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
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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
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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:
(i)
Irrigation and power
(ii)
Irrigation, power and navigation
(iii) Irrigation, power and water supply
(iv) Recreation, fisheries and wild life
(v)
Flood control and water supply
(vi) Power and water supply
(vii) Flood control, irrigation, power and water supply
– most common combination.
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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
No 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.
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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
Reservoir Pooling
Surcharge
Reservoir operating policies typically
Flood control
divide the storage capacity into several
Conservation
pools
according
to
the
intended
purposes.
Sediment
reserve
Inactive
Typical reservoir pooling for multipurpose
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Water Resources Planning and Management: M5L2
D Nagesh Kumar, IISc
Reservoir Pooling
 Water in the inactive pool or dead storage is not
Freeboard
Surcharge
utilized for any purpose
 It serves as a head for hydropower generation,
Flood control
recreation, fish habitat or sediment reserve
Conservation
 Conservation storage purposes include municipal
and
industrial
water
supply,
irrigation,
Sediment
reserve
Inactive
hydroelectric power, navigation etc.
 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.
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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,
Evaporation,EVt
Precipitation, Pt
Inflow, It
 Power generation and
 Recreation
Storage, St
Release
(irrigation+ water
supply), Rt
Hydropower
energy, HPt
 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
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Water Resources Planning and Management: M5L2
D Nagesh Kumar, IISc
Optimal sizing and Operation of a single
multipurpose reservoir
Evaporation,EVt
Precipitation, Pt
Inflow, It
Storage, St
Hydropower
energy, HPt
Objective function:

Release
(irrigation+ water
supply), Rt

Maximize NB   Bi Ti    Li ,t Di ,t   Gi ,t Ei ,t   C K 
i

t

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
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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
Evaporation,EVt
Precipitation, Pt
Release
(irrigation+ water
supply), Rt
Inflow, It
Storage, St
Hydropower
energy, HPt
(i) 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.
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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
Evaporation,EVt
Precipitation, Pt
Inflow, It
Storage, St
Release
(irrigation+ water
supply), Rt
Hydropower
energy, HPt
(ii) Constraints on total discharge and releases for various purposes
Rt
= Rirr,t + Rws,t + Rins,t
Rhp,t = Rins,t
for all 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.
(iii)Reservoir capacity
St ≤ K – Kd
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for all t, where Kd is the dead storage
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
(iv)Target allocation for irrigation
Rirr,t + Dirr,t - Eirr,t = Tirr,t
for all t.
(v) Target allocation for water supply release
Rws,t + Dws,t – Ews,t = Tws,t
for all t.
(vi)Target allocation for instream flow release
Rins,t + Dins,t – Eins,t = Tins,t
for all t.
(vii)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.
(viii)Target allocation for recreation
Kd + St + Drec,t – Erec,t = Trec,t
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Water Resources Planning and Management: M5L2
for all t.
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
1
HP1
2
HP2
R1,t
R2,t
by determining the optimal capacity
and release policy of each reservoir
HP3
3
R3,t
Power
demand
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Water Resources Planning and Management: M5L2
Rirr,t
Rws,t
Rins,t
D Nagesh Kumar, IISc
Optimal sizing and Operation of a multiple reservoir
systems…
Objective function for this optimization model:
 B R
  B R
irr ,t
irr ,3 ,t
t
ws ,t
ws ,3 ,t
,Tirr ,3 ,t , Dirr ,3 ,t , Eirr ,3 ,t 
,Tws ,3 ,t , Dws ,3 ,t , E ws ,3 ,t 
t
  Bhp ,t Rhp ,s ,t , S s ,t ,Thp ,t , Dhp ,t , E hp ,t 
Maximize NB =
1
HP1
2
HP2
R1,t
R2,t
t
  Bins,s ,t Rins,s ,t ,Tins,s ,t , Dins,s ,t , Eins,s ,t 
s
t
  Brec ,s ,t S s ,t ,Trec ,s ,t , Drec ,s ,t , E rec ,s ,t 
s
t
HP3
  C K s 
3
R3,t
Rirr,t
s
Power
demand
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Water Resources Planning and Management: M5L2
Rws,t
Rins,t
D Nagesh Kumar, IISc
Optimal sizing and Operation of a multiple reservoir
systems…
Constraints
Subject to
(i) 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.
(ii) 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.
(iii)Target allocation for irrigation
Rirr,3,t + Dirr,t - Eirr,t = Tirr,t
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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
(i) 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
(ii) 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.
(iii)Target allocation for irrigation
Rirr,3,t + Dirr,t - Eirr,t = Tirr,t
for all t.
(iv)Target allocation for water supply release
Rws,3,t + Dws,t – Ews,t = Tws,t
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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
(iv)Target allocation for water supply release
Rws,3,t + Dws,t – Ews,t = Tws,t
for all t.
(v) 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.
(vi)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.
(vii)Reservoir capacity
Ss,t ≤ Ks – Kd,s
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for s=1,2,3 and
Water Resources Planning and Management: M5L2
for all t.
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
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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.
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Water Resources Planning and Management: M5L2
D Nagesh Kumar, IISc
Thank You
Water Resources Planning and Management: M5L2
D Nagesh Kumar, IISc