Journal of Water Resource and Protection, 2013, 5, 28-34
http://dx.doi.org/10.4236/jwarp.2013.57A005 Published Online July 2013 (http://www.scirp.org/journal/jwarp)
Yield Estimation for a Single Purpose Multi-Reservoir
System Using LP Based Yield Model
Deepak V. Pattewar1, Kalpeshkumar M. Sharma1, P. D. Dahe2
1Civil Engineering Department, M. G. M’s College of Engineering, Nanded, India.
2Civil Water Management, S. G. G. S College of Engineering and Technology, Nanded, India
Email: deepak_pattewar@yahoo.co.in, kalpeshkumar_sharma@yahoo.com
Received April 17, 2013; revised May 22, 2013; accepted July 1, 2013
Copyright © 2013 Deepak V. Pattewar et al. This is an open access article distributed under the Creative Commons Attribution Li-
cense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
ABSTRACT
Application of optimization techniques for determining the optimal operation policy for reservoir is a major area in wa-
ter resources planning and management. Linear programming, ruled by evolution techniques, has become popular for
solving optimization problems in diversified fields of science. An LP-based yield model (YM) has been used to re-
evaluate the annual yield available from the reservoirs for irrigation. This paper extends the basic yield model and pre-
sents a yield model for a multiple-reservoir system consisting of single-purpose reservoirs. Optimum yield of reservoirs
system is calculated by yield model. The objective is to achieve prespecified reliability for irrigation and to incorporate
an allowable deficit in the annual irrigation target. The yield model is applied to a system of two reservoirs in the Manar
River in India. This model can act as a better screening tool in planning by providing outputs that can be very useful in
improving the efficiency and accuracy of detailed analysis methods such as simulation.
Keywords: Yield Model; Reservoir Operation; Irrigation Releases; Manar River
1. Introduction
Linear Programming (LP) is a commonly used optimiza-
tion approach in water resources management. It is con-
cerned with solving a special type of problem; one in
which all relations among the variables are linear, both in
constraints and the objective function to be optimized.
An application of LP to reservoir operations has varied
from simple straightforward allocation of resources to
complex situations of operation and management. In the
past, limitations of computing power meant that optimi-
zation was achieved by decomposing reservoir systems
in time and space. These early models were predomi-
nantly deterministic, that is, they did not take into ac-
count the stochastic nature of inflows but rather were
based on long-term average seasonal or monthly flows.
However, they have gradually been improved. For ex-
ample, Loucks [1] developed a stochastic LP technique
for a single reservoir subject to random, serially corre-
lated, flows. Subsequently, much more complicated sto-
chastic models have been developed to reflect more real-
istically stream flow stochasticity, evaporation losses and
more complex systems involving multiple reservoirs
(Dandy G.C., Connarty M.C and Locks D.P. [2]; William
W. G. Yeh [3]). Under certain assumptions, non-linear
problems can be linearized and LP equations solved by
iteration or approximation procedures. The program
MODSIM is a generic program based around LP ap-
proaches that has been developed specifically for model-
ing water resources systems and reservoir operation by
Labadiee [4]. Sinha A.K., Rao B.V. and Lall U. [5] have
studied optimal reservoir operation for irrigation, hydro-
power production which involved constrained linear op-
timization. Dahe P.D. and Srivastava D.K. [6] developed
the basic yield model and presented a multiple yield
model for a multiple reservoir system consisting of single
purpose and multipurpose reservoirs. The objective was
to achieve pre specified reliabilities for irrigation and
energy generation and to incorporate an allowable deficit
in the annual irrigation target. The results were analyzed
for four cases. Srivastava D.K. and Taymoor A. Awachi
[7] developed nested models which were applied in tan-
dem using linear programming (LP), dynamic program-
ming (DP), artificial neural networks (ANN), hedging
rules (HRs), and simulation. An LP-based yield model
(YM) has been used to reevaluate the annual yields avai-
lable from the Mula reservoir for water supply and irri-
gation.
C
opyright © 2013 SciRes. JWARP
D. V. PATTEWAR ET AL. 29
This study presents a methodology to optimize the de-
sign of the multi-reservoir irrigation system by taking
monthly inflow and initial storage and tries to predict the
maximum possible releases using Linear programming
based Yield model. The specific objectives of the present
study can be stated as fallows:
1) To develop a Linear Programming based yield mo-
del for reservoir operation for a monthly time step.
2) Comparison of yield model and actual irrigation re-
leases for single purpose irrigation reservoirs in Manar
River.
3) To draw the conclusions from the interpretation of
results obtained.
2. Reservoir Yield Model
The conceptualisation and details of the yield model on
which the present model development is based are pre-
sented in Loucks et al. [8]. When reservoir yield with
reliability lower than the maximum reliability is to be
determined, the extent of availability of yield (or the al-
lowable deficit in yield) during failure years can be
specified. This is achieved by specifying a failure frac-
tion for the yield during the failure years. The factor θp,j
is used in the model to define the extent of available
yield during failure years. The objective of this model is
to maximize the yield for given capacity of the reservoir.
Let p denotes the exceedence probability for the yield.
The index j refers to a year and index t refers to a
within-year period. In this model only the firm yield is
used.
The yield model given by Dahe and Srivastava [6] to
determine single yield from a reservoir is as follows.
The formulation of the yield model is as follows:
Objective function
Maximize
,,
12
f
pfp
Oy Oy
o
1, 1,
(1)
Constraint
1) Over-year storage continuity
o,
1, 11,1,,1,
fp
j
jpjjj j
s
IOy
 SpEls 
o
1, 2,
(2)
o,
2, 12,2,,2,2,
fp
j
jpj jjjj
s
IOySp
 ElSps 
,,
12
(3)
The over-the-year capacity is governed by the distri-
bution of annual stream flows and the annual yield to be
provided. The maximum of all the over-the-year storage
volumes is the over-the-year storage capacity. It is possi-
ble to specify a failure fraction to define the allowable
deficit in annual reservoir yield during the failure years
in a single-yield problem. In the above equation,
f
pfp
Oy Oy
o
1, 1
is the safe (firm) annual yield from Up-
per Manar reservoir and Lower Manar reservoir with
o
1,
and reliability p.
j
s
j
are the initial and the final
over-the-year active storages in year j for the Upper
Manar reservoir and similarly for the Lower Manar
o
2, 1
o
2,
j
s
and
j
s
respectively; 1,
j
I2,
and
j
I
1,
are the
inflows in year j (Upper Manar and Lower Manar in
Manar River); θp,j is the failure fraction defining the pro-
portion of the annual yield from reservoir to be made
available during the failure years to safeguard against the
risk of extreme water shortage during the critical dry
periods (θp,j lies between 0 and 1, i.e., for a complete
failure year θp,j =0, for a partial failure year 0 < θp,j <1,
and for a successful year θp,j =1);
j
El 2,
and
j
El
1,
=
evaporation loss in year j and
j
Sp 2,
and
j
Sp
o
1, 11j
excess-
release (spills) in year j;
2) Over-year active storage volume capacity
s
Y
o
2, 12j
(4)
s
Y
,
1,11,11,1, ,11,
wfp ttw
ttt fpt
t
(5)
The active over-year reservoir capacity (Y1) required
for delivering a safe or firm annual yield in Upper Manar
reservoir and active over-year reservoir capacity (Y2)
required for delivering a safe or firm annual yield in
Lower Manar reservoir
3) Within-year storage continuity
OyElOyEl s




,
2, 12,22,
2, ,21, ,2,
0.10
wfp
tt t
t
tt tw
s
(6)
f
pfpt
sOyEl
Oy ElOys





1, 12, 1
,
ww
tt
(7)
Any distribution of the within-the-year yields differing
from that of the within-the-year inflows may require ad-
ditional active reservoir capacity. The maximum of all
the within-the year storage volumes is the within-the-
s
s
year storage capacity. In the above equation,
1, 2,
,
ww
tt
and
s
s
1,t
are the initial and the final within-the-
year active storages at time t;
and 2,t
are the
ratio of the inflow in time t of the modelled critical year
of record to the total inflow in that year; and and
are the within-the-year evaporation losses during
1,t
El
2,t
El
time t. The inflows and the required releases are just in
balance. So, the reservoir neither fills nor empties during
the critical year.
4) Definition of estimated evaporation losses (Over-
year)
1, 11,
0
1,11, 11,1
10 2
ww
tt r
jj t
t
ss
EEsEl


 






(8)
Copyright © 2013 SciRes. JWARP
D. V. PATTEWAR ET AL.
30
2, 1
0
2,22, 1
2,
2,
10 2
ww
tt
jj t
t
ss
EEs


 






(9)
5) Definition of estimated evaporation losses (Within-
year)
2
r
El
1, 11,
o
11,1 1,1
10 2
tt
tr
tc
r t
EE
s El





(10)
ww
ss

2, 1
o
22,2
10 2
tc
r
EE
s

2,
2,2
ww
tt
tr
t
ss El



6) Total reservoir capacity
11,1 1
w
t
Ys Ya

2
Ya (13)
Sumear and the w
age capacities is equal to the active storage capacity of
the reservoir.
(11)
(12)
22,1
w
t
Ys
of the over-the-yithin-the-year stor-
7) Proportioning of yield in within-year periods

,
1, ,1,1
p
t
fp t
f
OyK Oy (14)

,
2, ,2,2
p
t
fp t
f
OyK Oy (15)
1,t
K
and 2,t
K
defines a predetetio
l reservoir yield to be supplied in the with
yield in period t.
The tions
, a tributary of Manjara River in Goda-
anar River
pper Manar
rmined fracn of an-
nuain-year
Equa (1) to (15) present the Multi-reservoir
yield model for Upper Manar and Lower Manar reservoir
in Manar River.
2.1. System Description: Manar River
The Manar River
vari basin, Maharashtra states in INDIA. In M
two medium project has constructed i.e. U
and Lower Manar reservoir for irrigation preposes Fig-
ure 1. Table 1 is the silent features of Upper Manar Pro-
1 2
1- Upper Mana r
for Irrigation
Purpose
2-Lo
Irrigat
wer Manar for
ion Purpose
Manar
river
Inflow
Evaporation
Evaporation
Sp
ill
Spil l
Figure 1. Line diagram of reservoir system on Manar River.
ject Limboti reservoir and Lower Manar Project-Barul
reservoir. A 37 years historic inflow data for the system
considered is available as shown in Figures 2 and 3.
2.1.1. Irrigation Parameters (Kt) for Upper Manar
Limboti Reservoir and Lower Manar Barul
Reservoir
The monthly proportions of the annual irrigation targets
(Kt values) are worked out by considering the cropping
patterns and irrigation intensities recommended by the
agricultural officer. Kt defines a predetermined fraction
of reservoir yield the within-year period t. The K value
t
are given in Table 2.
s
Figure 2. Inflow at Upper Manar-Limboti reservoir.
Figure 3. Inflow at Lower Manar-Barul reservoir.
Table 1. Silent features of Upper Manar and Lower Manar
project in Manar River.
Particulars Upper Manar Lower Manar
Irrigation Purpose Irrigation Purpose
Scope of Scheme
Location Manar River at
Limboti Manar River at Barul
Catchment area 987.60 Sq Km 1585.08 Sq Km
Gross storage capacity107.98 MCM 146.92 MCM
Capacity of Live
Storage 75.71 MCM 138.21 MCM
Capacity of
Dead Storage 32.27 MCM 8.71 MCM
75% dependable
yield(Project Report)162.50 MCM 205.76 MCM
Copyright © 2013 SciRes. JWARP
D. V. PATTEWAR ET AL.
Copyright © 2013 SciRes. JWARP
31
w approximation, irrigation and evaporatised iodel fo
Mana
Month
Table 2. Within-year infloon parameters un the yield mr reservoirs on
r River.
1,t
2,t
γt1 γt2Kt1 Kt2
June 0.0522 0.0560 0.133 0 0.04430.0215 0.1240
July 0.1580 0.1670 0.0686 0.0492 0.0151 0.0000
0589 0.0417 0.0186 0.0000
0.1490 0.1460 0.0575 0.0858 0.1618 0.0446
0.0363 0.0363 0.0540 0.0759 0.
M
Aug 0.2967 0.2953 0.
Sept 0.2289 0.2209 0.0568 0.0880 0.0299 0.0336
Oct
Nov 2347 0.1855
Dec 0.0186 0.0186 0.0450 0.0973 0.1794 0.1840
Jan 0.0160 0.0160 0.0547 0.0977 0.0842 0.1860
Feb 0.0103 0.0103 0.0623 0.1041 0.0634 0.0910
arch
Apr
0.0112 0.0112 0.1004 0.1173 0.0667 0.0293
il 0.0119 0.0119 0.1302 0.1400 0.0706 0.0230
May 0.0109 0.0109 0.1787 0.0585 0.0541 0.0990
2.1.2.ximationtical Withinnflows
Values for Uervoir
Lower Man Reservoi
βt vae based on monthly floe βt val-
ues baseon average hly flows fovoir are
given ble 2.
.1.3. Evaporation Parameters of Reservoirs γ
voirs
and an volume
at dead storage elevation for respec-
tive reservoirs. The storage-area and storage-elevation
e stor-
e values of te given in the
d Modelbserverical inflowanar
for 37(1969-2e used in ation
e yieldshe reseOut of the of 9
st flow 3rd, 4th, th, 18th, 23, 29th
th) 197172, 1985, 1986, 193,
and 20% of ths) were asss the
mon faiars in bervoir, de
the modified method of determining failure years by
el. Thus remaining 28 years were successful
roject reliability. Thus,
elve within year periods
s based on
for the
anal The within
Appro of Cri-Year I
(βt) pper Manar Limboti Res
andar Barul r
lues araveragews. Th
d
in Ta montr reser
Yiel. The od histos of M
River years 005) arcomput
of th from trvoirs.
th
se a set
rd th
lowe years (16 , 17, 25
and 36
1997
, 19
04 (25
4, 198
e year
91, 199
umed a
comlure yeoth restermined by
2t
The average monthly evaporation depth at all the reser-
is obtained from the Water Resources Department
vailable project reports. The evaporatio
loss due to dead storage E01 = 8.158 and E02 = 11.30 are
obtained by product of the average annual evaporation
depth and the area
relationship is taken for study. A linear fit for th
age-area data for each reservoir above the dead storage is
obtained from the storage area relationship. The evapora-
tion volume loss rate 10.2880
r
El and 20.413 9
r
El
are obtained by taking the product of the slope of the area
elevation curve linearized above dead storage and the
average annual evaporation depth at respective reservoirs.
The parameter γt (the fraction of the annual evaporation
volume loss that occurs in within-year period t) is com-
puted by taking the ratio of the average monthly evapo-
ration depth to the average annual evaporation depth at
respective reservoirs. Thhe γt ar
Table 2.
3. Analysis and Results
3.1. Application of the Yield Model in
Assessment of Manar River Yield
The approximate model which includes within year pe-
riods for only one modelled critical year is known as the
yield mod
years representing 75% annual p
thirty seven over year and tw
were considered for analysis. The value of βt
average monthly flows have been considered
ysis and are presented in the Table 2.
year yields from the reservoir for irrigation in a month
are represented as a fraction of its annual yield. With the
provision of θp,j , the extent of failure in the annual yield
from the reservoir during failure years was monitored as
clear guidelines were not established for deciding its
value. The value of θp,j for the project was determined
using the YM with an objective to minimize its value. In
Manar River, irrigation originally being the main project
target was considered as a single yield or firm yield from
the reservoir. The annual project reliability for irrigation
was kept equal to 75%. The value of θp,j was found to
increase with the decrease in the annual yield from the
reservoir. In Manar River two reservoirs (Upper Manar-
Limboti reservoir and Lower Manar-Barul reservoir) are
constructed for the irrigation purposes.
For Upper Manar-Limboti reservoir with active stor-
age capacity of 75.71 MCM and for Lower Manar-Barul
reservoir with active storage capacity of 95.71 MCM, the
yield is found out for Safe reservoir yield θp,j = 1 and θp,j
= 0.00 respectively. Calculated annual yield of Upper
Manar-Limboti reservoir by yield model is 52.44 and
D. V. PATTEWAR ET AL.
32
107.24 MCM respectively and for Lower Manar-Limboti
reservoir is 42.76 and 107.27 MCM respectively in Multi
reservoir yield model analysis. Within-period water re-
leases are shown in Table 3.
3.2. Comparison of YM and Actual Releases in
Lower Manar-Barul Reservoir
The main objective is to compute the yield that should be
released to fulfill the total demand. Comparison of actual
demand, releases and yield which are obtained from the
model used is as follows. Multi-reservoir yield model
based on the monthly inflow and monthly irrigation de-
mands of the reservoir operation system is considered for
Table 3. Representing the monthly water releases for irriga
Safe Reservoir Yield (MCM) θp,j = 1.00
the comparison. The Upper Manar-Limboti reservoir is
recently constructed and has started operating from Oc-
tober 2010. Water releases data is not available for it
hence only Lower Manar-Barul reservoir is taken for the
comparison.
Table 4 gives the output of the model used for 75%
reliable yield as well as demand and actual releases in the
years which are considered in Lower Manar-Barul res-
ervoir. The data available on actual releases of only 6
years is used for comparison. As per the Table 4 the ac-
tual release from the reservoir is maximum 98.79 MCM
in the year 2000-2001 and minimum is 68.49 MCM in
year 2003-2004. Figure 4 shows comparison between
tion by approximate YM (Multi-reservoir) in Manar River.
75% reliable Yield (MCM) θp,j = 0.00
Month Upper Manar
Limboti Reservoir Lower Manar
Barul Reservoir Total
Yield (MCM) Upper Manar
Limboti Reservoir Lower Manar
Barul Reservoir Total Yield (MCM)
June 1.127 5.302 6.429 2.305 13.301 15.606
July 0.719 0.000 0.719 1.619 0.000 1.619
1.994 0.000 1.994
Sep3.004 3.206 3.604 6.810
17.352 4.784 22.136
25.170 19.898 45.068
19.239 19.737 38.976
9.030 19.952 28.982
6.799 9.761 16.560
March 3.497 1.253 4.750 7.153 3.143 10.296
A
May 2.8070 5.801 16.420
Aug 0.975 0.000 0.975
t 1.568 1.436
Oct 8.485 1.907 10.392
Nov 12.308 7.932 20.240
Dec 9.408 7.868 17.276
Jan 4.415 7.954 12.369
Feb 3.324 3.891 7.215
pril 3.702 0.983 4.685 7.571 2.467 10.038
37 4.233 7.10.619
Total 52.437 42.765 95.207 107.239 107.27 214.515
Table 4. Values of actual demand, actuases and yield model (YM with 7able θp,j = 0.0
Actual Water Releases in ys 2000-2005 (MCM
l relea5% reli0).
ear)
SMonth M
)
Actual
Demand
(MCM) 2001 2001-2002-200320042004-2006
release
M)
r. No Multi Y
(MCM 2000-20022003-2005 2005-
Average
Water
(MC
6 June 25.51 25 111.57 8.5 10.1.63 95 13.30 12.1.29 46 110.
7 July 0 0 0 0 0 0 0 00
Aug 0 0 0 0 0 0 0 00
Sept 6.91 32 3.3.14 2.3 2.15 97
Oct 9.18 41 4.4.16 3.06 3.18 94
Nov 38.17 32 17.31 12.71 15.17.4 38
Dec 37.86 18 16.17.17 12.61 15.26 25
Jan 38.27 37 16.17.36 12.74 15.44 42
Feb 18.72 99 8.8.49 6.23 7.53
ar 6.03 9 2.73 2 2.2.75
4 Apr2.03
9.78 9.28
eld 98.79 91.78
0 0.
8 0 0.
9 3.60 3.06 83 3.2.
10 4.78 4.06 76 4.3.
11 19.89 18.16.9 65 16.
12 19.73 18.76 52 17.16.
1 19.95 18.94 69 17.16.
2
3 M
9.76
3.14
8.
2.
29
2.67
66 8.
47
8.03
2.59
il 2.46 4.73 2.27 2.09 2.15 1.56 1.94 2.16
5
Yi
May 10.61
107.26
20.37
205.75
01 9.24 6.78 8.35 9.
07 93.32 68.49 84.33 93.
8.74
88.30
Copyright © 2013 SciRes. JWARP
D. V. PATTEWAR ET AL. 33
Figure 4. Comparison of acreleases and releases
tai freld model.
ontater r, m deand ly
ield by yield model. From the figure it is very clear that
in the month of June, December and January the reser-
voir releases are comparable with the yield model, where
as the actual demand is very large as compared to the
actual releases from the reservoir except in the month
February, March and April. It can be seen from the Fig-
ure 4 that the releases are negligible in the period of
Kharif Crop i.e. June, July, August, September and mid
of October. Whereas the releases are more in the period
of Rabbi Crop (i.e. from October to February) and in Hot
Weather crop period (i.e. from February to May).
The Yield model can be used for yield assessment with
specified reliabilities and thus assists in the effectiv
istic full optimization model by the way of reduction in
l hoieo
whicfe r yield model and yield
l with reliabf flowmplete failure.
Ihe case of complete failure, the annual firm pro-
ved is zerouring thelure yearhe yieldel is
le of consideringreliabif annuad. It
in the coof come or partial failure
g the yearshe yield model for rvoir
m developed in thaddresses the
ts oforatiesiiabilitidif-
purpas we alldeficit criterion
nnualation target in a roir systcon-
g of a Mlti reserrrigatistem. It can act as
tter scrg tool ing. Being an option
el, no inlicy is eeded for the analysis of res-
ir syste mis gennough ould
ilar reir sys
REFERENCES
tual ob-
nedom yi
mhly weleasesonthlymand month
y
e
management and design of irrigation reservoir system.
Yield model provides a better alternative to the determi-
n
size and provides sufficiently accurate results. It also
allows determination of annual yield with a given reli-
ability less than the maximum reliability. There is also a
provision of determining the percentage of annual yield
to be supplied during failure years.
4. Conclusion
The study of multi-reservoir operation in Manar River is
carried out using LP based yield model. Identification
and screening of the feasible solution to provide potential
candidates for detailed evaluation is a crucial stage dur-
ing the search for optimal solution of real life problems.
Mathematical optimization models play a vital role in
this regard. The overall effort in handling real life sys-
tems can be significantly reduced with screening models
capable of better representing the system and providing
fewer and more accurate candidate solutions for detailed
evaluation which is proportional to the number of candi-
date solutions to be evaluated and their proximity to the
optimasolution. Te yield mdel is studd for the tw
cases,
mode
h are sa
75%
reservoi
ility o with co
n tyield
id d fais. T mod
capab the lity ol yiel
can alsocludencept plet
durin
syste
failure. T
is study successfully
rese
aspec incorpng the dred reles for
ferentoses, ll as anowable
for a irrigeservems
sistinuvoir ion sy
a beeeninin planntimiza
mod
ervo
itial po
ems. Th
n
odel eral eand c
be applied to simservotems.
[1] D. P. Loucks, “Computer Models for Reservoir Regula-
tion,” Journal of the Sanitary Engineering Division, Vol.
94, No. 4, 1968, pp. 657-671.
[2] G. C. Dandy, M. C. Connarty and D. P. Loucks, “Com-
parison of yield assessment of multiple reservoir sys-
tems,” Journal of Water Resources Planning and Man-
agement, Vol. 123, No. 6, 1997, pp. 350-358.
[3] W. W. G. Yeh, “Reservoir Management and Operation s
Models: A State-of-the-Art Review,” Water Resources
Research, Vol. 21, No.12, 1985, pp. 1797-1818.
doi:10.1029/WR021i012p01797
[4] J. W. Labadie, “Optimal Operation of Multireservoir
Systems: State-of-the-Art Review,” Journal of Water
Resources Planning and Management, Vol. 130, No. 2,
2004, pp. 93-111.
doi:10.1061/(ASCE)0733-9496(2004)130:2(93)
[5] A. K. Sinha, B. V. Rao and U. Lall, “Yield Model for
Screening Multi Purpose Reservoir Systems,” Journal of
Water Resources Planning and Management, Vol. 125,
No. 6, 1999, pp. 325-332.
doi:10.1061/(ASCE)0733-9496(1999)125:6(325)
[6] P. D. Dahe and D. K. Srivastava, “Multipurpose Multi-
yield Model with Allowable Deficit in Annual Yield,”
Journal of WRPM, Vol. 128, No. 6, 2002, pp. 406-414.
[7] D. K. Srivastav and T. A. Awchi, “Storage-Yield Evalua-
tion and Operation of Mula Reservoir, India,” Journal of
Water Resources Planning and Management, Vol. 135,
No. 6, 2009, pp. 414-425.
doi:10.1061/(ASCE)0733-9496(2009)135:6(414)
[8] D. P. Loucks, J. R. Stedinger and D. A. Haith, “Water
Resource Systems Planning and Analysis,” Prentice-Hall,
Inc., Englewood Cliffs, 1981.
Copyright © 2013 SciRes. JWARP
D. V. PATTEWAR ET AL.
34
Appendix: Notation
The following symbols are used in this paper:
,
1
fp
Oy Annual firm Upper Manar reservoir yield.
,
2
fp
Oy Annual firm Lower Manar reservoir yield.
o
1, 1j
s Initial storage of Upper Manar reservoir at
the beginning of year j.
Initial storage of Lower Manar reservoir at the
g of year j.
begin-
r at the
1, j
year j.
2, j
Sp Excess release in Lower Manar reservoir in
year j.
1, j
I Annual inflow at Upper Manar reservoir site in
year j.
2, j
I Annual inflow at Upper Manar reservoir site in
year j.
nin
o
1, j
s Final storage of Upper Manar reservoir at the
beginning of year j.
o
2, j
s Final storage of Lower Manar reservoi
beginning of year j.
Sp Excess release in Upper Manar reservoir in
1, 2,
,
j
j
ElEl = Annual evaporation volume loss from
reservoir in year j.
12
,
tt
ElEl = Evaporation volume loss from reservoir in
period t.
1, 2,
,
tt
= Fraction of total annual yield for assumed
critical period inflow in Uppe Manar reservoir and
Lower Manar reservoir.
12
,YY = Over-year storage capacity of Upper Manar
reservoir and Lower Manar re
,Ya Ya = Total active storage capacity of
servoir.
Upper
12
Manar reservoir and Lower Manar reservoir.
1, 2,
,
tt
K
K = Percentage fraction of annual irrigation
target in period t in Upper Manar and Lower Manar res-
ervoir.
1, 12, 1
,
ww
tt
s
s
= Initial within-year storage volume in pe-
riod t in Upper Manar reservoir and Lower Manar reser-
voir.
1,
w
t
s
= Final within-year storage volume in period t in
Upper Manar reservoir.
MCM = Million Cubic Meter.
Copyright © 2013 SciRes. JWARP