Optimal operation of a multi reservoir system and performance evaluation

International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 –
6480(Print), ISSN 0976 – 6499(Online) Volume 5, Issue 10, October (2014), pp. 165-174 © IAEME
165
OPTIMAL OPERATION OF A MULTI-RESERVOIR
SYSTEM AND PERFORMANCE EVALUATION
Safayat Ali Shaikh
(Visiting Faculty, Dept. of Civil Engineering, Indian Institute of Engineering, Science and
Technology,Shibpur, P.O.: Botanic Garden, Howrah- 711103, West Bengal, India)
ABSTRACT
An efficient algorithm to determine the initial trajectory required for Discrete Differential
Dynamic Programming to derive optimal policy for multi-reservoir operation is developed. These
techniques are applied to the operation problem of the Damodar Valley reservoir system in India with a
historical inflow sequence as input. Water supply for irrigation, municipal and industrial use is selected
as objective of operation. Two types of objective function used in this study: (1) when only deficit
ispenalized is termed as one sided loss function (2) when both the deficit and surplus are penalized is
termed as two-sided loss function. Concept of modern reliability parameters has been used to analysis
the system to obtain different reliability parameters for initial and optimal solutions.
Keywords: Initial Trajectory, Multi-Reservoir Operation, Optimal Operation of Reservoir, One-Sided
Loss Function, Two-Sided Loss Function, Performance Evaluation.
1. INTRODUCTION
Reservoir operation is a sequential decision-making process. Out of several mathematical
techniques ranging from simulation to optimization for optimal operation of reservoirs, Dynamic
Programming (DP) has been recognized as the most efficient solution procedure. The efficiency of DP
is due to the fact that the non-linear and stochastic features that characterize the operation problem can
very well be accommodated in its formulation. Moreover, it has the advantage of effectively
decomposing highly complex problems with a large number of variables into a series of sub-problems
with few variables.
But for multi-reservoir system it suffers from the so-called curse of dimensionality. With the
number of reservoirs in the system increases, computer storage and memory requirements increase
exponentially. This problem becomes more acute when the input in the system is considered as
stochastic. In the deterministic environment, different variants of the DP technique have been proposed
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Volume 5, Issue 10, October (2014), pp. 165-174
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166
to alleviate this dimensionality problem, e.g., Incremental DP (IDP), Differential DP (DDP), Discrete
Differential DP (DDDP) etc. In the past works, different researchers used these techniques with
different degree of success in curbing the dimensionality problem. In this study an efficient algorithm is
developed in conjunction with DDP to obtain initial trajectories then optimal trajectories have been
developed by using DDDP.
2. LITERATURE SURVEY
At the beginning, reservoir operation problem were limited to single reservoir with single
purpose only (Hall, 1964). Techniques were then extended to incorporate the multiple objectives in to
the objective function (Hall and Roefs,1966, Hall et al., 1968). After the introduction of state increment
DP (Larson, 1968), IDP (Hall et al., 1969b), DDP (Jacobson and Mayne, 1970) and DDDP (Heidari et
al., 1971) the multi-reservoir operation problems were taken up.
Heidari et al. [1971] demonstrated the computational efficiency of the DDDP algorithm by
solving the four reservoir problem introduced by Larson [1968]. Meredith [1975] applied DDDP to a
serially linked multi-reservoir system and verified the convergence for different initial policies and
observed that the number of iterations and amount of computation time required for convergence
decreased for an initial policy closer to the optimal policy. Fults et al. [1976] used incremental DP to
optimize the monthly operation of a four reservoir system within the California Central Valley Project
for an annual forecast. The objective function was to maximize the project's energy generation, while
satisfying water supply and firm water commitments. Chung and Helweg [1985] used conventional DP
to determine initial trajetories. Then DDDP used to determine optimal operating policies Lake Oroville
and San Louis, USA. Mousavi and Karamouz [2003] presented a DP optimization model for long term
planning of a four reservoir system in Iran.
Hashimoto et al. [1982] and Moy et al. [1986] observed that as system reliability is increased
when the maximum length of consecutive deficits decreases (resiliency increases), the vulnerability of
the system to larger deficits increases. Maier et al [2001] has introduced an efficient method for
estimating reliability, resiliency and vulnerability which was based on the First-Order Reliability
Method (FORM) and demonstrated for the case study of managing water quality in the
Willametrteriver, Oregon.Kjeldsen and Rosbjerg [2004] have thoroughly investigated reliability,
resiliency and vulnerability of water resources and commented that the strong correlation between
resilience and vulnerability may suggest that resilience should not be explicitly accounted for.
3. DESCRIPTION OF THE SYSTEM UNDER STUDY
Damodar Valley (DV) is extended from 23°30´ N to 24°30´ N latitude and 84°30´ E to 88°30´
E longitude. DV reservoir system is an integrated multipurpose multi-reservoir system in India. The
multiple purposes include water supply for irrigation, municipal and industrial use, hydroelectric power
generation, flood control and navigation. The system consists of reservoirs Konar and Panchet on river
Damodar, reservoirs Tilaiya andMaithon on river Barakar and a barrage at Durgapur. In this study
Konar, Tilaiya, Panchet and Maithon is termed as Reservoir 1, Reservoir 2, Reservoir 3 and Reservoir 4
respective as shown in Fig.1The area between (a) Reservoir 1 and 3 (b) Reservoir2 and 4 (c)
downstream of reservoirs 3 and 4 upto Durgapur barrage and (d) downstream of Durgapur barrage are
defined as ‘Reach 1’, ‘Reach 2’, ‘Reach 3’ and ‘Reach 4’ respectively. In Reach 1, Reach 2 and Reach 3
there are only municipal and industrial (M&I) demands and in Reach 4 there only irrigation demands
for three major crop seasons: Kharif Season (June to October), Rabi Season (November to January) and
Hot Weather Crop Season (February to May). Downstream of reservoir 2, 3 and 4, there are
hydroelectric power plant.

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The mean annual evaporation rates for the reservoirs Konar, Tilaiya, Panchet and Mithon are
1.496m, 1.773m, 1.547m, and 1.389m respectively. Of this, about 50% were recorded in the four hot
months of March, April, May and June. Based on these data, monthly evaporation rates for each of the
four reservoirs are computed. Enroute losses are considered at the rate of 5% of the flows during
monsoon period (June-September) and at 10% during non-monsoon months (October to May).
Fig.1: Schematic Diagram of DV System
4. FORMULATION OF PROBLEM
4.1 Application of DP in DV reservoir system
4.1.1 Objective Function
In DV reservoir system, water supply for irrigation, municipal and industrial use is selected as
the objective of operation, while other purposes are treated as binding constraints on the system
variables. In this optimization model, a loss function is used for this purpose in the form of penalty
function. The penalty is associated with the deviation from the target and the objective is to minimize
the penalty incurred during the years of operation. If ‫ܦ‬௧and ‫ݑ‬௧ represent the target release and
corresponding reservoir release respectively during stage ‫ ,ݐ‬then the generic form of single stage loss
function to be used in the DP recursive equation, and objective function can be expressed as;
‫ܮ‬௧ሺ‫ݑ‬௧ሻ = ۤሺ‫ܦ‬௧ − ‫ݑ‬௧ሻ/‫ܦ‬௧‫ۥ‬ఉ
… ሺ‫ݓ‬ℎ݁‫2 ݋ݐ ݈ܽݑݍ݁ ݀݊ܽ ݐ݊݁݊݋݌ݔ݁ ݊ܽ ݏ݅ ߚ ݁ݎ‬ሻ (1)
In Eq. (1) when numerator is negative, i.e., surplus occurs and is positive for deficit, if not zero.
If only the deficit values are to be penalized then ‫ܮ‬௧becomes an one-sided loss function (OSLF), other

168
hand if both deficit and surplus are penalized then ‫ܮ‬௧will be termed as two-sided loss function(TSLF).
In the present study, both forms of loss functions are used and their relative performances are compared.
4.1.2 DV System Dynamics
The flow of water through the systems of reservoir during the stage ‫,ݐ‬ can be described by the
following set of equations;
‫ݔ‬௧ାଵሺ1ሻ = ‫ݔ‬௧ሺ1ሻ + ‫ݕ‬௧ሺ1ሻ − ‫ݑ‬௧ሺ1ሻ − ݁‫ݒ‬௧ሺ1ሻ (2a)
‫ݔ‬௧ାଵሺ2ሻ = ‫ݔ‬௧ሺ2ሻ + ‫ݕ‬௧ሺ2ሻ − ‫ݑ‬௧ሺ2ሻ − ݁‫ݒ‬௧ሺ2ሻ (2b)
‫ݔ‬௧ାଵሺ3ሻ = ‫ݔ‬௧ሺ3ሻ + ‫ݕ‬௧ሺ3ሻ + ‫ݑ‬௧ሺ1ሻ − ݁݊௧ሺ3ሻ − ‫ݑ‬௧ሺ3ሻ − ݁‫ݒ‬௧ሺ3ሻ (2c)
‫ݔ‬௧ାଵሺ4ሻ = ‫ݔ‬௧ሺ4ሻ + ‫ݕ‬௧ሺ4ሻ + ‫ݑ‬௧ሺ2ሻ − ݁݊௧ሺ4ሻ − ‫ݑ‬௧ሺ4ሻ − ݁‫ݒ‬௧ሺ4ሻ (2d)
In Eq.(2a) to Eq.(2d), ‫ݔ‬௧ሺ݅ሻ, ‫ݑ‬௧ሺ݅ሻ and ‫ݕ‬௧ሺ݅ሻ represent the volume of storage, volume of release
and volume of inflow to the ݅ − ‫ݐ‬ℎ reservoir during time period ‫ .ݐ‬The term݁‫ݒ‬௧ሺ݅ሻ represents the
evaporation loss from the ݅ − ‫ݐ‬ℎ reservoir and is computed from plot of storage capacity versus surface
area from the ݅ − ‫ݐ‬ℎ reservoir. Theterm ݁݊௧ሺ݅ሻ represents en-route loss in the reach between ሺ݅ − 2ሻ‫ݐ‬ℎ
reservoir and ݅ − ‫ݐ‬ℎ reservoir.This en-route loss also includes the part of water consumed in the
municipal and industrial uses. Throughout the analysis, this consumptive use is assumed as 40% of the
demand.
4.1.3 DV reservoir system constraints
On storage; ࡯௠௜௡ ≤ ࢞௧ ≤ ࡯௠௔௫ (3a)
On release: ࢛௧,௠௜௡ ≤ ࢛௧ ≤ ࢛௧,௠௔௫ (3b)
Terms ࡯௠௜௡and ࡯௠௔௫represent vectors of minimum and maximum storage capacities of the four
reservoirs where as ࢛௧,௠௜௡and ࢛௧,௠௔௫ are the vector of minimum mandatory releases and maximum
permissible releases based on channel capacities.
4.1.4 DV reservoir system recursive equation
The state space and the control space are discretized by finite set of vectors {࢞௧
௜
}௜ୀଵ
ூ
and
{࢛௧
௝
}௝ୀଵ
௃
respectively. Then discretized version of recursive equation (4) is;
ܸ௧൫࢞௧
௜
൯ = min
௨೟
ೕ
[ ‫ܮ‬௧ + ܸ௧ାଵሺ࢞௧ାଵ
௞
ሻ] … ݂‫ݐ ݎ݋‬ = ܰ, ܰ − 1, … 1 (4)
where ܸேାଵ൫࢞ேାଵ
௞
൯ = 0 ݂‫,݇ ݈݈ܽ ݎ݋‬ ܽ݊݀ ࢛௝
௧
, ࢞௝
௧
, ࢞௞
௧ାଵ
= ݂௧൫࢞௜
௧
, ࢛௝
௧
൯ ݂݁ܽ‫.݈ܾ݁݅ݏ‬The term ‫ܮ‬௧in Eq.(4) is
the single stage loss function and is defined as;
‫ܮ‬௧ = ۤሺ‫ܦ‬௧ሺ1ሻ − ‫ݑ‬௧ሺ1ሻሻ/‫ܦ‬௧ሺ1ሻ‫ۥ‬ଶ
+ ۤሺ‫ܦ‬௧ሺ2ሻ − ‫ݑ‬௧ሺ2ሻሻ/‫ܦ‬௧ሺ2ሻ‫ۥ‬ଶ
+ ۤሺ‫ܦ‬௧
௖
− ‫ݑ‬௧ሺ3ሻ − ‫ݑ‬௧ሺ4ሻሻ/‫ܦ‬௧
௖‫ۥ‬ଶ
(5)
where ‫ܦ‬௧ሺ1ሻand ‫ܦ‬௧ሺ2ሻrepresent the water supply target levels for reservoirs 1 and 2
respectively and ‫ܦ‬௧
௖
represents the combined water supply target level for reservoirs 3 and 4.

169
The main difficulty associated with the discrete dynamic programming (DDP) approach is the
excessive computational burden imposed by the discretization of the state vector and the control
vector.In order to avoid this excessive computational burden, optimal policies have been derived using
Discrete Differential Dynamic Programming (DDDP) as proposed by Heidari et al. [1971].
4.2 Application of DDDP to DV reservoir system
DDDP is an iterative procedure which determines the optimal solution using the
recursiveequation(4) within a small neighborhood of a specified trajectory called initial trajectory,
instead of whole state space. As mentioned by Heidari et al.[1971], DDDP technique is particularly
effective in the system where there are as many as control variables as state variables, and the system is
called invertible.For an invertible system, control variable ࢛௧ can be explicitly computed from ࢞௧and
࢞௧ାଵ and recursive equation (Eq. 4) can be written as;
ܸ௧൫࢞௧
௜
൯ = min
࢞೟శభ
ೖ
[ ‫ܮ‬௧ + ܸ௧ାଵሺ࢞௧ାଵ
௞
ሻ] … ݂‫ݐ ݎ݋‬ = ܰ, ܰ − 1, … 1 (6)
Where ܸேାଵ൫࢞ேାଵ
௞
൯ = 0 ݂‫,݇ ݈݈ܽ ݎ݋‬ ܽ݊݀ ࢛௝
௧
, ࢞௝
௧
, ࢛௧ = ݂௧൫࢞௧
௜
, ࢞௧ାଵ
௞
൯ ݂݁ܽ‫.݈ܾ݁݅ݏ‬As there are 4
reservoir in the system, so the total number of computation that will be needed to determine the
minimum value of ܸ, for 3ସ
node points, becomes 3ସ
× 3ସ
= 3଼
instead of 100଼
in DDP.
5. DETERMINATION OF INITIAL TRAJECTORY OF ALL THE RESERVOIRS OF DV
Fig.2: Decomposition of DV Reservoir System
All the four reservoirs have been decomposed into four number of individual reservoir (as
shown in Fig.2) maintaining the continuityof the system and then algorithm mentioned in section 4, is

170
applied to one reservoir at a time.When that algorithm applied to Reservoir 1 (sub-problem 1) the
recursive equation;
ܸ௧ሺ‫ݔ‬௧ሺ1ሻሻ = min [
௫೟శభሺଵሻ
‫ܮ‬௧ ሺ1ሻ + ܸ௧ሺ‫ݔ‬௧ାଵሺ1ሻሻሻ] … … … ݂‫ݐ ݎ݋‬ = ܰ, ܰ − 1, … ,1
subject to the constraints;
On storage; ࡯௠௜௡ሺ1ሻ ≤ ࢞௧ ≤ ࡯௠௔௫ሺ1ሻ
On storage; ࢛௧,௠௜௡ሺ1ሻ ≤ ࢛௧ ≤ ࢛௧,௠௔௫ሺ1ሻ
and system dynamics;
‫ݔ‬௧ାଵሺ1ሻ = ‫ݔ‬௧ሺ1ሻ + ‫ݕ‬௧ሺ1ሻ − ‫ݑ‬௧ሺ1ሻ − ݁‫ݒ‬௧ሺ1ሻ
with
‫ܮ‬௧ሺ1ሻ = ۤሺ‫ܦ‬௧ሺ1ሻ − ‫ݑ‬௧ሺ1ሻሻ/‫ܦ‬௧ሺ1ሻ‫ۥ‬ଶ
ܸேାଵሺ‫ݔ‬ேାଵሻ = 0and specified initial state ‫ݔ‬ଵ. The resulting optimum trajectory and control
sequences are denoted as {‫ݔ‬௧ഥ ሺ1ሻ} and {‫ݑ‬௧തതതሺ1ሻ} respectively. The next optimization problem is solved
for Reservoir2 in a similar manner and the resulting optimal trajectory and control sequences are stored
as {‫ݔ‬௧ഥ ሺ2ሻ} and {‫ݑ‬௧തതതሺ2ሻ} respectively.
After solving for reservoirs 1 and 2, reservoirs 3 and 4 are considered subsequently. However
instead of having separate individual targets ‫ܦ‬௧ሺ3ሻ and ‫ܦ‬௧ሺ4ሻ for Reservoir 3 and 4 respectively, there
is a combined target ‫ܦ‬௧
௖
which is to be met with the combined release {‫ݑ‬௧ሺ3ሻ + ‫ݑ‬௧ሺ4ሻ} from two
downstream reservoirs. As one reservoir at time approach has been followed, here the combined target
demand is to be divided in a suitable way, such that Reservoir 3 and 4 can have their individual targets
‫ܦ‬௧ሺ3ሻ and ‫ܦ‬௧ሺ4ሻ defined in order to solve the problem. Here the demand on the Reservoir 3 is taken as
fraction of ‫ܦ‬௧
௖
. The algorithm then solves the recursive equation for Reservoir 3;
ܸ௧ሺ‫ݔ‬௧ሺ3ሻሻ = min [
௫೟శభሺଷሻ
‫ܮ‬௧ ሺ3ሻ + ܸ௧ሺ‫ݔ‬௧ାଵሺ3ሻሻሻ] … … … ݂‫ݐ ݎ݋‬ = ܰ, ܰ − 1, … ,1
‫ݔ‬௧ାଵሺ3ሻ = ‫ݔ‬௧ሺ3ሻ + ‫ݕ‬௧ሺ3ሻ + ‫ݑ‬௧ሺ1ሻ − ݁݊௧ሺ3ሻ − ‫ݑ‬௧ሺ3ሻ − ݁‫ݒ‬௧ሺ3ሻ
with
‫ܮ‬௧ሺ3ሻ = ۤሺ‫ܦ‬௧ሺ3ሻ − ‫ݑ‬௧ሺ3ሻሻ/‫ܦ‬௧ሺ3ሻ‫ۥ‬ଶ
‫ݓ‬ℎ݁‫ܦ ݁ݎ‬௧ሺ3ሻ = 0.6‫ܦ‬௧
௖
and respective constraints applicable to
{‫ݔ‬௧ሺ3ሻ} and {‫ݑ‬௧ሺ3ሻ}. The resulting optimal trajectory and control sequences are denoted as {‫ݔ‬௧ഥ ሺ3ሻ} and
{‫ݑ‬௧തതതሺ3ሻ}respectively.
Reservoir 4 is now considered and optimization is carried out in a manner similar to that of
Reservoir 3 with ‫ܦ‬௧ሺ4ሻ = ‫ܦ‬௧
௖
− ‫ݑ‬௧തതതሺ3ሻ where ‫ݑ‬௧തതത(3) is the optimal release from reservoir 3 already
determined. Then the single stage loss function for Reservoir 4 is ‫ܮ‬௧ሺ4ሻ = ۤሺ‫ܦ‬௧
௖
− ‫ݑ‬௧തതതሺ3ሻ −
‫ݑ‬௧ሺ4ሻሻ/‫ܦ‬௧
௖
‫ۥ‬ଶ
. The resulting optimal trajectory and control sequences are denoted as {‫ݔ‬௧ഥ ሺ4ሻ} and
{‫ݑ‬௧തതതሺ4ሻ} respectively.
After determining the initial trajectories, DDDP is used to determine the optimal policies, for the
integrated operation of four reservoirs. Eq.(4) with the single stage loss function as defined in Eq.(5),
system dynamics mentioned in Eq.(2a) to Eq. (2d) and system constraints in Eq.(3) and Eq.(4) are used
to derive optimal policies for 44 years (1961 to 2005) historical monthly inflow data.

171
6. FRAME WORK FOR ANALYSIS
In this study, the operation problem of the DV reservoir system is solved in the deterministic
frame work. As the main objective of operation is water supply, potential of the DV system is judged by
evaluating the performance of the system in terms of modern concept of reliability parameters:
reliability, resiliency, vulnerability and total deficit. Reliability is defined as number of periods of deficit
(‘ND’), Resiliency is the number of consecutive deficits (‘Res’), Vulnerability is the maximum number
of deficit (‘Vul’) and sum of total amount of deficit during the period of operation is total deficit
(‘TotD’). These reliability parameters are computed from a sequence of release.
7. RESULTS AND DISCUSSION
7.1 Initial Solution
Initial solution has been derived based on the algorithm as discussed in the section 5. Here
trajectories obtained are optimal with reference to that particular single reservoir framework.
7.2 Optimal Solution
Optimal solution for the DV system is derived for the present reach wise demand as given in
section 3 for a control variable discretization level ‫ܫ‬ = 100 both for OSLF as well as TSLF.
Performance analysis of different release policies (both initial and optimal) in terms of reliability
parameters are presented in Tab.1 other than Reservoir 2 as all the reliability parameters are zero for
Reservoir 2. As no deficit (‘ND’) occurs for Rabi Crop Irrigation, therefore ‘Res’, ‘Vul’ and ‘TotD’ are
zero.
Table 1: Reliability parameters for initial and optimal solution
Policies
One-Sided Loss Function (OSLF) Two-Sided Loss Function (TSLF)
ND Res Vul TotD ND Res Vul TotD
Initial
Optimal
Reservoir 1
10 6 2.585 14.786 10 6 2.585 14.79
27 5 1.566 16.376 13 7 2.486 15.91
Initial
Optimal
Reservoir 3 and Reservoir 4, M & I Demand
4 1 3.115 7.076 11 1 4.699 25.19
8 1 0.819 4.277 8 1 0.998 3.16
Initial
Optimal
Reservoir 3 and Reservoir 4, Irrigation Demand
28 3 595.3 6429.2 24 3 543.05 4549.90
23 2 532.6 3866.5 23 2 553.22 4479.76
Initial
Optimal
For Rabi Crop Irrigation
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
Initial
Optimal
For Boro Paddy Irrigation
4 3 15.87 37.998 4 3 12.282 19.623
2 2 11.19 19.820 3 2 10.70 20.366
Initial
Optimal
For Kharif Crop Irrigation
24 2 595.3 6391.2 20 2 543.05 4530.4
21 2 532.6 3846.6 20 2 553.22 4459.4
Comparative plots of initial and optimal trajectories are shown in the Fig.3a to Fig.3d. In these
figures, the term ‫ݔ‬௧
′
ሺ= ‫ݔ‬௧/ሺ‫ܥ‬௠௔௫ − ‫ܥ‬௠௜௡ሻሻ indicates non-dimensionalized storage, ‘IT’ represent initial
trajectory and ‘OT’ represent optimum trajectories. For convenience of presentation, these plots are
shown for the first seven years only. This period contains most of the deficits observed in the solution.

172
Fig.3: Comparative Plot of Trajectories for OSLF
Fig.4: Comparative Plot of Trajectories for TSLF

173
Tab.1 shows that performance of the optimal policy is much satisfactory as the reliability
parameters are well below their permissible values. Improvement in the performance of the downstream
reservoirs is achieved by the joint operation of reservoir 3 and 4 and also by drawing additional water
from the upstream reservoirs which is reflected in the corresponding storage behavior (Fig.3a to Fig.3d).
It can be seen in the Fig.3a and Fig.3b, that for OSLF, the optimal storage state is depleted below the
initial storage state, indicating additional release from the reservoirs. This is of course made without
appreciably affecting the performance of the reservoirs. Fig.3c and Fig.3d shows that the actual sharing
of combined demand ‫ܦ‬௧
௖
is rather different than that defined by the sharing factor.
Fig.4a to Fig.4d show that in case of TSLF, initial trajectories and optimal trajectories are almost
identical for all the reservoirs. This is due to the fact that TSLF is target independent. Drawing
additional water from upstream reservoirs increased, but that did not affect the policies much.
Improvement of optimal policies is partly due to joint operation of downstream reservoirs. From these
observations, it may be concluded that the optimal solution provided by the DDDP model achieved the
defined objective of operation of the DV system.
8. CONCLUSION
The algorithm used in determination of initial trajectories, is very efficient procedure to
determine near optimal trajectories to be used with the DDDP algorithm. This algorithm determines the
initial trajectories very close to the optimal trajectories. The results show that this algorithm itself can
well be used for preliminary analysis. Both OSLF and TSLF yielded equally acceptable results. The
OSLF is sensitive to the target level and hence can well be used for examining the storage behavior with
the change in demand magnitude. In case of TSLF, as the solutions are almost target independent,
policy obtained for a particular target level can be effectively used for other target levels without solving
again. The additional releases over the target level made by the policies obtained from TSLF, can
effectively be used for other water uses indirectly which are not included in the objective function.
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