8. Modelling Uncertainty

Transcription

1 8. Modellng Uncertanty. Introducton. Generatng Values From Known Probablty Dstrbutons. Monte Carlo Smulaton 4. Chance Constraned Models 5 5. Markov Processes and Transton Probabltes 6 6. Stochastc Optmzaton Probabltes of Decsons A Numercal Example Conclusons 5 8. References 5

2 8 Modellng Uncertanty Decson makers are ncreasngly wllng to consder the uncertanty assocated wth model predctons of the mpacts of ther possble decsons. Informaton on uncertanty does not make decson makng easer, but to gnore t s to gnore realty. Incorporatng what s known about the uncertanty nto nput parameters and varables used n optmzaton and smulaton models can help n quantfyng the uncertanty n the resultng model predctons the model output. Ths chapter dscusses and llustrates some approaches for dong ths.. Introducton Water resources planners and managers work n an envronment of change and uncertanty. Water supples are always uncertan, f not n the short term at least n the long term. Water demands and the multple purposes and servces water provde are always changng, and these changes cannot always be predcted. Many of the values of parameters of models used to predct the multple hydrologcal, economc, envronmental, ecologcal and socal mpacts are also changng and uncertan. Indeed models used to predct these mpacts are, at least n part, based on many mprecse assumptons. Plannng and managng, gven ths uncertanty, cannot be avoded. To the extent that probabltes can be assgned to varous uncertan nputs or parameter values, some of ths uncertanty can be ncorporated nto models. These models are called probablstc or stochastc models. Most probablstc models provde a range of possble values for each output varable along wth ther probabltes. Stochastc models attempt to model the random processes that occur over tme, and provde alternatve tme seres of outputs along wth ther probabltes. In other cases, senstvty analyses (solvng models under dfferent assumptons) can be carred out to estmate the mpact of any uncertanty on the decsons beng consdered. In some stuatons, uncertanty may not sgnfcantly affect the decsons that should be made. In other stuatons t wll. Senstvty analyses can help estmate the extent to whch we need to try to reduce that uncertanty. Model senstvty and uncertanty analyss s dscussed n more detal n Chapter 9. Ths chapter ntroduces a number of approaches to probablstc optmzaton and smulaton modellng. Probablstc models wll be developed and appled to some of the same water resources management problems used to llustrate determnstc modellng n prevous chapters. They can be, and have been, appled to numerous other water resources plannng and management problems as well. The purpose here, however, s smply to llustrate some of the commonly used approaches to the probablstc modellng of water resources system desgn and operatng problems.. Generatng Values From Known Probablty Dstrbutons Varables whose values cannot be predcted wth certanty are called random varables. Often, nputs to hydrologcal smulaton models are observed or synthetcally generated values of ranfall or streamflow. Other examples of such random varables could be evaporaton losses, pont and non-pont source wastewater dscharges, demands for water, spot prces for energy that may mpact the amount of hydropower producton, and so on. For random processes that are statonary that s, the statstcal attrbutes of the process are not changng and f there s

3 Water Resources Systems Plannng and Management f R () r p* Ea r* r Fgure 8.. Probablty densty dstrbuton of a random varable R. The probablty that R s less than or equal r* s p*. no seral correlaton n the spatal or temporal sequence of observed values, then such random processes can be characterzed by sngle probablty dstrbutons. These probablty dstrbutons are often based on past observatons of the random varables. These observatons or measurements are used ether to defne the probablty dstrbuton tself or to estmate parameter values of an assumed type of dstrbuton. Let R be a random varable whose probablty densty dstrbuton, f R (r), s as shown n Fgure 8.. Ths dstrbuton ndcates the probablty or lkelhood of an observed value of the random varable R beng between any two values of r on the horzontal axs. For example, the probablty of an observed value of R beng between and r* s p*, the shaded area to the left of r*. The entre shaded area of a probablty densty dstrbuton, such as shown n Fgure 8., s. Integratng ths functon over r converts the densty functon to a cumulatve dstrbuton functon, F R (r*), rangng from to, as llustrated n Fgure 8.. r* f () r dr Pr( R r*) F (*) r R R (8.) Gven any value of p* from to, one can fnd ts correspondng varable value r* from the nverse of the cumulatve dstrbuton functon. F R (p*) r* (8.) From the dstrbuton shown n Fgure 8., t s obvous that the lkelhood of dfferent values of the random varable vares; ones n the vcnty of r* are much more lkely to occur than are values at the tals of the dstrbuton. A unform dstrbuton s one that looks lke a rectangle; any value of the random varable between ts Fgure 8.. Cumulatve dstrbuton functon of a random varable R showng the probablty of any observed value of R beng less than or equal to a gven value r. The probablty of an observed value of R beng less than or equal to r* s p*. lower and upper lmts s equally lkely. Usng Equaton 8., together wth a seres of unformly dstrbuted (all equally lkely) values of p* over the range from to (that s, along the vertcal axs of Fgure 8.), one can generate a correspondng seres of varable values, r*, assocated wth any dstrbuton. These random varable values wll have a cumulatve dstrbuton as shown n Fgure 8., and hence a densty dstrbuton as shown n Fgure 8., regardless of the types or shapes of those dstrbutons. The mean, varance and other moments of the dstrbutons wll be mantaned. The mean and varance of contnuous dstrbutons are: r f R(r)dr E[R] (8.) (r E[R]) f R (r)dr Var[R] (8.4) The mean and varance of dscrete dstrbutons havng possble values denoted by r wth probabltes p are: rp r E[ R] p Var[ R] (8.5) (8.6) If a tme seres of T random varable values, r t, from the same statonary random varable, R, exsts, then the seral or autocorrelatons of r t and r t k n ths tme seres for any postve nteger k can be estmated usng: ρˆ R(k) ER [ ] [( rτ E[ R])( rτ k E[ R])] ( rt E[ R]) t T (8.7) τ T k

4 Modellng Uncertanty The probablty densty and correspondng cumulatve probablty dstrbutons can be of any shape, not just those named dstrbutons commonly found n probablty and statstcs books. The process of generatng a tme sequence t,, of nputs, r t, from the probablty dstrbuton of a random varable R where the lag seral correlaton, ρ R () ρ, s to be preserved s a lttle more complex. The expected value of the random varable R t depends on the observed value, r t, of the random varable R t, together wth the mean of the dstrbuton, E[R], and the correlaton coeffcent ρ. If there s no correlaton (ρ s ), then the expected value of R t s the mean of the populaton, E[R]. If there s perfect correlaton (ρ s ), then the expected value of R t s r t. In general, the expected value of R t gven an observed value r t of R t s: ER [ t Rt rt] ER [ ] ρ( rt ER [ ]) (8.8) The varance of the random varable R t depends on the varance of the dstrbuton, Var[R], and the lag correlaton coeffcent, ρ. Var[ Rt Rt rt] Var[ R]( ρ ) (8.9) If there s perfect correlaton (ρ ), then the process s determnstc, there s no varance, and r t r t. The value for r t s r t. If there s no correlaton that s, seral correlaton does not exst (ρ ) then the generated value for r t s ts mean, E[R], plus some randomly generated devaton from a normal dstrbuton havng a mean of and a standard devaton of, denoted as N(, ). In ths case the value r t s not dependent on r t. When the seral correlaton s more than but less than, then both the correlaton and the standard devaton (the square root of the varance) nfluence the value of r t. A sequence of random varable values from a multvarate normal dstrbuton that preserves the mean, E[R]; overall varance, Var[R]; standard devaton σ, and lag correlaton ρ; can be obtaned from. rt E[ R] ρ( rt E[ R]) Zσ( ρ ) / (8.) The term Z n Equaton 8. s a random number generated from a normal dstrbuton havng a mean of and a varance of. The process nvolves selectng a random number from a unform dstrbuton rangng from to, and usng t n Equaton 8. for an N(, ) dstrbuton to obtan a value of random number for Ec r t+ E[ R] + ρ (r t *-E[ R]) E[ R] use n Equaton 8.. Ths postve or negatve number s substtuted for the term Z n Equaton 8. to obtan a value r t. Ths s shown on the graph n Fgure 8.. Smulaton models that have random nputs, such as a seres of r t values, wll generally produce random outputs. After many smulatons, the probablty dstrbutons of each random output varable value can be defned. These can be used to estmate relabltes and other statstcal characterstcs of those output dstrbutons. Ths process of generatng multple random nputs for multple smulatons to obtan multple random outputs s called Monte Carlo smulaton.. Monte Carlo Smulaton ρ To llustrate Monte Carlo smulaton, consder the water allocaton problem nvolvng three frms, each of whch receves a beneft, B (x t ), from the amount of water, x t, allocated to t n each perod t. Ths stuaton s shown n Fgure 8.4. Monte Carlo smulaton can be used to fnd the probablty dstrbuton of the benefts to each frm assocated wth the frm s allocaton polcy. Suppose the polcy s to keep the frst two unts of flow n the stream, to allocate the next three unts to Frm, and the next four unts to Frms and equally. The remanng flow s to be allocated to each of the three frms equally up to the lmts desred by each frm, namely.,., and 8. respectvely. Any excess flow wll reman n the stream. The plots n Fgure 8.5 llustrate ths polcy. Each allocaton plot reflects the prortes gven to the three frms and the users further downstream. r t * N(, Var[R] (-ρ )) E[R t+ R t r t* ] Fgure 8.. Dagram showng the calculaton of a sequence of values of the random varable R from a multvarate normal dstrbuton n a way that preserves the mean, varance and correlaton of the random varable. rt

5 4 Water Resources Systems Plannng and Management Fgure 8.4. Streamflow allocatons n each perod t result n benefts, B (x t ), to each frm. The flows, Q t, at each dverson ste are the random flows Q t less the upstream wthdrawals, f any. Q t frm B ( x 6x -x t ) t t xt Q t Q t x t frm B ( x 7x -.5x t ) t t frm B ( x 8x -.5x t ) t t xt Q t Ed allocaton at dverson ste : x t allocaton at dverson ste : x t Ee streamflow Q t Ef streamflow Q t Fgure 8.5a. Water allocaton polcy for Frm based on the flow at ts dverson ste. Ths polcy apples for each perod t. Fgure 8.5b. Water allocaton polcy for Frm based on the flow at ts dverson ste for that frm. Ths polcy apples for each perod t. Eg allocaton at dverson ste : x t streamflow Q t Fgure 8.5c. Water allocaton polcy for Frm based on the flow at ts dverson ste. Ths polcy apples for each perod t. A smulaton model can be created. In each of a seres of dscrete tme perods t, the flows Q t are drawn from a probablty dstrbuton, such as from Fgure 8. usng Equaton 8.. Once ths flow s determned, each successve allocaton, x t, s computed. Once an allocaton s made t s subtracted from the streamflow and the next allocaton s made on the bass of that reduced E h streamflow at ste streamflow Q t Fgure 8.5d. Streamflow downstream of ste gven the streamflow Q t at ste before the dverson. Ths apples for each perod t. streamflow, n accordance wth the allocaton polcy defned n Fgures 8.5a d. After numerous tme steps, the probablty dstrbutons of the allocatons to each of the frms can be defned. Fgure 8.6 shows a flow chart for ths smulaton model.

6 Modellng Uncertanty 5 start set: t T max t t+ compute Qt (Eq.) Q t Q t compute x t (fg. 5a) Fgure 8.6. Monte Carlo smulaton to determne probablty dstrbutons of allocatons to each of three water users, as llustrated n Fgure 8.4. The dashed lnes represent nformaton (data) flows. Q t Q t - x t compute x t (fg. 5b) data storage Q t Q t - x t compute x t (fg. 5c) no t T max yes calculate and plot probablty dstrbutons Ej stop Havng defned the probablty dstrbuton of the allocatons, based on the allocaton polcy, one can consder each of the allocatons as random varables, X, X, and X for Frms, and respectvely. 4. Chance Constraned Models For models that nclude random varables, t may be approprate n some stuatons to consder constrants that do not have to be satsfed all the tme. Chance constrants specfy the probablty of a constrant beng satsfed, or the fracton of the tme a constrant has to apply. Consder, for example, the allocaton problem shown n Fgure 8.4. For plannng purposes, the three frms may want to set allocaton targets, not expectng to have those targets met % of the tme. To ensure, for example, that an allocaton target, T, of frm wll be met at least 9% of the tme, one could wrte the chance constrant: Pr{T X }.9, and (8.) In ths constrant, the allocaton target T s an unknown decson-varable, and X s a random varable whose dstrbuton has just been computed and s known.

7 6 Water Resources Systems Plannng and Management f x (X t ). Pr( Q q t )F Q (q t ).9. Ek. X t X t Em. q t q t Fgure 8.7. Probablty densty dstrbuton of the random allocaton X to frm. The partcular allocaton value x t has a 9% chance of beng equalled or exceeded, as ndcated by the shaded regon. Fgure 8.8. Example cumulatve probablty dstrbuton showng the partcular value of the random varable, q t, that s equalled or exceeded 9% of the tme. To nclude chance constrants n optmzaton models, ther determnstc equvalents must be defned. The determnstc equvalents of the three chance constrants n Equaton 8. are: T x t, and (8.) where x t s the partcular value of the random varable X that s equalled or exceeded 9% of the tme. Ths value s shown on the probablty dstrbuton for X n Fgure 8.7. To modfy the allocaton problem somewhat, assume the beneft obtaned by each frm s a functon of ts target allocaton and that the same allocaton target apples n each tme perod t. The equpment and labour used n the frm s presumably based on the target allocatons. Once the target s set, assume there are no benefts ganed by excess water allocatons. If the benefts obtaned are to be based on the target allocatons rather than the actual allocatons, then the optmzaton problem s one of fndng the values of the three targets that maxmze the total benefts obtaned wth a relablty of, say, at least 9%. Maxmze 6T T 7T.5T 8T.5T (8.) subject to: Pr{T T T [Q t mn(q t, )]}.9 for all perods t (8.4) where Q t s the random streamflow varable upstream of all dverson stes. If the same uncondtonal probablty dstrbuton of Q t apples for each perod t, then only one Equaton 8.4 s needed. Assumng the value of the streamflow, q t, that s equalled or exceeded 9% of the tme, s greater than (the amount that must reman n the stream), the determnstc equvalent of chance constrant Equaton 8.4 s: T T T q t mn q t, (8.5) The value of the flow that s equal to or exceeds 9% of the tme, q t, can be obtaned from the cumulatve dstrbuton of flows as llustrated n Fgure 8.8. Assume ths 9% relable flow s 8. The determnstc equvalent of the chance constrant Equaton 8.9 for all perods t s smply T T T 6. The optmal soluton of the chance-constraned target allocaton model, Equatons 8.8 and 8.9, s, as seen before, T, T and T 4. The next step would be to smulate ths problem to see what the actual relabltes mght be for varous sequences of flows q t. 5. Markov Processes and Transton Probabltes Tme-seres correlatons can be ncorporated nto models usng transton probabltes. To llustrate ths process, consder the observed flow sequence shown n Table 8.. The estmated mean, varance and correlaton coeffcent of the observed flows shown n Table 8. can be calculated usng Equatons 8.6, 8.7 and 8.8.

8 Modellng Uncertanty 7 perod t flow Q t perod t flow Qt.8.5 perod t flow Qt.8. E87w probablty En q t Fgure 8.9. Hstogram showng an equal / probablty that the values of the random varable Q t wll be n any one of the three two-flow unt ntervals. Table 8.. Sequence of flows for thrty-one tme perods t. flow nterval n : t flow nterval n t + j E87x E[Q] q t /.55 (8.6) Var[Q] (q t.55) /.95 (8.7) Lag-one correlaton coeffcent ρ ( qt. 55)( qt. 55) ( q. 55). 5 t (8.8) The probablty dstrbuton of the flows n Table 8. can be approxmated by a hstogram. Hstograms can be created by subdvdng the entre range of random varable values, such as flows, nto dscrete ntervals. For example, let each nterval be two unts of flow. Countng the number of flows n each nterval and then dvdng those nterval counts by the total number of counts results n the hstogram shown n Fgure 8.9. In ths case, just to compare ths wth what wll be calculated later, the frst flow, q, s gnored. Fgure 8.9 shows a unform uncondtonal probablty dstrbuton of the flow beng n any of the possble dscrete flow ntervals. It does not show the possble dependency of the probabltes of the random varable Table 8.. Matrx showng the number of tmes a flow n nterval n perod t was followed by a flow n nterval j n perod t. value, q t, n perod t on the observed random varable value, q t, n perod t. It s possble that the probablty of beng n a flow nterval j n perod t depends on the actual observed flow nterval n perod t. To see f the probablty of beng n any gven nterval of flows s dependent on the past flow nterval, one can create a matrx. The rows of the matrx are the flow ntervals n perod t. The columns are the flow ntervals j n the followng perod t. Such a matrx s shown n Table 8.. The numbers n the matrx are based on the flows n Table 8. and ndcate the number of tmes a flow n nterval j followed a flow n nterval. Gven an observed flow n an nterval n perod t, the probabltes of beng n one of the possble ntervals j n the next perod t must sum to. Thus, each number n each row of the matrx n Table 8. can be dvded by the total number of flow transtons n that row (the sum

9 8 Water Resources Systems Plannng and Management flow nterval n t: flow nterval n t + : j.5.. Table 8.. Matrx showng the probabltes P j of havng a flow n nterval j n perod t gven an observed flow n nterval n perod t. of the number of flows n the row) to obtan the probabltes of beng n each nterval j n t gven a flow n nterval n perod t. In ths case there are ten flows that followed each flow nterval, hence by dvdng each number n each row of the matrx by defnes the transton probabltes P j. P j Pr{Q t n nterval j Q t n nterval } (8.9) These condtonal or transton probabltes, shown n Table 8., correspond to the number of transtons shown n Table 8.. Table 8. s a matrx of transton probabltes. The sum of the probabltes n each row equals. Matrces of transton probabltes whose rows sum to are also called stochastc matrces or frst-order Markov chans. If each row s probabltes were the same, ths would ndcate that the probablty of observng any flow nterval n the future s ndependent of the value of prevous flows. Each row would have the same probabltes as the uncondtonal dstrbuton shown n Fgure 8.9. In ths example the probabltes n each row dffer, showng that low flows are more lkely to follow low flows, and hgh flows are more lkely to follow hgh flows. Thus the flows n Table 8. are postvely correlated, as ndeed has already determned from Equaton 8.8. Usng the nformaton n Table 8., one can compute the probablty of observng a flow n any nterval at any perod on nto the future gven the present flow nterval. Ths can be done one perod at a tme. For example assume the flow n the current tme perod t s n nterval. The probabltes, PQ j,, of beng n any of the three E9b ntervals n the followng tme perod t are the probabltes shown n the thrd row of the matrx n Table 8.. The probabltes of beng n an nterval j n the followng tme perod t s the sum over all ntervals of the jont probabltes of beng n nterval n perod t and makng a transton to nterval j n perod t. Pr{Q n nterval j} PQ j, Pr{Q n nterval } Pr{Q n nterval j Q n nterval } (8.) The last term n Equaton 8. s the transton probablty, from Table 8., that n ths example remans the same for all tme perods t. These transton probabltes, Pr{Q t n nterval j Q t n nterval } can be denoted as P j. Referrng to Equaton 8.9, Equaton 8. can be wrtten n a general form as: PQ j,t PQ t P j for all ntervals j and perods t (8.) Ths operaton can be contnued to any future tme perod. Table 8.4 llustrates the results of such calculatons for tme perod t flow nterval probablty PQ t Table 8.4. Probabltes of observng a flow n any flow nterval n a future tme perod t gven a current flow n nterval. These probabltes are derved usng the transton probabltes P j n Table 8. n Equaton 8. and assumng the flow nterval observed n Perod s n Interval. E87y

10 Modellng Uncertanty 9 up to sx future perods, gven a present perod (t ) flow n nterval. Note that as the future tme perod t ncreases, the flow nterval probabltes are convergng to the uncondtonal probabltes n ths example /, /, / as shown n Fgure 8.9. The predcted probablty of observng a future flow n any partcular nterval at some tme n the future becomes less and less dependent on the current flow nterval as the number of tme perods ncreases between the current perod and that future tme perod. When these uncondtonal probabltes are reached, PQ t wll equal PQ,t for each flow nterval. To fnd these uncondtonal probabltes drectly, Equaton 8. can be wrtten as: PQ j PQ P j for all ntervals j (8.) Equaton 8. (less one) along wth Equaton 8. can be used to calculate all the uncondtonal probabltes PQ drectly. PQ (8.) Condtonal or transton probabltes can be ncorporated nto stochastc optmzaton models of water resources systems. 6. Stochastc Optmzaton To llustrate the development and use of stochastc optmzaton models, consder frst the allocaton of water to a sngle user. Assume the flow n the stream where the dverson takes place s not regulated and can be descrbed by a known probablty dstrbuton based on hstorcal records. Clearly, the user cannot dvert more water than s avalable n the stream. A determnstc model would nclude the constrant that the dverson x cannot exceed the avalable water Q. But Q s a random varable. Some target value, q, of the random varable Q wll have to be selected, knowng that there s some probablty that n realty, or n a smulaton model, the actual flow may be less than the selected value q. Hence, f the constrant x q s bndng, the actual allocaton may be less than the value of the allocaton or dverson varable x produced by the optmzaton model. If the value of x affects one of the system s performance ndcators, such as the net benefts, B(x), to the user, a more accurate estmate of the user s net benefts wll be obtaned from consderng a range of possble allocatons x, dependng on the range of possble values of the random flow Q. One way to do ths s to dvde the known probablty dstrbuton of flows q nto dscrete ranges, each range havng a known probablty PQ. Desgnate a dscrete flow q for each range. Assocated wth each specfed flow q s an unknown allocaton x. Now the sngle determnstc constrant x q can be replaced wth the set of determnstc constrants x q, and the term B(x) n the orgnal objectve functon can be replaced by ts expected value, PQ B(x ). Note, when dvdng a contnuous known probablty dstrbuton nto dscrete ranges, the dscrete flows q, selected to represent each range havng a gven probablty PQ, should be selected so as to mantan at least the mean and varance of that known dstrbuton as defned by Equatons 8.5 and 8.6. To llustrate ths, consder a slghtly more complex example nvolvng the allocaton of water to consumers upstream and downstream of a reservor. Both the polces for allocatng water to each user and the reservor release polcy are to be determned. Ths example problem s shown n Fgure 8.. If the allocaton of water to each user s to be based on a common objectve, such as the mnmzaton of the total sum, over tme, of squared devatons from pre-specfed target allocatons, each allocaton n each tme perod wll depend n part on the reservor storage volume. Eo allocaton u t flow Q t user U ntal storage S t reservor capacty K release R t user D allocaton d t Fgure 8.. Example water resources system nvolvng water dversons from a rver both upstream and downstream of a reservor of known capacty.

11 Water Resources Systems Plannng and Management Consder frst a determnstc model of the above problem, assumng known rver flows Q t and upstream and downstream user allocaton targets UT t and DT t n each of T wthn-year perods t n a year. Assume the objectve s to mnmze the sum of squared devatons from actual allocatons, u t and d t, and ther respectve target allocatons, UT t and DT t n each wthn-year perod t. T Mnmze {(UT t u t ) (DT t d t ) } (8.4) t The constrants nclude: a) Contnuty of storage nvolvng ntal storage volumes S t, net nflows Q t u t, and releases R t. Assumng no losses: S t Q t u t R t S t for each perod t, T (8.5) b) Reservor capacty lmtatons. Assumng a known actve storage capacty K: S t K for each perod t (8.6) c) Allocaton restrctons. For each perod t: u t Q t (8.7) d t R t (8.8) Equatons 8.5 and 8.8 could be combned to elmnate the release varable R t, snce n ths problem knowledge of the total release n each perod t s not requred. In ths case, Equaton 8.5 would become an nequalty. The soluton of ths model, Equatons , would depend on the known varables (the targets UT t and DT t, flows Q t and reservor capacty K). It would dentfy the partcular upstream and downstream allocatons and reservor releases n each perod t. It would not provde a polcy that defnes what allocatons and releases to make for a range of dfferent nflows and ntal storage volumes n each perod t. A backward-movng dynamc programmng model can provde such a polcy. Ths polcy wll dentfy the allocatons and releases to make based on varous ntal storage volumes, S t, and flows, Q t, as dscussed n Chapter 4. Ths determnstc dscrete dynamc programmng allocaton and reservor operaton model can be wrtten for dfferent dscrete values of S t from S t capacty K as: F t n (S t, Q t ) mn (UT t u t ) (DT t d t ) F n t (S t, Q t ) The mnmzaton s over all feasble u t, R t, d t : u t Q t R t S t Q t u t R t S t Q t u t K d t R t S t S t Q t u t R t (8.9) There are three varables to be determned at each stage or tme perod t n the above dynamc programmng model. These three varables are the allocatons u t and d t and the reservor release R t. Each decson nvolves three dscrete decson-varable values. The functons F t n (S t, Q t ) defne the mnmum sum of squared devatons gven an ntal storage volume S t and streamflow Q t n tme perod or season t wth n tme perods remanng untl the end of reservor operaton. One can reduce ths three decson-varable model to a sngle varable model by realzng that, for any fxed dscrete par of ntal and fnal storage volume states, there can be a drect tradeoff between the upstream and downstream allocatons, gven the partcular streamflow n each perod t. Increasng the upstream allocaton wll decrease the resultng reservor nflow, and ths n turn wll reduce the release by the same amount. Ths reduces the amount of water avalable to allocate to the downstream use. Hence, for ths example problem nvolvng these upstream and downstream allocatons, a local optmzaton can be performed at each tme step t for each combnaton of storage states S t and S t. Ths optmzaton fnds the allocaton decson-varables u t and d t that mnmze(ut t u t ) (DT t d t ) (8.) where u t Q t (8.) d t S t Q t u t S t (8.) Ths local optmzaton can be solved to dentfy the u t and d t allocatons for each feasble combnaton of S t and S t n each perod t. Gven these optmal allocatons, the dynamc programmng model can be smplfed to nclude only one dscrete decson-varable, ether R t or S t. If the decson varable S t s used n each perod t, the releases R t n

12 Modellng Uncertanty 4 those perods t do not need to be consdered. Thus the dynamc programmng model expressed by Equatons 8.9 can be wrtten for all dscrete storage volumes S t from to K and for all dscrete flows Q t as: F t n (S t, Q t ) mn (UT t u t (S t, S t )) (DT t d t (S t, S t )) F n t (S t, Q t ) The mnmzaton s over all feasble dscrete values of S t, S t K (8.) where the functons u t (S t, S t ) and d t (S t, S t ) have been determned usng Equatons As the total number of perods remanng, n, ncreases, the soluton of ths dynamc programmng model wll converge to a steady or statonary state. The best fnal storage volume S t gven an ntal storage volume S t wll probably dffer for each wthn-year perod or season t, but for a gven season t t wll be the same n successve years. In addton, for each storage volume S t, streamflow, Q t, and wthn-year perod t, the dfference between F t n T (S t, Q t ) and F t n (S t, Q t ) wll be the same constant regardless of the storage volume S t and perod t. Ths constant s the optmal, n ths case mnmum, annual value of the objectve functon, Equaton 8.4. There could be addtonal lmts mposed on storage varables and release varables, such as for flood control storage or mnmum downstream flows, as mght be approprate n specfc stuatons. The above determnstc dynamc programmng model (Equaton. 8.) can be converted to a stochastc model. Stochastc models consder multple dscrete flows as well as multple dscrete storage volumes, and ther probabltes, n each perod t. A common way to do ths s to assume that the sequence of flows follow a frst-order Markov process. Such a process nvolves the use of transton or condtonal probabltes of flows as defned by Equaton 8.. To develop these stochastc optmzaton models, t s convenent to ntroduce some addtonal ndces or subscrpts. Let the ndex k denote dfferent ntal storage volume ntervals. These dscrete ntervals dvde the contnuous range of storage volume values from to the actve reservor capacty K. Each S kt s a dscrete storage volume that represents the range of storage volumes n nterval k at the begnnng of each perod t. Let the followng letter l be the ndex denotng dfferent fnal storage volume ntervals. Each S l,t s a dscrete volume that represents the storage volume nterval l at the end of each perod t or equvalently at the begnnng of perod t. As prevously defned, let the ndces and j denote the dfferent flow ntervals, and each dscrete flow q t and q j,t represent those flow ntervals and j n perods t and t respectvely. These subscrpts and the volume or flow ntervals they represent are llustrated n Fgure 8.. Wth ths notaton, t s now possble to develop a stochastc dynamc programmng model that wll dentfy the allocatons and releases that are to be made gven both the ntal storage volume, S kt, and the flow, q t. It follows the same structure as the determnstc models defned by Equatons 8. through 8., and 8.. To dentfy the optmal allocatons n each perod t for each par of feasble ntal and fnal storage volumes S kt and S l,t, and nflows q t, one can solve Equatons 8.4 through 8.6. mnmze (UT t u kt ) (DT t d klt ) (8.4) where u kt q t k,, t. (8.5) d klt S kt q t u kt S l,t feasble k,, l, t. (8.6) The soluton to these equatons for each feasble combnaton of ntervals k,, l, and perod t defnes the optmal allocatons that can be expressed as u t (k, ) and d t (k,, l). The stochastc verson of Model 8., agan expressed n a form sutable for backward-movng dscrete dynamc programmng, can be wrtten for dfferent dscrete values of S kt from to K and for all q t as: Ft n ( Skt, qt ) mn ( UTt ut( k, t)) ( DTt dt( k,, l)) PF j t n t ( S, t, qjt, ) j The mnmzaton s over all feasble dscrete values of S l,t S l,t K S l,t S kt q t (8.7)

13 4 Water Resources Systems Plannng and Management Fgure 8.. Dscretzaton of streamflows and reservor storage volumes. The area wthn each flow nterval below the probablty densty dstrbuton curve s the uncondtonal probablty, PQ t, assocated wth the dscrete flow q t. flow probablty flow nterval Pr{Q t } PQ t Pr{Q t } PQ t storage volume nterval k 5 k 4 k k k S5t S4t St St St Ep q q q q q q q q q t t t 4t 5t 6t 7t 8t 9t streamflow Q t Each P j n the above recursve equaton s the known condtonal or transton probablty of a flow q j,t wthn nterval j n perod t gven a flow of q t wthn nterval n perod t. P t j Pr{flow q j,t wthn nterval j n t flow of q t wthn nterval n t} The sum over all flow ntervals j of these condtonal probabltes tmes the F n t (S l,t, q j,t ) values s the expected mnmum sum of future squared devatons from allocaton targets wth n perods remanng gven an ntal storage volume of S kt and flow of q t and fnal storage volume of S l,t. The value F n t (S kt, q t ) s the expected mnmum sum of squared devatons from the allocaton targets wth n perods remanng gven an ntal storage volume of S kt and flow of q t. Stochastc models such as these provde expected values of objectve functons. Another way to wrte the recurson equatons of ths model, Equaton 8.7, s by usng just the ndces k and l to denote the dscrete storage volume varables S kt and S l,t and ndces and j to denote the dscrete flow varables q t and q j,t : Ft n (,) k mn( UTt ut(,)) k t ( DTt dt(,,)) k l l PF j t n t (, l j) j such that S l,t K S l,t S kt q t (8.8) The steady-state soluton of ths dynamc programmng model wll dentfy the preferred fnal storage volume S l,t n perod t gven the partcular dscrete ntal storage volume S kt and flow q t. Ths optmal polcy can be expressed as a functon that dentfes the best nterval l gven ntervals k, and perod t. l (k,, t) (8.9) All values of l gven k, and t, defned by Equaton 8.9, can be expressed n a matrx, one for each perod t. Knowng the best fnal storage volume nterval l gven an ntal storage volume nterval k and flow nterval, the

14 Modellng Uncertanty 4 optmal downstream allocaton, d t (k, ), can, lke the upstream allocaton, be expressed n terms of only k and n each perod t. Thus, knowng the ntal storage volume S kt and flow q t s suffcent to defne the optmal allocatons u t (k, ) and d t (k, ), fnal storage volume S l,t, and hence the release R t (k, ). S kt q t u t (k, ) R t (k, ) S l,t k,, t where l (k,, t) (8.) 6.. Probabltes of Decsons Knowng the functon l (k,, t) permts a calculaton of the probabltes of the dfferent dscrete storage volumes, allocatons, and flows. Let PS kt the unknown probablty of an ntal storage volume S kt beng wthn some nterval k n perod t; PQ t the steady-state uncondtonal probablty of flow q t wthn nterval n perod t; and P kt the unknown probablty of the upstream and downstream allocatons u t (k, ) and d t (k, ) and reservor release R t (k, ) n perod t. As prevously defned, P t j the known condtonal or transton probablty of a flow wthn nterval j n perod t gven a flow wthn nterval n perod t. t These transton probabltes P j can be dsplayed n matrces, smlar to Table 8., but as a separate matrx (Markov chan) for each perod t. The jont probabltes of an ntal storage nterval k, an nflow n the nterval, P kt n each perod t must satsfy two condtons. Just as the ntal storage volume n perod t s the same as the fnal storage volume n perod t, the probabltes of these same respectve dscrete storage volumes must also be equal. Thus, j P P l t,, jt kt, k (8.4) where the sums n the rght hand sde of Equaton 8.4 are over only those combnatons of k and that result n a fnal volume nterval l. Ths relatonshp s defned by Equaton 8.9 (l (k,, t)). Whle Equaton 8.4 must apply, t s not suffcent. The jont probablty of a fnal storage volume n nterval l n perod t and an nflow j n perod t must equal the jont probablty of an ntal storage volume n the same nterval l and an nflow n the same nterval j n perod t. Multplyng the jont probablty P kt tmes the condtonal probablty P j and then summng over all k and t that results n a fnal storage nterval l defnes the former, and the jont probablty P l,j,t defnes the latter. P l,j,t t P kt P j l, j, t l (k,, t) (8.4) Once agan the sums n Equaton 8.4 are over all combnatons of k and that result n the desgnated storage volume nterval l as defned by the polcy (k,, t). Fnally, the sum of all jont probabltes P kt n each perod t must equal. k P kt t (8.4) Note the smlarty of Equatons 8.4 and 8.4 to the Markov steady-state flow Equatons 8. and 8.. Instead of only one flow nterval ndex consdered n Equatons 8. and 8., Equatons 8.4 and 8.4 nclude two ndces, one for storage volume ntervals and the other for flow ntervals. In both cases, one of Equatons 8. and 8.4 can be omtted n each perod t snce t s redundant wth that perod s Equatons 8. and 8.4 respectvely. The uncondtonal probabltes PS kt and PQ t can be derved from the jont probabltes P kt. PS kt P kt k, t (8.44) PQ t P kt, t (8.45) Each of these uncondtonal jont or margnal probabltes, when summed over all ther volume and flow ndces, wll equal. For example, k k k PS kt PQ t (8.46) Note that these probabltes are determned only on the bass of the relatonshps among flow and storage ntervals as defned by Equaton 8.9, l (k,, t) n each perod t, and the Markov chans defnng the flow nterval transton or condtonal probabltes, P t j. It s not necessary to know the actual dscrete storage values representng those ntervals. Thus assumng any relatonshp among the storage volume and flow nterval ndces, l (k,, t) and a

15 44 Water Resources Systems Plannng and Management knowledge of the flow nterval transton probabltes P t j, one can determne the jont probabltes P kt and ther margnal or uncondtonal probabltes PS kt. One does not need to know what those storage ntervals are to calculate ther probabltes. Gven the values of these jont probabltes P kt, the determnstc model defned by Equatons 8.4 to 8.8 can be converted to a stochastc model to dentfy the best storage and allocaton decson-varable values assocated wth each storage nterval k and flow nterval n each perod t. Mnmze Pkt{( UTt ukt) ( DTt dkt) } k (8.) The constrants nclude: a) Contnuty of storage nvolvng ntal storage volumes S kt, net nflows q t u kt, and at least partal releases d kt. Agan assumng no losses: S kt q t u kt d kt S l,t k,, t l (k,, t) (8.48) b) Reservor capacty lmtatons. S kt K k,, t (8.49) c) Allocaton restrctons. u kt q t k,, t (8.5) More detal on these and other stochastc modellng approaches can be found n Faber and Stednger (); Gablnger and Loucks (97); Huang et al. (99); Km and Palmer (997); Loucks and Falkson (97); Stednger et al. (984); Su and Dennger (974); Tejada- Gubert et al. (99 995); and Yakowtz (98). 6.. A Numercal Example T τ A smple numercal example may help to llustrate how these stochastc models can be developed wthout gettng bured n detal. Consder two wthn-year perods each year. The random flows Q t n each perod t are dvded nto two ntervals. These flow ntervals are represented by dscrete flows of and volume unts per second n the frst perod, and and 6 volume unts per second n the second perod. Ther transton probabltes are shown n Table 8.5. flow n t Q Assumng equal wthn-year perod duratons, these three dscrete flow rates are equvalent to about 6, and 95 mllon volume unts per perod. Assume the actve storage volume capacty K n the reservor equals 5 mllon volume unts. Ths capacty can be dvded nto dfferent ntervals of storage. For ths smple example, assume three storage volume ntervals represented by, 5 and mllon volume unts. Assume the allocaton targets reman the same n each perod at both the upstream and downstream stes. The upstream allocaton target s approxmately volume unts per second or mllon volume unts n each perod. The downstream allocaton target s approxmately 5 volume unts per second or 8 mllon volume unts n each perod. Wth these data we can use Equatons to determne the allocatons that mnmze the sum of squared devatons from targets and what that sum s, for all feasble combnatons of ntal and fnal storage volumes and flows. Table 8.6 shows the results of these optmzatons. These results wll be used n the dynamc programmng model to determne the best fnal storage volumes gven ntal volumes and flows. Wth the nformaton n Tables 8.5 and 8.6, the dynamc programmng model, Equaton 8.8 or as expressed n Equaton 8.5, can be solved to fnd the optmal fnal storage volumes, gven an ntal storage volume and flow. The teratons of the recursve equaton, suffcent to reach a steady state, are shown n Table 8.7. Ft n k SDkl Pj t n (,) mn Ft (, l j) l j such that Q j flow n t Q j flow n t 6 S l,t K S l,t S kt Q t (8.5) flow n t Table 8.5. Transton probabltes for two ranges of flows n two wthn-year perods. Q E89a

16 Modellng Uncertanty 45 ntal storage S k flow Q fnal storage nterval ndces S k,, l 5 5 5,,,,,,,,,,,,,,,, upstream allocaton uk downstream allocaton dkl sum squared devaton SDkl E89b Table 8.6. Optmal allocatons assocated wth gven ntal storage, S k, flow, Q, and fnal storage, S l, volumes. These allocatons u k and d kl mnmze the sum of squared devatons, DS kl ( u k ) (8 d kl ), from upstream and downstream targets, and 8 respectvely, subject to u k flow Q, and d kl release (S k Q u k S l ) ,,,,,,,,,,,,,,,,,, ,,,,,,,,,,,,,,,,,, Ths process can contnue untl a steady-state polcy s defned. Table 8.8 summarzes the next fve teratons. At ths stage, the annual dfferences n the objectve values assocated wth a partcular state and season have come close to a common constant value. Whle the dfferences between correspondng F t n T and F t n have not yet reached a common constant value to the nearest unt devaton (they range from, 5.5 to 497. for an average of 485.7), the polcy has converged to that shown n Tables 8.8 and 8.9. Gven ths operatng polcy, the probabltes of beng n any of these volume and flow ntervals can be determned by solvng Equatons 8.4 through Table 8. shows the results of these equatons appled to the data n Tables 8.5 and 8.8. It s obvous that f the polcy from Table 8.9 s followed, the steady-state probabltes of beng n storage Interval n Perod and n Interval n Perod are. Multplyng these jont probabltes by the correspondng SD kt values n the last column of Table 8.6 provdes the annual expected squared devatons, assocated wth the selected dscrete storage volumes and flows. Ths s done n Table 8. for those combnatons of k,, and l that are contaned n the optmal soluton as lsted n Table 8.9. The sum of products of the last two columns n Table 8. for each perod t equals the expected squared devatons n the perod. For perod t, the expected

17 46 Water Resources Systems Plannng and Management Table 8.7. Frst four teratons of dynamc programmng model, Equatons 8.5, movng backward n successve perods n, begnnng n season t wth n. The teratons stop when the fnal storage polcy gven any ntal storage volume and flow repeats tself n two successve years. Intally, wth no more perods remanng, F (k, ) for all k and. storage perod t, n & flow SD kl + Σ j P t j F n- t+ (l, j) F n t (k, ) optmal k, l, l l 4. + l , l l l E89c, l l l , l l l , l l l , l l l , storage & flow k, perod t, n SD kl + Σ j P j t F t+ n- (l, j) F t n (k, ) optmal l E89d, l (989.) +.4 (.5) l (5.) +.4 ( ) 78. l nfeasble , l l l (989.) +.7 (.5) (5.) +.7 ( ) ( 544.5) +.7 ( ) , l l l +.6 (989.) +.4 (.5) (5.) +.4 ( ) ( 544.5) +.4 ( ) , l l l (989.) +.7 (.5) (5.) +.7 ( ) ( 544.5) +.7 ( ) , l l l (989.) +.4 (.5) +.6 (5.) +.4 ( ) ( 544.5) +.4 ( ) , l l l (989.) +.7 (.5) (5.) +.7 ( ) ( 544.5) +.7 ( ) (contd.)

18 Modellng Uncertanty storage & flow k, perod t, n SD kl + Σ j P j t F t+ n- (l, j) F t n (k, ) optmal l E89e Table 8.7. Concluded., l l l (64.4) +. (664.5) (459.4) +. (87.5) (94.4) +. (9.9) , l l l.5 +. (64.4) +.8 (664.5) (459.4) +.8 (87.5).5 +. (94.4) +.8 (9.9) , l l l (64.4) +. (664.5) (459.4) +. (87.5) (94.4) +. (9.9) , l l l +. (64.4) +.8 (664.5).5 +. (459.4) +.8 (87.5) (94.4) +.8 (9.9) , l l l (64.4) +. (664.5) (459.4) +. (87.5) (94.4) +. (9.9) , l l l +. (64.4) +.8 (664.5) +. (459.4) +.8 (87.5).5 +. (94.4) +.8 (9.9) storage & flow k, perod t, n 4 SD kl + Σ j P j t F t+ n- (l, j) F t n (k, ) optmal l E89f, l (699.8) +.4 (6.).8.8 l l (574.8) +.4 (84.8) nfeasble ---.8, l l l (699.8) +.7 (6.) (574.8) +.7 (84.8) (466.) +.7 (7.) , l l l +.6 (699.8) +.4 (6.) (574.8) +.4 (84.8) (466.) +.4 (7.) , l l l (699.8) +.7 (6.) (574.8) +.7 (84.8) (466.) +.7 (7.) , l l l (699.8) +.4 (6.) +.6 (574.8) +.4 (84.8) (466.) +.4 (7.) , l l l (699.8) +.7 (6.) (574.8) +.7 (84.8) (466.) +.7 (7.)

19 48 Water Resources Systems Plannng and Management Table 8.8. Summary of objectve functon values F t n (k, ) and optmal decsons for stages n 5 to 9 perods remanng. storage t, n 5 & flow F n t (k, ) l* k, t, n 6 t, n 7 t, n 8 t, n 9 F n t (k, ) l* F n t (k, ) l* F n t (k, ) l* F n t (k, ) l* E89g,, 69.7, ,,, ,,, sum of squared devatons are 89. and for t they are 59.. The total annual expected squared devatons are Ths compares wth the expected squared devatons derved from the dynamc programmng model, after 9 teratons, rangng from 5.5 to 497. (as calculated from data n Table 8.8). These upstream allocaton polces can be dsplayed n plots, as shown n Fgure 8.. The polcy for reservor releases s a functon not only of the ntal storage volumes, but also of the current nflow, n other words, the total water avalable n the perod. Reservor release rule curves such as shown n Fgures 4.6 or 4.8 now must become two-dmensonal. However, the nflow for each perod usually cannot be predcted wth certanty at the begnnng of each perod. In stuatons where the release cannot be adjusted durng the perod as the nflow becomes more predctable, the reservor release polcy has to be expressed n a way that can be followed wthout knowledge of the current nflow. One way to do ths s to compute the expected value of the release for each dscrete storage volume, and show t n a release rule. Ths s done n Fgure 8.. The probablty of each dscrete release assocated wth each dscrete rver flow s the probablty of the flow tself. Thus, n Perod when the storage volume s, the expected release s 46(.4) 56(.59) 5. These dscrete expected releases can be used to defne a contnuous range of releases for the contnuous range of storage volumes from to full capacty, 5. Fgure 8. also shows the hedgng that mght take place as the reservor storage volume decreases. Another approach to defnng the releases n each perod n a manner that s not dependent on knowledge of the current nflow, even though the model used assumes ths, s to attempt to defne ether release targets wth constrants on fnal storage volumes, or fnal storage targets wth constrants on total releases. Obvously, such polces wll not guarantee constant releases throughout each perod. For example, consder the optmal polcy shown n Table 8.9. The releases (or fnal storage volumes) n each perod are dependent on the ntal storage and current nflow. However, ths operatng polcy can be expressed as: If n perod, the fnal storage target should be n nterval. Yet the total release cannot exceed the flow n nterval. If n perod and the ntal storage s n nterval, the release should be n nterval. If n perod and the ntal storage s n nterval, the release should be n nterval. If n perod and the ntal storage s n nterval, the release should equal the nflow. Ths polcy can be followed wthout any forecast of current nflow. It wll provde the releases and fnal storage volumes that would be obtaned wth a perfect nflow forecast at the begnnng of each perod.

20 Modellng Uncertanty 49 perod ntal storage volume and flow nterval fnal storage volume nterval t k l E89h Table 8.9. Optmal reservor polcy l (k,, t) for the example problem. uncondtonal probabltes PQ t of flow ntervals n the tme perods t PQ (, ).6 PQ (, ).4594 PQ (, ).5885 PQ (, ).5766 E89j Table 8.. Probabltes of flow and storage volume ntervals assocated wth the polcy as defned n Table 8.9 for the example problem. uncondtonal probabltes PS kt of storage ntervals k n the tme perods t PS (, ) PS (, ).5885 PS (, ).4594 PS (, ) PS (, ).5766 PS (, ) jont probabltes Pkt of storage volume ntervals k and flow ntervals n the tme perods t P (,, ) P (,, ) P (,, ) P (,, ).56 P (,, ).966 P (,, ).859 P (,, ).7588 P (,, ).85 P (,, ).594 P (,, ) P (,, ) P (,, )

21 5 Water Resources Systems Plannng and Management Table 8.. The optmal operatng polcy and the probablty of each state and decson. ntal storage S k flow Q fnal storage nterval ndces S k,, l tme perod ukt optmal allocaton dkt sum squared devatons SDkt jont probablty P kt E89k ,,,,,,,,,,,, sum ,,,,,,,,,,,, sum. upstream user allocaton n perod upstream user allocaton n perod rver flow 9 rver flow ndependent of storage volume storage volume Eq storage volume perod reservor releases perod Fgure 8.. Reservor release rule showng an nterpolated release, ncreasng as storage volumes ncrease Er Fgure 8.. Upstream user allocaton polces. In Perod they are ndependent of the downstream ntal storage volumes. In Perod the operator would nterpolate between the three allocaton functons gven for the three dscrete ntal reservor storage volumes.