MODULE 1 LECTURE NOTES 3

Transcription

1 Water Resources Systems Plag ad Maagemet: Itroducto ad Basc Cocepts: Optmzato ad Smulato MODULE LECTURE NOTES 3 OPTIMIZATION AND SIMULATION INTRODUCTION I the prevous lecture we studed the bascs of a optmzato problem ad ts formulato as a mathematcal programmg problem. I ths lecture we look at the varous crtera for classfcato of optmzato problems, ecoomc cosderatos ad challeges water resources. MODELLING TECHNIQUES The modellg or system aalyss techques were developed durg the Secod World War to deploy lmted resources a optmum maer. Sce, these techques were aded for mltary operatos, these were kow as operato research techques. The popular operatos research techques clude optmzato methods, smulato, game theory, queug theory etc. Amog, these, the popular oes water resources feld are optmzato ad smulato. OPTIMIZATION Optmzato s the scece of choosg the best amogst a umber of possble alteratves. There may be umber of possble solutos for may egeerg problems. It s requred to detfy the best through evaluato. The drvg force the optmzato s the obectve fucto (or fuctos). The optmal soluto s the oe whch gves the best (ether maxmum or mmum) soluto uder all assumptos ad costrats. Optmzato theory s defed as the brach of mathematcs dealg wth techques for maxmzg or mmzg a obectve fucto subect to lear, o-lear ad teger costrats. A optmzato model ca be stated as: Obectve fucto: Maxmze (or Mmze) f(x) Subect to the costrats g (X) 0, =,2,..,m h (X) = 0, = m+, m+2,.., p where X s the vector of decso varables, g(x) are the equalty costrats ad h(x) are the equalty costrats. Classfcato of Optmzato Techques Optmzato problems ca be classfed based o the type of costrats, ature of desg varables, physcal structure of the problem, ature of the equatos volved, permssble D Nagesh Kumar, IISc, Bagalore

2 Water Resources Systems Plag ad Maagemet: Itroducto ad Basc Cocepts: Optmzato ad Smulato 2 value of the desg varables, determstc/ stochastc ature of the varables, separablty of the fuctos ad umber of obectve fuctos. These methods are brefly dscussed below. Classfcato based o exstece of costrats Uder ths category optmzato problems ca be classfed to two groups as follows: Costraed optmzato problems: whch are subect to oe or more costrats. Ucostraed optmzato problems: whch o costrats exst. Classfcato based o the physcal structure of the problem Based o the physcal structure, optmzato problems are classfed as optmal cotrol ad o-optmal cotrol problems. () Optmal cotrol problems A Optmal cotrol (OC) problem s a mathematcal programmg problem volvg a umber of stages, where each stage evolves from the precedg stage a prescrbed maer. It s defed by two types of varables: the cotrol or desg ad state varables. The cotrol varables defe the system ad cotrols how oe stage evolves to the ext. The state varables descrbe the behavor or status of the system at ay stage. The problem s to fd a set of cotrol varables such that the total obectve fucto (also kow as the performace dex, PI) over all stages s mmzed, subect to a set of costrats o the cotrol ad state varables. A OC problem ca be stated as follows: Subect to the costrats Fd X whch mmzes f(x) = f ( x, y ) ( x, y ) y y l q =, 2,., l g x ) 0, =, 2,., l ( h y ) 0, k =, 2,., l k ( k Where x s the th cotrol varable, y s the th state varable, ad f s the cotrbuto of the th stage to the total obectve fucto. g, h k, ad q are the fuctos of x, y ; x k, y k ad x, y respectvely, ad l s the total umber of states. The cotrol ad state varables x ad y ca be vectors some cases. () Problems whch are ot optmal cotrol problems are called o-optmal cotrol problems. D Nagesh Kumar, IISc, Bagalore

3 Water Resources Systems Plag ad Maagemet: Itroducto ad Basc Cocepts: Optmzato ad Smulato 3 Classfcato based o the ature of the equatos volved Based o the ature of equatos for the obectve fucto ad the costrats, optmzato problems ca be classfed as lear, o-lear, geometrc or quadratc programmg problems. Ths classfcato s much useful from a computatoal pot of vew sce may predefed specal methods are avalable for effectve soluto of a partcular type of problem. () Lear programmg problem If the obectve fucto ad all the costrats are lear fuctos of the desg varables, the optmzato problem s called a Lear Programmg Problem (LPP). A lear programmg problem s ofte stated the stadard form: Fd X = x x 2.. x Subect to the costrats Whch maxmzes f(x) = c x a x b, =, 2,..., m x 0, =, 2,..., m where c, a, ad b are costats. () No-lear programmg problem If ay of the fuctos amog the obectves ad costrat fuctos s o-lear, the problem s called a No-Lear Programmg (NLP) problem. Ths s the most geeral form of a programmg problem ad all other problems ca be cosdered as specal cases of the NLP problem. () Geometrc programmg problem A Geometrc Programmg Problem (GPP) s oe whch the obectve fucto ad costrats are expressed as polyomals X. A fucto h(x) s called a polyomal (wth m terms) f h ca be expressed as h( X ) c x a x a2 a a2 a22 a2 am a2m 2 x c2x x2 x cm x x2 x am D Nagesh Kumar, IISc, Bagalore

4 Water Resources Systems Plag ad Maagemet: Itroducto ad Basc Cocepts: Optmzato ad Smulato 4 where c (,, m ) ad a (,, ad,, m ) are costats wth c 0 ad x 0. Thus GPP problems ca be posed as follows: Fd X whch mmzes N 0 f(x) = c x, c > 0, x > 0 a subect to Nk k g k (X) = a k x 0, a k > 0, q k > 0, x > 0, k =,2,..,m q where N 0 ad N k deote the umber of terms the obectve fucto ad the k th costrat fucto, respectvely. (v) Quadratc programmg problem A quadratc programmg problem s the best behaved o-lear programmg problem wth a quadratc obectve fucto ad lear costrats ad s cocave (for maxmzato problems). It ca be solved by sutably modfyg the lear programmg techques. It s usually formulated as follows: f(x) = c q x Q x x Subect to a x b, =,2,.,m x 0, =,2,., where c, q, Q, a, ad b are costats. Classfcato based o the permssble values of the decso varables Uder ths classfcato, obectve fuctos ca be classfed as teger ad real-valued programmg problems. () Iteger programmg problem If some or all of the desg varables of a optmzato problem are restrcted to take oly teger (or dscrete) values, the problem s called a teger programmg problem. For example, the optmzato problem s to fd umber of artcles eeded for a operato wth D Nagesh Kumar, IISc, Bagalore

5 Water Resources Systems Plag ad Maagemet: Itroducto ad Basc Cocepts: Optmzato ad Smulato 5 least effort. Thus, mmzato of the effort requred for the operato beg the obectve, the decso varables,.e., the umber of artcles used ca take oly teger values. Other restrctos o mmum ad maxmum umber of usable resources may be mposed. () Real-valued programmg problem A real-valued problem s that whch t s sought to mmze (or maxmze) a real fucto by systematcally choosg the values of real varables from wth a allowed set. Whe the allowed set cotas oly real values, t s called a real-valued programmg problem. Classfcato based o determstc/ stochastc ature of the varables Uder ths classfcato, optmzato problems ca be classfed as determstc or stochastc programmg problems. () Determstc programmg problem I a determstc system, for the same put, the system wll produce the same output always. I ths type of problems all the desg varables are determstc. () Stochastc programmg problem I ths type of a optmzato problem, some or all the desg varables are expressed probablstcally (o-determstc or stochastc). Classfcato based o separablty of the fuctos Based o ths classfcato, optmzato problems ca be classfed as separable ad oseparable programmg problems based o the separablty of the obectve ad costrat fuctos. () Separable programmg problems I ths type of problem, the obectve fucto ad the costrats are separable. A fucto s sad to be separable f t ca be expressed as the sum of sgle-varable fuctos, f,... x, f 2 x2 f x,.e. f ( X ) f x ad separable programmg problem ca be expressed stadard form as : Fd X whch mmzes f ( X ) f x subect to g ( X ) g x b, =,2,..., m where b s a costat. D Nagesh Kumar, IISc, Bagalore

6 Water Resources Systems Plag ad Maagemet: Itroducto ad Basc Cocepts: Optmzato ad Smulato 6 Classfcato based o the umber of obectve fuctos Uder ths classfcato, obectve fuctos ca be classfed as sgle-obectve ad multobectve programmg problems. () Sgle-obectve programmg problem whch there s oly oe obectve fucto. () Mult-obectve programmg problem A multobectve programmg problem ca be stated as follows: Fd X whch maxmzes/mmzes Subect to f X, f 2 X,... f g (X) 0, =, 2,..., m where f, f 2,... f k deote the obectve fuctos to be maxmzed/mmzed smultaeously. k X SIMULATION Smulato process duplcates the system s behavour by desgg a model of the system ad coductg expermets for a better uderstadg of the system fuctog varous probable scearos. Smulato reproduces the respose of the system to ay mposed future codtos. The ma advatage of smulato s ts ablty to accurately descrbe the realty. The operatg polces ca be tested through smulato before mplemetg actual stuatos. Ofte, water resources systems are too complex to be expressed ay aalytcal expresso. A smulato model duplcates the system s operato wth a defed operatoal polcy, parameters, tme seres of flows, demads etc. The desg parameters or operato polcy are usually evaluated through the obectve fucto or some relablty measures. Steps Smulato. Problem defto: Defe the goals of the study 2. System defto: Idetfy the water resources system compoets ad ts hydrologcal aspects. Idetfy the performace measures to be aalysed. 3. Model desg: Uderstad the behavor of actual system. Decde the model structure by determg the varables descrbg the system, ts teracto ad varous parameters of structures. Decde the puts (tme seres of flows, demads of the system, operato polces etc) ad outputs (hydrologcal varables ad desg varables). 4. Data Collecto: Determe the type of data to be collected. New/ Old data s collected/ gathered. D Nagesh Kumar, IISc, Bagalore

7 Water Resources Systems Plag ad Maagemet: Itroducto ad Basc Cocepts: Optmzato ad Smulato 7 5. Valdato: Test the model ad apply the model to the problem Classfcato of Smulato models Smulato models ca be Physcal (e.g. a scale model of a spllway), aalog (system of electrcal compoets such as resstors or capactors arraged to act as a aalog to the hydrologcal compoets) or mathematcal (acto of a system expressed as equatos or logcal statemets. Smulato models ca ether be statc (fxed parameters ad operatoal polcy) or be dyamc (takes to accout the chage the parameters of the system ad the operatoal polcy wth tme) ature. Sce may hydrologcal varables are stochastc character, smulato models ca be determstc or stochastc depedg o the way ths stochastcty s accouted for. Smulato models ca be statstcal or process oreted, or a mxture of both. Pure statstcal models are based solely o data (feld measuremets). Regresso ad artfcal eural etworks are examples of purely statstcal models. Pure process oreted models are based o kowledge of the fudametal processes that are takg place. I ths, calbrato usg feld data s requred to estmate the parameter values the process relatoshps. Hybrd models corporate some process relatoshps to regresso models or eural etworks. Comparso betwee Optmzato ad Smulato Optmzato models elmate the worst solutos. Smulato tools evaluate the performace for varous cofguratos of the system; but they are ot effectve for choosg the best cofgurato. Smulato smply addresses what-f scearos what may happe f a partcular scearo s assumed or f a partcular decso s made. The users have to specfy the value of desg or decso varables for coductg smulato. Smulato s ot feasble whe there are too may alteratves for decso varables, whch demads a eormous computatoal effort. O the other had, optmzato wll determe the best decso; but the soluto s ofte based o may lmtg assumptos. Hece, to take full advatage of systems techques, optmzato should be used to defe a relatvely small umber of good alteratves that ca later be tested, evaluated ad mproved by meas of smulato. ECONOMICS IN WATER RESOURCES Ecoomcs egeerg deals wth applyg ecoomc crtera to select the best soluto from a group of feasble alteratves or evolvg the best ecoomc polcy for plag ad D Nagesh Kumar, IISc, Bagalore

8 Water Resources Systems Plag ad Maagemet: Itroducto ad Basc Cocepts: Optmzato ad Smulato 8 maagemet of a egeerg proect. The rakg ad selecto of alteratves are doe based o the prcples of egeerg ecoomcs. The magtude of cosequeces expected from employg each alteratve are assessed ad coverted to commesurable uts for comparso. Cocepts used ecoomc aalyss: Cash flow dagram: Assess the cosequeces of each alteratve ad assg a moetary value for each cosequece. The graphc represetato of each moetary value wth tme s called a cash flow dagram. The beefts are represeted as upward arrows ad costs as dowward arrows. It s draw to covert the tme stream of moetary value to a equvalet sgle umber. All cash flows are combed to a equvalet sgle lump sum at the ed of a perod. A example cash flow dagram s show below. At the begg, a large expedture s made. Beefts are receved thereafter every year. Beefts Costs Cash flow dagram Dscout factors: The amouts at dfferet tmes have dfferet values. I order to compare them, all moetary values are coverted to equvalet amouts at some defte tme usg dscout factors. May dscout factors are used as gve below. Compoud amout A amout P vested at the begg of frst year grows to Q at the factor ed of years, Q = P(+) Preset worth factor Iverse of the above, gves the preset value of a future amout, P = Q/(+) Skg fud factor The amout X that wll be receved at the ed of each year to get Q at the ed of years, D Nagesh Kumar, IISc, Bagalore

9 Water Resources Systems Plag ad Maagemet: Itroducto ad Basc Cocepts: Optmzato ad Smulato 9 Captal recovery factor Seres compoud factor Seres preset worth factor X = Q / [(+) -] The amout X that should be vested at the ed of each year, f amout P s vested at the begg of frst year, X = P (+) / [(+) -] The amout Q that wll be receved at the ed of th year, f a amout X s vested at the ed of each year, Q = X [(+) -] / The preset value of P f a amout X s vested at the ed of each year, P = X [(+) -] / (+) Suk cost: The moey spet already whch has o ecoomc relevace decdg future alteratves Salvage value: The value of the uused lfe of a elemet at the ed of the perod of aalyss. The salvage value, S = I ( U/L), where I = tal value, U = uused lfe ad L = total lfe. Dscoutg techques Dscoutg techques are used to fd the feasble oe amog varous alteratves. Commoly used dscoutg techques are () Beeft-cost rato method (2) Preset worth method (3) Rate of retur method ad (4) Aual cost method. The frst two methods are explaed here. Beeft Cost (BC) rato method BC rato, R s defed as the rato of the preset worth of beefts ad the preset worth of cost. It ca be expressed as R B C t t 0 0 t C B t where B t ad C t are the moetary values of beefts ad costs curred at tme t respectvely, s the dscout rate ad s the lfe of the proect tme steps (years or moths or weeks). The steps to be followed for choosg the best alteratve are: () Calculate the BC value for each alteratve t t D Nagesh Kumar, IISc, Bagalore

10 Water Resources Systems Plag ad Maagemet: Itroducto ad Basc Cocepts: Optmzato ad Smulato 0 () Reta all alteratves wth BC> ad reect the rest. If sets of mutually exclusve alteratves are volved the go to steps (), (v) ad (v). () Rak the set of mutually exclusve alteratves the order of creasg cost. Calculate the BC rato usg cremetal cost ad cremetal beeft of the ext alteratve above the least costly alteratve. (v) Choose the more costly alteratve of the cremetal BC >. Otherwse choose the less costly alteratve. (v) Repeat the aalyss for all alteratves the order of rak. Example Two alteratve plas are feasble. The estmated cost of st pla s 70 lakhs ad the correspodg beeft s 80 lakhs. The cost for the 2 d pla s 85 lakhs ad beeft s 00 lakhs. Whch pla should be selected? Soluto BC rato for st pla = 80/70 =.4 BC rato for 2 d pla = 00/85 =.76 Sce BC ratos are >, accordg to steps () ad (v), rak the plas based o cost. Pla s raked ad pla 2 s raked 2. Now, cremetal cost (2 d pla over st pla) = = 5 Icremetal beeft (2 d pla over st pla) = =20 Icremetal BC rato = 20/5 =.33. Sce Icremetal BC rato >, the more costly alteratve should be selected. Therefore, pla 2 s chose. Preset worth method I ths method, the et worth (beeft cost ) for each year s computed ad dscouted to the preset wth usg the preset worth factor. Ther sum s the Net Preset Value (NPV). NPV B 0 C 0 0 B where B t ad C t are the moetary values of beefts ad costs curred at tme t respectvely, s the dscout rate ad s the lfe of the proect. The steps for selectg the best alteratve are: () Determe the NPV of each alteratve. C... B C D Nagesh Kumar, IISc, Bagalore

11 Water Resources Systems Plag ad Maagemet: Itroducto ad Basc Cocepts: Optmzato ad Smulato () Reta those alteratves wth NPV > 0 ad reect the rest. If there s ay mutually exclusve alteratve, the proceed to steps () ad (v). Otherwse, stop. () Choose the oe wth greatest NPV from the set of mutually exclusve alteratves. (v) If a set of mutually exclusve alteratves, some have beefts that caot be quatfed but are approxmately equal, the choose the oe wth least cost. CHALLENGES IN WATER SECTOR The maor challeges water sector as per World Water Forum are lsted below: Meetg basc eeds 20% of the world populato do ot have access to adequate safe drkg water. 3-4 mllo people de due to water-bore dseases every year. Access to water s a basc huma eed. Whle the partcpato of commuty s essetal to esure secure ad sustaable supples of water, wome as the custodas of health ad hygee hold the key such effort. Protectg ecosystems Aquatc ecosystems are repostores of bodversty ad form or costtute a crucal part of hydrologcal cycle. The decle ther area ad ther qualty wll tur rvers to ope sewers wthout ay aquatc lfe ad also reduce the bologcal dversty. A balace betwee huma eeds ad the trsc value of ecosystems eeds to be esured. Securg food supply Agrculture s the largest user of water (aroud 90%). The demad for food gras s expected to crease by % the ext 25 years due to the crease populato ad also chage the cosumpto patters. Sharg water resources Rver bass as the ma source of freshwater, eed a cooperatve maagemet. Upper states should cosder the terests of other rpara states the same bas. Dealg wth hazards Too much water ad also too lttle water affects people ad property. Floods ad droughts are recurret pheomea that caot be preveted. Govermet s resposble to provde securty from such hazards. Both structural (embakmets, dams etc) ad o-structural (forecastg systems, cotgecy plas etc) methods are to be adopted for a sustaable flood ad drought maagemet. Valuato of water It s ecessary to uderstad the ecoomc, socal ad cultural values of water. Wastage decreases whe the prce s hgh. For effectve water maagemet, the cocered ageces D Nagesh Kumar, IISc, Bagalore

12 Water Resources Systems Plag ad Maagemet: Itroducto ad Basc Cocepts: Optmzato ad Smulato 2 must have adequate resources. The valuato of water resources ad chargg for water servces eed actve partcpato of stakeholders. Goverg water wsely Rver bas maagemet requres a sutable sttutoal framework. Goverg bodes should allocate water effectvely ad also maage water resources based o legtmate requremets as per agreed polces ad laws. A tegrato of prvate ad publc sectors ad also frequet teractos betwee stakeholders wll create a eablg evromet. The ma actos to be take to solve problems water sector are: a) Itegrated maagemet of rver bass b) Partcpato of commuty the maagemet c) Improved agrcultural practces d) Extesve ad relable database of formato dssemato e) Proper valuato of water BIBLIOGRAPHY/ FURTHER READING:. Global Water Partershp (GWP), Itegrated Water Resources Maagemet, Backgroud Papers No. 4, Techcal Advsory Commttee (TAC), Ja, S.K. ad V.P. Sgh, Water Resources Systems Plag ad Maagemet, Vol. 5, Elsever Scece, Hames, Herarchcal Aalyses of Water Resources Systems: Modelg ad Optmzato of Largescale systems, McGraw-Hll, New York, Loucks D.P. ad va Beek E., Water Resources Systems Plag ad Maagemet, UNESCO Publshg, The Netherlads, Loucks, D.P., J.R. Stedger, ad D.A. Hath, Water Resources Systems Plag ad Aalyss, Pretce-Hall, N.J., Mays, L.W. ad K. Tug, Hydrosystems Egeerg ad Maagemet, McGraw-Hll Ic., New York, 992. D Nagesh Kumar, IISc, Bagalore