MODULE 3 LECTURE NOTES 6 RESERVOIR OPERATION AND RESERVOIR SIZING USING LP

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1 Waer Resources Sysems Planning and Managemen: Linear Programming and Applicaions: Reservoir Operaion and Reservoir Sizing using LP 1 MODULE 3 LECTURE NOTES 6 RESERVOIR OPERATION AND RESERVOIR SIZING USING LP INTRODUCTION In he previous lecures, we discussed applicaions of LP in deciding he opimal irrigaion allocaion and waer qualiy managemen. In his lecure we will discuss abou he applicaions of LP in modeling reservoir operaion and reservoir sizing. RESERVOIR OPERATION Reservoir operaion policies are developed o enable he operaor o ake appropriae decision. The reservoir operaion policy indicaes he amoun of waer o be released based on he sae of he reservoir, demands and he likely inflow o he reservoir. The release from a single purpose reservoir can be done wih he objecive of maximizing he benefis. For muli-purpose reservoirs, here is a need o opimally allocae he releases among purposes. The simples of he operaion policies is he sandard operaion policy (SOP). According o SOP, if he waer available (sorage, S + inflow, I ) a a paricular period is less han he demand D, hen all he available waer is released. If he available waer is more han he demand bu less han demand + sorage capaciy K, hen release is equal o he demand. If afer releasing he demands, here is no space for exra waer, hen he excess waer is also released. This is shown graphically in figure 1. Release C D A B 45 0 O D D + K Available waer = Sorage + Inflow Fig. 1 Sandard Operaing Policy

2 Waer Resources Sysems Planning and Managemen: Linear Programming and Applicaions: Reservoir Operaion and Reservoir Sizing using LP 2 Along OA: Release = waer available; reservoir will be empy afer release. Along AB: Release = demand; excess waer is sored in he reservoir (filling phase). A A: Reservoir is empy afer release. A B: Reservoir is full afer release. Along BC: Release = demand + excess of availabiliy over he capaciy (spill) The releases according o he SOP need no be opimum. The opimizaion of reservoir operaion is done ofen by linear programming (LP) and dynamic programming (DP). DP will be explained in he nex module. Derivaion of opimal operaing policy using LP Consider a reservoir of capaciy K. The opimizaion problem is o deermine he releases R ha opimize an objecive funcion saisfying all he consrains. The objecive funcion can be a funcion of sorage volume or release. The ypical consrains in a reservoir opimizaion model include conservaion of mass and oher hydrological and hydraulic consrains, minimum and maximum sorage and release, hydropower and waer requiremens as well as hydropower generaion limiaions. Evaporaion, EV Inflow, I Sorage, S Release (irrigaion+ waer supply), R Fig. 2 Single reservoir operaion Consider he objecive of meeing he demands o he exen possible i.e., maximizing he releases. The opimizaion model can be formulaed as: Maximize R Subjec o (i) Hydraulic consrains as defined by he reservoir coninuiy equaion S +1 = S + I EV - R - O for all where O is he ouflow. The consrains for ouflow are

3 Waer Resources Sysems Planning and Managemen: Linear Programming and Applicaions: Reservoir Operaion and Reservoir Sizing using LP 3 O = 0 if S + I EV - R K = K [S + I EV - R ] if S + I EV - R > K (ii) Reservoir capaciy S K K d for all, where K d is he dead sorage or simply S K (iii) Targe demand S 0 for all. R D for all. R 0 for all. Large LP problems can be solved very efficienly using LINGO - Language for INeracive General Opimizaion, LINDO Sysems Inc, USA Example Derive an opimal operaing policy for a reservoir o mee a long-erm objecive. Single reservoir operaion wih deerminisic inflows. K = 400. Table 1. Inflow, evaporaion and demand values of he reservoir Inflows Evaporaion Demand Soluion Objecive funcion Subjec o Maximize R S +1 = S + I EV - R - O for all

4 Waer Resources Sysems Planning and Managemen: Linear Programming and Applicaions: Reservoir Operaion and Reservoir Sizing using LP 4 where O is he ouflow O = 0 if S + I EV - R K = K [S + I EV - R ] if S + I EV - R > K S 400 ; S 0; R D ; R 0 for all. The problem is solved using LINGO and he soluion is given in able 2. Table2. LP soluion S I D R EV S +1 O The rule curve derived is shown in figure 3. Fig. 3 Rule curve The opimal operaion of a mulipurpose single and muliple reservoir sysems are discussed in module 5.

5 Cumulaive inflow Waer Resources Sysems Planning and Managemen: Linear Programming and Applicaions: Reservoir Operaion and Reservoir Sizing using LP 5 RESERVOIR SIZING In many siuaions, annual demand may be less han he oal inflow o a paricular sie. However, he ime disribuion of demand and inflows may no mach, which in urn resul in surplus in some periods and defici in some oher periods. Hence, here is a need of sorage srucure i.e., reservoir o sore waer in periods of excess flow and make i available when here is a defici. In order o enable regulaion of he sorage o bes mee he specified demands, he reservoir sorage capaciy should be enough. The problem of reservoir sizing involves deerminaion of he required sorage capaciy of he reservoir when inflows and demands in a sequence of periods are given. Reservoir capaciy can be deermined using wo mehods: Mass curve mehod and Sequen peak algorihm mehod. Mass diagram mehod I was developed by W. Rippl (1883). A mass curve is a plo of he cumulaive flow volumes as a funcion of ime. Mass curve analysis is done using a graphical mehod called Ripple s mehod. I involves finding he maximum posiive cumulaive difference beween a sequence of pre-specified (desired) reservoir releases R and known inflows Q (as shown in figure 4). One can visualize his as saring wih a full reservoir, and going hrough a sequence of simulaions in which he inflows and releases are added and subraced from ha iniial sorage volume value. Doing his over wo cycles of he record of inflows will idenify he maximum defici volume associaed wih hose inflows and releases. This is he required reservoir sorage. Release rae Fig. 4 Typical mass curve Time,

6 Waer Resources Sysems Planning and Managemen: Linear Programming and Applicaions: Reservoir Operaion and Reservoir Sizing using LP 6 Sequen Peak Algorihm This algorihm compues he cumulaive sum of differences beween he inflows and reservoir releases for all periods over he ime inerval [0, T]. Le K be he maximum oal sorage requiremen needed for periods 1 hrough period and R be he required release in period, and Q be he inflow in ha period. Seing K 0 equal o 0, he procedure involves calculaing K using equaion below for upo wice he oal lengh of record. Algebraically, K R 0 Q K 1 if posiive oherwise The maximum of all K is he required sorage capaciy for he specified releases R and inflows, Q. Formulaion of reservoir sizing using LP Linear Programming can be used o obain reservoir capaciy more eleganly by considering variable demands and evaporaion raes. The opimizaion problem is Minimize K a where K a is he acive sorage capaciy Subjec o (i) Hydraulic consrains as defined by he reservoir coninuiy equaion S +1 = S + I EV - R - O for all (ii) Reservoir capaciy S K a S T+1 = S for all where T is he las period. (iii) Targe demands R D for all. STORAGE YIELD A complemenary problem o reservoir capaciy esimaion can be done by maximizing he yield. Firm yield is he consan (or larges) quaniy of flow ha can be released a all imes. I is he flow magniude ha is equaled or exceeded 100% of ime for a hisorical sequence of flows. Linear Programming can be used o maximize he yield, R (per period) from a reservoir of given capaciy, K. The opimizaion problem can be saed as:

7 Waer Resources Sysems Planning and Managemen: Linear Programming and Applicaions: Reservoir Operaion and Reservoir Sizing using LP 7 Maximize R Subjec o (i) Sorage coninuiy equaion S +1 = S + I EV - R - O for all (ii) Reservoir capaciy S K a S T+1 = S for all where T is he las period. BIBLIOGRAPHY / FURTHER READING: 1. Dennis T.L. and L.B. Dennis, Microcompuer Models for Managemen Decision Making, Wes Publishing Company, Loucks, D.P., J.R. Sedinger, and D.A. Haih, Waer Resources Sysems Planning and Analysis, Prenice-Hall, N.J., Mays, L.W. and K. Tung, Hydrosysems Engineering and Managemen, Waer Resources Publicaion, Rao S.S., Engineering Opimizaion Theory and Pracice, Fourh Ediion, John Wiley and Sons, Taha H.A., Operaions Research An Inroducion, 8 h ediion, Pearson Educaion India, Vedula S., and P.P. Mujumdar, Waer Resources Sysems: Modelling Techniques and Analysis, Taa McGraw Hill, New Delhi, Rippl., W., The capaciy of sorage reservoirs for waer supply, Proceedings of he Insiuion of Civil Engineers, 71: