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

2. 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

3. 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

4. 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

5. 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

6. 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

7. 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

8. 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

9. 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

10. 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

11. 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

12. 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

13. 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

14. 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

15. 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

16. 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

17. 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

18. 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

19. 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

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