perod E[Q ] ad subtractg the ed-of-perod storage S f that satsfes (2): R = S E[Q ] S (3) 0 f such that the sum of releases equals a curret-perod total dowstream release target R OT =1 R = R OT (4) Ths ofte requres a search over to fd the proper total release. Water Supply Whe the ut value of water h s the same amog reservors provdg water supply, h s corporated to the costat ad thus drops from the equato Pr[CQ K S ]=, (5) f Water Qualty Whe the qualty of water vares betwee reservors, such as varyg total dssolved or suspeded solds cocetratos the probabltes of spll are adjusted by h, the margal value of water use mus ts treatmet cost for each reservor. Thus, f the margal value of treated water use dowstream s $500/ acre-ft ad the cost of treatmet for water from reservor 1 s $50/acre-ft, h 1 = $450. Such a formulato mght also be appled to provde a balacg rule that preferetally stores water for dowstream fsh flows. Eergy Storage For eergy storage applcatos, the probabltes of the potetal eergy of spll are equated. Therefore the probabltes of spll are adjusted by h (2), the full head level of each reservor, represetg the margal value of eergy lost from spll. The NYC space rules apply to the refll seaso of systems of parallel reservors ad attempt to mmze the expected value of splled water. The prmary dffcultes are specfcato of flow probabltes, computatoal mplemetato of the rule (ow a mor problem), ad potetally the absece of cosderg future refll seaso demads o the flows to the system. Space Rule The space rule seeks to leave more space reservors where greater flows are expected, or where greater potetal eergy of flows are expected the case of eergy storage (Bower et al. 1966). It s a specal case of the NYC rule, seekg to mmze the total volume of splls. The same codtos for applcablty ad optmalty apply. The NYC rule becomes the space rule whe the dstrbutoal forms of flows to each parallel reservor are the same, wth dstrbutos scaled by ther expected value [.e., the dstrbuto f (CQ /EV(CQ )), 146 / JOURNAL OF WATER RESOURCES PLANNING AND MANAGEMENT / MAY/JUNE 1999

5 where EV( ) s the expected value operator, s detcal for all reservors] (Sad 1984). Ths dervato appears Appedx III. The advatage of the space rule over the NYC rule s ts drect computato. Lke the NYC rule, the spll mmzg objectve mples that ths rule s applcable to the system s refll seaso. Lke the NYC rule, the space rule has bee foud to behave optmally or ear-optmally for a wde varety of operatg codtos ad system cofguratos (Sad 1984; Wu 1988). Johso et al. (1991) examed the applcato of space rules for operatg the Cetral Valley Project Norther Calfora. I ths system, power output s maxmzed whle matag hgh levels of water supply relablty. Ths applcato appeared to offer mprovemets over the smulated operato of the system. The partcular form of the equal rato space rule depeds o the reservor purpose beg examed for the system. Space rules have bee developed for water supply storage ad eergy storage purposes. Water Supply For water supply purposes, mplemetg the space rule cossts of settg target storages each reservor so that the rato of space remag at the ed of the curret perod to the expected value of remag refll seaso flow for each reservor s detcal (Johso et al. 1991). Ths s expressed mathematcally as follows: wth f =1 K V K S =, (6) EV(CQ ) EV(CQ ) =1 V = (S EV(Q )) R (7) 0 OT =1 where V = total water storage of the system at the ed of the curret tme-step ad all other terms are as defed the NYC rule. Usg the above equato, releases a parallel system of reservors cotag equally valued uts of water are determed smlarly to mplemetg the NYC rule as follows: S f = K EV(CQ ) (8) =1 =1 K V EV(CQ ) Eergy Storage Whe prevetg eergy splls, the space rule equato used for water supply s modfed by replacg reservor capactes, avalable storage, ad expected flows wth ther potetal eergy couterparts. These cosst of maxmum potetal eergy that ca be stored or the capacty of the turbes, avalable potetal eergy storage, ad potetal eergy of expected flows, respectvely (Johso et al. 1991). The substtuto of these elemets yelds the followg equato: f =1 KE E EV(CE ) KE E =, (9) EV(CE ) =1 where KE = maxmum eergy cotet of reservor ; E f = target eergy cotet of reservor at the ed of the curret tme-step; CE = cumulatve eergy flow to reservor ; E = total target eergy cotet of the reservor system at the ed of the curret tme-step; ad EV( ) = expected value operator. If releases or storages fall outsde ther permtted upper or lower bouds, the decso varables ca be set to those bouds whle the remag varables are balaced accordg to the space rule (Stedger et al. 1983). Johso et al. (1991) used a quadratc program to mplemet ther eergy space rule for each tme-step for ther Calfora Cetral Valley Project smulato model. Space rules are smpler to mplemet tha NYC rules. However, they rest upo dstrbutoal assumptos that mght ot always hold. The mportace of these dstrbutoal assumptos ca be tested for partcular stuatos usg log-term smulato modelg. LP-NYC Rule Smlar to the two prevous balacg rule forms, the LP- NYC rule also reles o hstorcal data to determe expected flows. However, rather tha producg cumulatve flow dstrbutos for CI, the flow data are etered drectly to a LP. Spll, or the value of spll, s mmzed by cosderg all dvdual cumulatve flows from each perod to the ed of the refll cycle past years. Compared to the NYC rules, the prmary advatage of LP-NYC rules s ts ablty to corporate other (lear) short-term reservor operato costrats to the rule. Such addtoal costrats mght clude mmum or maxmum flows dowstream of each reservor or requred dversos below a subset of reservors. The same codtos requred for the NYC rule apply to the LP-NYC rule. LP-NYC rules are slghtly more geeral tha NYC rules that they ca also corporate other lear operatg costrats the settg of short-term storage targets for each reservor ad could also corporate other operatg purposes the objectve fucto (Johso et al. 1991). However, mplemetato of LP-NYC rules requres greater computatoal effort. For each tme-step, the lear program descrbed Appedx III would be solved. The values of h the formulato deped o whether water supply, water qualty, or eergy storage ssues are beg cosdered. Thus, water supply, water qualty, ad eergy storage versos of the LP- NYC rule ca be developed. Flood Cotrol Balacg Rule The approach take for flood cotrol parallel reservors s to mata balace betwee reservors terms of occuped capactes ad flood ruoff from draage areas. If a reducto outflows s requred, t s made from the reservor wth the least percetage occupacy or smallest flood ruoff. Whe a crease releases s possble, t s made from the reservor wth the greatest capacty occuped or where relatvely hgher flood ruoff s occurrg. Hgher releases from reservors recevg greater flood ruoff may thus be couterbalaced by reducg releases from reservors recevg lesser ruoff (Ghosh 1986). The tet of the flood cotrol balacg rule s to operate the parallel reservors to balace the amout of flood cotrol storage avalable, whle maxmzg udamagg releases from the system. Although the prcple of balacg flood cotrol storage o parallel reservors should be clear, the operato used to meet ths objectve s ot a exact oe. If the objectve were to mmze the expected value of damagg splls above the dowstream chael capacty, the flood cotrol rules could be developed aalogous to the NYC rules. Ufortuately, the objectve of flood cotrol s more lkely to be mmzato of peak dowstream flood flows durg the refll seaso, where peak flows to the system ca arrve durg a JOURNAL OF WATER RESOURCES PLANNING AND MANAGEMENT / MAY/JUNE 1999 / 147

6 very short tme. Ths stuato s less rgorously represeted by the NYC rule approach. The USACE ( Flood 1976) suggested the followg method for allocatg flood cotrol space betwee two parallel reservors (A ad B) wth a sgle dowstream flood damage locato: 1. Route the project desg flood ad other observed floods wth a maxmum amout of ruoff occurrg above reservor A ad wth maxmum odamagg releases from reservor A. Allow reservor B to make the remag releases, up to the maxmum odamagg level. Plot the space requred at reservor A versus total space requred. 2. Perform the same exercse for reservor B, wth the maxmum desg ad observed flood flows eterg above reservor B. Plot the space requred at reservor B versus total space requred. 3. The rato for balacg flood storage betwee the two reservors should le betwee these two curves. Whle the cocept of a flood cotrol balacg rule appears coceptually soud, exact formulatos rema uclear. Water Supply Drawdow For drawdow of water supply reservors parallel, Wu (1988) suggested a rule that equalzes the probablty of each reservor beg empty at the ed of the drawdow seaso. For systems of reservors parallel wth sde demads, depedg o releases from a specfc dvdual reservor, such a operato s meat to avod the possblty of havg to reduce a sde demad whe water s avalable to meet other demads. Wu fashoed a drawdow rule alog these les smlar to the space rule. I smulato tests, he foud a combato of space rule ad ths drawdow rule to be smple to mplemet ad provde ear-optmal performace relatve to other rule forms. Hydropower Producto Rules for Reservors Parallel Steady-State Hydropower Producto Rule For steady-state hydropower producto at reservors parallel, the hydropower producto rule for small tme-steps derved Appedx II reduces to m max P = easi (10) j j j j j=1 where the subscrpt j refers to a dvdual parallel reservor; ad varables are defed as for (1). Takg the frst dervatve P/S j = e j a j I j = V j, or the hydropower storage effectveess of parallel reservor j. The resultg rule s as follows: Empty parallel reservors sequetally, begg wth those wth the smallest V j. Fll the reverse order. Power Producto ad Eergy Drawdow Rules for Parallel Reservors Sheer (1986) provded a rule for drawg dow reservors wth the mmum mpact o log-term hydropower producto (e.g., lost potetal eergy) for reservors that wll fll before they empty. Reservors whch wthdrawal results the smallest reducto potetal eergy should be draw dow frst. Ths rule s derved ad elaborated by Lud (upublshed paper,.d.). Some of these results for parallel reservors appear below. Assumg the reservors do ot empty before they refll, the most effcet drawdow of water volume from the system would mmze total reducto aual hydropower producto. The margal ecoomc value of hydropower release s calculated for each reservor as follows: z H (S 0) = e P0H(S 0) PR r PH(K f ) (11) T T where e = effcecy of power geerato at reservor ; T = volume of storage to be released from reservor the preset tme-step; H (S 0 ) = expected flow-weghted hydropower head for reservor utl refll wth preset reservor storage S 0 ; H (K ) = hydropower head wth storage at refll capacty K ; P 0 = preset prce of eergy; P r = flow-weghted average prce of eergy expected utl the reservor reflls; P f = expected prce of eergy whe the reservor s flled; ad = margal proporto of addtoal storage the preset tme-step that would ot be splled durg the refll seaso for reservor (f =0.9, 10% of ay addtoal storage ow s expected to be splled). The rule the s to draw dow reservors wth the greatest values of z/t frst ad to refll them the reverse order. (Note that the last two terms are egatve ad a postve z/ T dcates creased hydropower value wth creased release.) Ths rule s partcularly applcable where the wthdrawals are beg made to supply some dowstream water supply volume requremet. If the curret drawdow s teded to supply a eergy demad or cotract, the above rule s the modfed somewhat. The most effcet drawdow of potetal eergy from the system would mmze total reducto aual hydropower producto. The margal ecoomc value of eergy release from each reservor s estmated, z/e, where E = H (S 0 )e T. Ths leads to the followg: z P rr H (S 0) PH(K f ) = P0 (12) E H(S ) T H(S ) 0 0 The rule the s to draw dow reservors wth the greatest values of z/e frst ad to refll them the reverse order. Both hydropower producto rules should apply well where the reservors refll most years ad do ot empty. Uder these crcumstaces, eergy splls mght be commo uless suffcet turbe flow capacty exsts to pass commo hgh refll-seaso flows. Thus, the coeffcet ca be mportat. Alteratve rules are developed for systems where reservors are expected to empty before they refll (Lud, upublshed paper,.d.). Because the value of hydropower producto ofte vares seasoally, these formulatos also allow cosderato of relatve eergy prces dfferet perods. STORAGE ALLOCATION FOR RESERVOIR RECREATION Where reservor recreato s the predomat purpose for operatos durg a tme-step, how should a gve total storage be allocated amog reservors? For dvdual reservors, recreato potetal usually vares dscotuously aroud storage levels correspodg to the elevatos of docks, boat ramps, ad beaches. However, wth these rages reservor recreato potetal s lkely to be roughly proportoal to reservor surface area A, perhaps weghted by some costat represetg accessblty or recreatoal facltes r at each reservor. Thus, the systemwde reservor recreato objectve s to maxmze total weghted surface area over all reservors subject to max A = r A(S ) (13) T =1 148 / JOURNAL OF WATER RESOURCES PLANNING AND MANAGEMENT / MAY/JUNE 1999

7 =1 S = S (14) Solvg ths problem usg Lagrage multplers gves the codto that for ay two reservors ad j A (S ) A j(s j) r = r j (15) S S or, each reservor s margal storage cotrbuto to recreato should be equal. Ths becomes a storage allocato rule for a arbtrary system of reservors predomatly used for reservor recreato. Because reservor surface area s usually a cocave fucto of storage, the rule usually should roughly balace storage amog all reservors, skewed by the weghtg factor r. Ths type of storage effectveess rule s lkely to be of greatest use durg the drawdow seaso, whe storage s decreasg ad the dstrbuto of remag storage becomes a mportat recreatoal ssue. Durg the drawdow seaso, operatos for hydropower ad recreato mght be made more compatble by varyg the developmet of reservor recreato facltes ad access. Ths would vary r to make the outcomes of hydropower ad recreato rules more closely agree. SDP-BASED RULES Rules for operato of multreservor systems ca also be developed by SDP. Here, a explct characterzato of streamflow probabltes s used together wth a explct loss fucto ad defto of system cofgurato ad costrats to umercally derve optmal reservor operatg polces. Ths approach has log bee explored ad developed (Lttle 1955; Tejada-Gubert et al. 1995; Km ad Palmer 1997). Varous formulatos have bee developed to clude probablstc streamflow forecasts ad mprovemets ad approxmatos to mprove computatoal speed. SDP approaches to dervg operatg rules are rgorous ad coceptually flexble. However, they suffer from extreme computatoal demads for large problems ad requre explct probablstc characterzato of umpared streamflows. Attempts to reduce computatoal requremets by creasg the coarseess of storage ad flow dscretzatos lead to results beg more approxmate (Klemes 1977). I most cases, t s also dffcult to have cofdece a specfc explct probablstc characterzato of a rego s flows, ad there seems to be a lmted varety of approaches avalable for represetg streamflow relatoshps wth a SDP format. SDP-based rules cotue to show creasg promse but rema somewhat prohbtvely dffcult to apply practce. A few tests of SDP versus other operatg rules have bee performed (Karamouz ad Houck 1987; Wu 1988; Johso et al. 1991). These tests show that well-crafted covetoal operatg rules typcally perform early as well, ad sometmes better, tha SDP-based rules. COMMENTARY Refll versus Drawdow Perods The rules preseted here apply mostly to ether perods of refll or drawdow. NYC ad space rules for reservors parallel ad water supply, eergy storage, ad flood cotrol rules for reservors seres typcally apply to refll seasos. Hydropower producto rules apply mostly to drawdow perods. Wth each type of perod, t s relatvely easy to determe desrable sgle-purpose operatg polces for the system. It s far more dffcult to determe operatoally whe these perods beg ad ed, ad thus how operatos should make the trasto betwee these perods. Rules for determg whch seaso apples ca be based o tme of year or streamflow j forecasts ad evaluated usg hstorcal probabltes, explct optmzato, mplct optmzato, ad/or hstorcally-based smulato studes. Forecastg s lkely to be mportat here. Multpurpose ad Geerc Multreservor Rules Most reservor systems are multpurpose. Fortuately, may of the operatg rules preseted here show compatblty betwee dfferet reservor purposes, such as flood cotrol, water supply, ad eergy storage for refll perods o reservors seres. I complex cases, optmzato approaches ad tradtoal smulato may be used to calbrate the rule forms detfed above or parameterzed smplfcatos of such multreservor rules. A wde varety of parameterzed forms of these ad other multreservor operatg rules are avalable, cludg the USACE s hghly flexble dex-level method for allocato of total storage amog multple reservors ( Reservor 1977; Nalbats ad Koutsoyas 1997). Tradtoally, teratve smulato methods are used to calbrate operatg rules to perform well for multple purposes, ad they are the fal aalyss methods used to refe ad test operatg rules ( Reservor 1977). Performace-based optmzato models ca also sort through the operatos of complex multpurpose systems. Where mplct stochastc optmzato s used for computatoal coveece (Lud ad Ferrera 1996), some of the rule forms suggested here mght be useful for dsetaglg the results. Some of these rule forms may also be useful for other smulato-based reservor optmzato studes, such as geetc algorthms (Olvera ad Loucks 1997). Reservor Aggregato Multreservor Systems For systems cosstg of may reservors, t s commo to aggregate some of the reservors to reduce the computatoal or data storage demads of large smulato ad optmzato models (Saad et al. 1994). A fal use of the rules preseted here s to help aggregate reservors ad uderstad how to dsaggregate operatos of aggregated reservors. I ths regard, reservors seres are typcally easer to aggregate ad dsaggregate tha reservors parallel. A related use of these rules s to derve mproved startg-pot solutos for applcato of large-scale optmzato methods or more refed smulato studes. Smulato ad Optmzato I almost every case, smulato modelg s the stadard by whch operatg rules are refed ad tested. Smulato models ca provde more realstc ad detaled represetato of reservor system operatos ad much lower computatoal demads tha optmzato models for all but the most straghtforward cases. Smulato models are also more commo practce ad, therefore, are more lkely to be trusted as a stadard of comparso. Comparso of proposed operatg rules by smulato modelg provdes may beefts. I may cases, smulato results wll show that several sets of operatg procedures wll provde approxmately equvalet performace (Wu 1988). I other cases, smulato results wll demostrate the trade-offs of multple performace objectves wth dfferet operatg polces. I a few cases, the ablty to demostrate trade-offs wll ad egotatos over how the system should best be operated. For all of these purposes, t s useful to have a wde varety of potetal operatg rule forms avalable for examato. CONCLUSIONS Derved operatg rules avalable for reservors seres ad parallel have bee revewed. These rules are summarzed JOURNAL OF WATER RESOURCES PLANNING AND MANAGEMENT / MAY/JUNE 1999 / 149

8 Tables 1 ad 2. A wde varety of operatg rules are avalable for smulato modelg of reservors seres ad parallel. These provde a great deal of flexblty the specfcato of system operatos uder varous flow, storage, ad demad codtos. Fortuately, for may sgle-purpose cases, a derved bass exsts to help egeers arrow the search for approprate operatg rules. A partcular set of operatg rules ca be supported techcally a umber of ways. Some rules are based o smple egeerg prcples for reservor operatos, such as keepg reservors full for water supply or empty for flood cotrol. Several rules are derved from more formal optmzato prcples, such as the NYC space rules ad hydropower producto ad eergy storage rules. However, may rules practce are based largely o emprcal or expermetal successes, ether from actual operatoal performace, performace smulato studes, or optmzato results. These expermetally supported rules are commo for large multpurpose projects. May opportutes exst for the use of formal optmzato methods wth reservor smulato models. Examples clude mplemetg storage allocato rules for reservors seres ad parallel, as well as geeral pealty-mmzg operatos, ad allocatg water amog uses wth a gve tmestep. Effectve meas of employg ad hybrdzg these rules are lkely to be requred for multpurpose systems. Smlarly, rules for shftg operatg rule sets accordg to drawdow or refll codtos are a mportat area for further work. APPENDIX I. STORAGE EFFECTIVENESS INDEX RULES FOR HYDROPOWER PRODUCTION The storage effectveess dex has bee developed by the USACE for maxmzg frm hydropower producto durg the drawdow seaso (Egeerg 1985). For each reservor, a storage effectveess dex s calculated for each tme-step, usg forecast flows ad power demads for the curret tme-step ad remag tme-steps the drawdow seaso. Reservors wth a low dex value are draw dow frst. Step 1. Fd the frm eergy requremet for the curret tme-step E f. Step 2. Estmate the shortfall of frm hydropower producto due to suffcet flows to the system 720 f f U S = E I H(S )e =1 where S f = eergy shortage for the curret tme-step; I U = flow upstream of reservor durg the curret tme-step; H (S ) = hydropower head as a fucto of reservor storage for reservor ; S = curret reservor storage for reservor ; e = hydropower producto effcecy of reservor ; ad the costat s a coverso factor for I U (cfs), H (ft), ad S f (Kw h). Ths assumes that all flows ca be utlzed through the turbes. Step 3. For each reservor, estmate the drawdow requred for that reservor to dvdually elmate the shortfall 720 S f = SHe 11.81(59.5) where 1/59.5 = coverso of cubc feet per secod to acrefoot draft per moth; S = drawdow (acre-ft); ad H = average head correspodg to the drawdow (ofte foud teratvely). Solvg for S 11.81(59.5) S = S f (He) 720 Step 4. For each reservor, estmate the eergy loss the remader of the drawdow seaso due to a drawdow of S durg ths tme-step (moth) 720 E = (CI V )H (S S )e L U p where E L = drawdow seaso power loss due to drawdow of reservor by S ; CI U = cumulatve atural flow upstream of reservor for the remader of the refll seaso; ad V p = volume (acre-ft) of upstream storage to be empted durg the remader of the drawdow seaso. Step 5. Calculate the storage effectveess rato (SER) for each reservor SER = E L/Sf Reservors wth the lowest ratos are to be draw dow frst. APPENDIX II. DERIVATION OF HYDROPOWER PRODUCTION RULES FOR RESERVOIRS IN SERIES LP Short-Term Storage Allocato The objectve s to fd the allocato of a total storage volume that maxmzes hydropower producto for oe perod, subject to flow forecasts for each reservor, reservor storage capactes, ad a total storage target. As a equlbrum aalyss, chages reservor storage are eglected. Ths s expressed mathematcally as follows: subject to max P = H (S )Q e (16) =1 Q = I (17) 1 1 Q = Q I, > 1 (18) 1 S K, (19) =1 S = S (20) where P = sum total of eergy produced by all reservors; = umber of reservors system; H = level of head reservor ; S = storage target for reservor ; S = total system storage target; Q = total flow ad release for reservor ; I = drect flows to reservor ; K = storage capacty of reservor ; e = effcecy of turbes reservor ; V = chage overall power producto P wth chage storage reservor ; ad = ut weght of water. It should be oted that reservor = 1 s the most upstream reservor seres. For short-term allocato, the head-storage relatoshp ca ofte be learzed, or H (S )=a S. The followg LP results: max P = aesq (21) =1 subject to costrat equatos (17) (20), where a s a costat relatg chage head wth chage of storage reservor (for small chages head). H (S ) s commoly olear but may be pecewse learzed ad solved wth a LP, because head-storage relatoshps for most reservors are cocave. Dervato of Hydropower Producto Rule The above LP formulato ca typcally be smplfed, where the head-storage relatoshp ca be learzed. Usg the objectve fucto of (21) ad substtutg t (17) ad (18) results the smpler LP j =1 j=1 max P = ae I S (22) 150 / JOURNAL OF WATER RESOURCES PLANNING AND MANAGEMENT / MAY/JUNE 1999

9 subject to costrat equatos (19) ad (20). Ths problem s solved by fdg the slope of the objectve fucto wth respect to storage for each reservor j S j=1 P ae I = V (23) Rule. Fll reservors order of hghest to lowest V utl total storage S s flled. Empty the reverse order. LP Short-Term Eergy Storage Allocato Ths problem s slghtly modfed from that dscussed above that stead of seekg to maxmze hydropower producto subject to a gve total water storage, t s desred to maxmze hydropower producto of reservors seres subject to a gve total eergy storage. For ths problem, the objectve ad costrats (19) ad (22) are restated below, ad (20) s modfed to a eergy storage costrat subject to j =1 j=1 max P = ae I S (24) S K, (25) =1 E (S )=E (26) where E = total eergy storage sought; ad E (S ) = eergy cotet of each reservor as a fucto of ts water storage. For small chages reservor storage, the fucto E (S ) ca probably be learzed to the form b S (where b s a costat), allowg (26) to be made lear ad (24) (26) to be employed as a LP to allocate storage amog reservors seres to maxmze hydropower producto, subject to a total eergy storage level. For large chages storage, aother soluto method s lkely to be requred, as E (S ) s covex. APPENDIX III. DERIVATIONS OF PARALLEL BALANCING RULES These dervatos of the NYC ad space rules are adapted from dervatos preseted by Sad (1984) ad Johso et al. (1991). These dervatos are further exteded to exame eergy storage ad water qualty applcatos. Basc Dervato of NYC Rule Deftos of varables used ths appedx are as follows: z = value of objectve fucto; h = ut value of water reservor ; = umber of reservors the system; S f = edof-perod storage for the curret perod for reservor ; S 0 = begg of curret perod storage for reservor ; CQ = cumulatve flow to reservor from the ed of the curret perod to the ed of the refll cycle; K = storage capacty of reservor, assumed to be the same every perod; V = total volume of water storage at the ed of the curret perod; I = flow to reservor for the curret perod; D = demad for curret perod (release for curret perod); ad f (CQ ) = probablty desty of CQ. The objectve of the NYC balacg rule s to mmze the expected value EV( ) of total cumulatve spll from all of the parallel reservors at the ed of the refll seaso. Ths s reflected the objectve fucto (1). I (1), the term h represets the relatve value of water stored each reservor. Varato the value of water ca reflect varato pumpg costs, treatmet costs, or eergy cotet betwee the varous reservors, as dscussed later the dervato. To mplemet ths rule, ths optmzato problem s solved for each tmestep durg the refll seaso =1 m z =EV h m(0, S CQ K ) (27) subject to the followg costrats: =1 f S = V (28) f V = (S I ) D (29) =1 0 Costrat equato (29) dcates that the total water avalable should equal the sum of avalable water (curret storage plus curret perod flows) mus dowstream water demads for the curret perod. Expadg the expected value fucto the objectve fucto [(27)] yelds f =1 KSf m z = h (S CQ K ) f (CQ ) dcq (30) subject to costrat equatos (28) ad (29). The Lagraga for ths problem s f =1 KSf f f =1 =1 KSf f K S =1 L = h (S CQ K ) f (CQ ) dcq or S V = h (S K ) f (CQ ) dcq CQ f (CQ ) dcq S V f (31) KS f f =1 0 KSf f =1 L = h (S K ) 1 f (CQ ) dcq CQ CQ f (CQ ) dcq S V 0 (32) where CQ = expected value of cumulatve flows for reservor durg the remader of the refll seaso. The frst-order codtos for solvg ths problem are KSf L =0=h 1 f (CQ ) dcq S f 0 or or or (S K )(f (CQ = K S )) f f (K S ) f (CQ = K S ) f f (33) KSf h 1 f (CQ ) dcq = (34) 0 h Pr(CQ > K S )=, (35) f h Pr(ay spll reservor ) = (36) JOURNAL OF WATER RESOURCES PLANNING AND MANAGEMENT / MAY/JUNE 1999 / 151

10 Ths geeral result dcates that the storage targets for all reservors should have the same probablty of spll, weghted by the value of water for each reservor h. The use of the NYC rule for water supply, water qualty, ad eergy storage purposes all follow (35) ad (36), wth dfferet terpretatos of h. NYC Water Supply Rule For smple water supply purposes, h has the same value for all, ad so (35) becomes Pr(CQ > K S )=, (37) f NYC Rule for Water Qualty For smple water supply purposes wth mportat water qualty dffereces (e.g., total dssolved solds or perhaps water temperature, or total suspeded solds) betwee reservors, h vares betwee reservors ad ca be terpreted as the margal value of water use mus ts water treatmet cost for reservor. I ths case (35) ad (36) rema the same but wth ths et water value varyg wth splls of dfferet water qualtes. NYC Rule for Eergy Storage Here, the objectve s to mmze the expected value of potetal eergy splled rather tha physcal water splled. Here, h has the terpretato of the eergy cotet of water stored reservor. Eqs. (35) ad (36) rema the same ad apply to ths terpretato. Dervato of Space Rules Returg to (35) the cetral result of the NYC rule the assumpto s made that the dstrbutos f (CQ ) have the same dstrbutoal form, except that they are scaled by the average cumulatve flow of the bas CQ. Where ths assumpto holds, the the dstrbutos f (CQ /CQ )=f (CQ /CQ ) (38) j j j for ay two reservors ad j. Where ths s the case, the rato (K S f)/cq becomes a stadard devate for all dstrbutos, havg the same probablty of exceedece for all reservors. If ths rato s set so that t equals the same rato at the bas-wde scale =1 =1 K V EV(CQ ) (39) the the reservors are all balaced terms of mmzg expected value of spll ad maxmzg capture of curret flows. By replacg water flows, water storage capactes, ad water storage levels wth eergy flows, eergy storage capactes, ad eergy storage levels, the space rule ca be adapted to eergy storage purposes as the NYC rule ca be adapted to other operatg purposes (Johso et al. 1991). Dervato of LP-NYC Rules Dervato of the LP space rules begs wth (27) (29) used the dervato of the NYC rules. Addtoal costrats ca also be added to the LP-NYC rule problem, as log as the addtoal costrats are lear. I ths case the expected value operator (27) s replaced by use of the weghed summato of spll values that would result from each year of the hstorcal record or the probabltes of several represetatve year-types. Gve hstorcal streamflows of equal record legth >m for each of reservors, m represetatve refll seasos ca be ferred. Ths yelds the followg LP: subject to m j j j=1 =1 m z = p h L (40) =1 S = V (41) f L E = CQ K S, ad j (42) j j j f V = (S Q ) D (43) =1 f addto to ay other lear costrats o preset-perod operatos, where m = umber of equally probable refll seasos; p j = probablty of hydrologc year-type j; CQ j = hydrologc year j, the expected cumulatve flow to reservor from the ed of the curret perod to the ed of the refll cycle; L j = spll from reservor uder hydrologc year j; ad E j = empty storage capacty reservor uder hydrologc year j. ACKNOWLEDGMENTS Ths work was supported by the USACE Hydrologc Egeerg Ceter ad beefted cosderably from dscussos wth Da Barcellos, Rchard Hayes, Morrs Israel, Ke Krby, Da Sheer, Jery Stedger, ad Mm Jeks. The wrters wsh to thak the two revewers of the paper for ther helpful commets. APPENDIX IV. REFERENCES Bower, B. T., Hufschmdt, M. M., ad Reedy, W. W. (1966). Operatg procedures: Ther role the desg of water-resource systems by smulato aalyses. 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