Multi-objective Optimization Multipurpose Reservoir Operation Water Resources Planning and Management: M5L2 D Nagesh Kumar, IISc

Objectives To discuss the common purposes of reservoirs To learn the planning of multi-purpose reservoirs To formulate the operation of multipurpose single and multiple reservoir systems Water Resources Planning and Management: M5L2 D Nagesh Kumar, IISc

Introduction Reservoir operation - Important element in the field of water resources planning and management Different objectives: Flood control, hydropower generation and water allocation to different users etc. Several control variables in order to define the operation strategies for guiding a sequence of releases to meet the demands Often, these objectives are conflicting and unequal Makes reservoir operation a difficult task Balanced solutions between the conflicting objectives are needed to optimize reservoir operation Water Resources Planning and Management: M5L2 D Nagesh Kumar, IISc

Combinations of multipurpose reservoir For effective utilization of water, some of the purposes are combined often The preferred combinations are: Irrigation and power Irrigation, power and navigation Irrigation, power and water supply Recreation, fisheries and wild life Flood control and water supply Power and water supply Flood control, irrigation, power and water supply – most common combination. Water Resources Planning and Management: M5L2 D Nagesh Kumar, IISc

Planning of multipurpose reservoir Purposes of a reservoir may not be compatible to one another Unique feature of multipurpose design is an operation plan which effectively compromises the various purposes There are two possible extremes in reservoir storage allocation: No storage is jointly used All storage is jointly used Total storage requirement is the sum of storage requirements from all purposes This can be economically obtained when the unit cost of storage is constant or the unit cost decreases as the total storage increases. Water Resources Planning and Management: M5L2 D Nagesh Kumar, IISc

Planning of multipurpose reservoir… All storage is jointly used Storage required is not greater than that necessary for any one of the many purposes Gives maximum economy Usually a multipurpose reservoir is designed in between these extremes. Freeboard Surcharge Flood control Conservation Inactive Sediment reserve Reservoir Pooling Reservoir operating policies typically divide the storage capacity into several pools according to the intended purposes. Typical reservoir pooling for multipurpose Water Resources Planning and Management: M5L2 D Nagesh Kumar, IISc

Reservoir Pooling Water in the inactive pool or dead storage is not utilized for any purpose It serves as a head for hydropower generation, recreation, fish habitat or sediment reserve Conservation storage purposes include municipal and industrial water supply, irrigation, hydroelectric power, navigation etc. Freeboard Surcharge Flood control Conservation Inactive Sediment reserve Flood control pool remains empty, except during and immediately after a flood event Operation procedures include emptying the flood control pools as quickly as possible after a flood event so as to be prepared for accomodating next flood. Releases should be made by ensuring not to cause downstream flooding. Surcharge pool is uncontrolled storage capacity above the flood control pool. Water Resources Planning and Management: M5L2 D Nagesh Kumar, IISc

Formulation of Multi-purpose Reservoir System Optimal sizing and Operation of a single multipurpose reservoir Consider a multipurpose reservoir designed for Water supply, Irrigation, Power generation and Recreation Optimization problem: To determine both the capacity and operation of the reservoir that maximizes the annual net benefit Primary decision variables: Reservoir storage and the releases at particular periods to various needs Inflow, It Evaporation,EVt Precipitation, Pt Storage, St Release (irrigation+ water supply), Rt Hydropower energy, HPt Water Resources Planning and Management: M5L2 D Nagesh Kumar, IISc

Optimal sizing and Operation of a single multipurpose reservoir Objective function: Maximize where NB is the annual net benefit, Bi(Ti) is the benefit from target allocation Ti to ith user. Di,t and Ei,t are the deficit and excess with respect to Ti for user i in period t The corresponding loss and gain functions are Li,t and Gi,t C(K) is the annual cost for the reservoir of capacity K Inflow, It Evaporation,EVt Precipitation, Pt Storage, St Release (irrigation+ water supply), Rt Hydropower energy, HPt Water Resources Planning and Management: M5L2 D Nagesh Kumar, IISc

Optimal sizing and Operation of a single multipurpose reservoir… Typical constraints in a reservoir optimization model Hydraulic constraints as defined by the reservoir continuity equation: St+1 = St + It + Pt – EVt -Rt for t = 1,2,…,N where St+1 is storage at time step t+1 St is storage at time step t It is the reservoir net inflow at time step t (including reservoir inflow, precipitation and evaporation) Rt is the reservoir outflow at time step t N is the total number of time steps in the considered period. Inflow, It Evaporation,EVt Precipitation, Pt Storage, St Release (irrigation+ water supply), Rt Hydropower energy, HPt Water Resources Planning and Management: M5L2 D Nagesh Kumar, IISc

Optimal sizing and Operation of a single multipurpose reservoir… Typical constraints in a reservoir optimization model Constraints on total discharge and releases for various purposes Rt = Rirr,t + Rws,t + Rins,t for all t Rhp,t = Rins,t for all t where Rirr,t , Rws,t , Rins,t and Rhp,t are releases for irrigation, water supply, instream flow requirement and power generation respectively. These relations are problem specific. Reservoir capacity St ≤ K – Kd for all t, where Kd is the dead storage Inflow, It Evaporation,EVt Precipitation, Pt Storage, St Release (irrigation+ water supply), Rt Hydropower energy, HPt Water Resources Planning and Management: M5L2 D Nagesh Kumar, IISc

Optimal sizing and Operation of a single multipurpose reservoir… Typical constraints in a reservoir optimization model Target allocation for irrigation Rirr,t + Dirr,t - Eirr,t = Tirr,t for all t. Target allocation for water supply release Rws,t + Dws,t – Ews,t = Tws,t for all t. Target allocation for instream flow release Rins,t + Dins,t – Eins,t = Tins,t for all t. Target allocation for power supply ξ Rhp,t h(Kd + St, Kd + St+1) + Dhp,t – Ehp,t = Thp,t for all t. where ξ is the plant efficiency. Target allocation for recreation Kd + St + Drec,t – Erec,t = Trec,t for all t. Water Resources Planning and Management: M5L2 D Nagesh Kumar, IISc

Optimal sizing and Operation of a multiple reservoir systems Consider a three reservoir system in which all reservoirs are multipurpose The purposes are same as that of the previous problem. The hydropower generation is done by taking advantage of the head drop. No additional release is made for generating hydropower. Objective: Maximize the net benefit by determining the optimal capacity and release policy of each reservoir 1 2 3 HP1 HP2 HP3 R1,t R2,t R3,t Rirr,t Rws,t Rins,t Power demand Water Resources Planning and Management: M5L2 D Nagesh Kumar, IISc

Optimal sizing and Operation of a multiple reservoir systems… Objective function for this optimization model: Maximize NB = 1 2 3 HP1 HP2 HP3 R1,t R2,t R3,t Rirr,t Rws,t Rins,t Power demand Water Resources Planning and Management: M5L2 D Nagesh Kumar, IISc

Optimal sizing and Operation of a multiple reservoir systems… Constraints Subject to Mass balance for three reservoirs Ss,t+1 = S s,t + I s,t + P s,t – EV s,t - R s,t for s = 1,2 and t = 1,2,…,N S3,t+1 = S 3,t + I3,t + P3,t – EV3,t + R1,t + R2,t - R3,t for t = 1,2,…,N Rins,s,t = Rs,t for s=1,2 and for all t Rins,3,t = R3,t – Rws,3,t - Rirr,3,t for all t Rhp,s,t = Rs,t for s=1,2,3 and for all t. Hydropower generation ξs Rs,t hs,t(Kd,s + Ss,t , Kd,s + Ss,t+1) + Dhp,t – Ehp,t = Thp,t for all t. Target allocation for irrigation Rirr,3,t + Dirr,t - Eirr,t = Tirr,t for all t. Water Resources Planning and Management: M5L2 D Nagesh Kumar, IISc

Optimal sizing and Operation of a multiple reservoir systems… Constraints Subject to Mass balance for three reservoirs Ss,t+1 = S s,t + I s,t + P s,t – EV s,t - R s,t for s = 1,2 and t = 1,2,…,N S3,t+1 = S 3,t + I3,t + P3,t – EV3,t + R1,t + R2,t - R3,t for t = 1,2,…,N Hydropower generation ξs Rs,t hs,t(Kd,s + Ss,t , Kd,s + Ss,t+1) + Dhp,t – Ehp,t = Thp,t for all t. Target allocation for irrigation Rirr,3,t + Dirr,t - Eirr,t = Tirr,t for all t. Target allocation for water supply release Rws,3,t + Dws,t – Ews,t = Tws,t for all t. Water Resources Planning and Management: M5L2 D Nagesh Kumar, IISc

Optimal sizing and Operation of a multiple reservoir systems… Constraints Subject to Target allocation for water supply release Rws,3,t + Dws,t – Ews,t = Tws,t for all t. Target allocation for instream flow release in each stream section Rs,t + Dins,s,t – Eins,s,t = Tins,s,t for s=1,2 and for all t. R3,t – Rws,3,t – Rirr,3,t + Dins,3,t – Eins,3,t = Tins,3,t for all t. Target allocation for recreation Kd,s + Ss,t + Drec,s,t – Erec,s,t = Trec,s,t for s=1,2,3 and for all t. Reservoir capacity Ss,t ≤ Ks – Kd,s for s=1,2,3 and for all t. Water Resources Planning and Management: M5L2 D Nagesh Kumar, IISc

Operation of multi-objective multipurpose reservoir In cases where operation objectives have trade-offs, single-objective optimization cannot provide a unique optimum solution. Some objectives can be improved by sacrificing the others. Concept of “non-inferiority” (explained in the previous lecture) replaces the single- objective optimization problem (either maximization or minimization) The most suitable solution is chosen by the operator according to the preferences Water Resources Planning and Management: M5L2 D Nagesh Kumar, IISc

Operation of multi-objective multipurpose reservoir… A multi-objective reservoir operation problem can be formulated as follows Maximize Z(X) = [Z1(X), Z2(X), …, Zn(X)] Subject to gi(X) ≥ 0 for i = 1, 2, …, m. where X is a vector of decision variables; Zj(X), j=1,…, n are the objective functions and gi(X) ), i=1,…,m are the constraints that define the feasible solutions. Water Resources Planning and Management: M5L2 D Nagesh Kumar, IISc

Thank You Water Resources Planning and Management: M5L2 D Nagesh Kumar, IISc