Hydrological forecasting - Prévisions hydrologiques (Proceedings of the Oxford Symposium,
April 1980; Actes du Colloque d'Oxford, avril 1980): IAHS-AISH Publ. no. 129.
A forecasting model for the optimal scheduling of a reservoir supplying
an irrigated area in an arid environment
OTTO SCHMIDT and ERICH J. PLATE
Institut Wasserbau III, Karlsruhe, FR Germany
Abstract. A forecasting model is developed for the optimal scheduling of reservoir releases used
for irrigation in an arid region. The forecast consists of expected values based on the application of
dynamic programming techniques as well as on the use of generated inflows. The data generation
model for rivers in arid areas (wadis) is briefly described. The generation model consists of three
components. The first determines if the following year will be dry or wet. The second generates
a sequence of days in which a flow may occur, and the third calculates the magnitude of such a
flow. The third component depends directly on the first. The optimization problem has two state
variables and one decision variable, and is solved by dynamic programming. The state variables are
the reservoir content and the soil moisture of the irrigated area. The decision variable is that
quantity of water which should be applied for irrigation. The evaporation losses from the reservoir
are calculated on a daily basis because of the stochastic nature of the inflows into the reservoir.
The process is non-stationary because of the continuing reservoir sedimentation.
Modèle de prévision pour la gestion optimale d'un réservoir utilisé pour l'irrigation d'une région en
zone aride
Résumé. Un modèle de prévision a été mis au point pour la gestion optimale des éclusées d'un
réservoir utilisés pour l'irrigation d'une région en zone aride. La prévision consiste en un ensemble
de valeurs possibles obtenues par application des techniques de programmation dynamique ainsi
que par simulation des apports entrants dans le réservoir. Le modèle générateur de données pour les
rivières des régions arides (ouadis) est décrit brièvement. Le modèle se décompose en trois parties.
La première détermine si l'année sera sèche ou non. La seconde détermine une série de jours
pendant lesquels un écoulement peut se produire et la troisième calcule l'importance d'un tel
écoulement. La troisième partie repose directement sur la première. Le problème d'optimisation
comporte deux variables d'états et une variable de décision et est résolu par les techniques de
programmation dynamique. Les variables d'état sont le contenu du réservoir et l'humidité du sol
de la région à irriguer. La variable de décision est le niveau d'humidité du sol à partir duquel
l'irrigation devient nécessaire. Les pertes par evaporation sont calculées pour le réservoir sur une
base journalière à cause de la nature stochastique des apports y entrant. Le processus est non
stationnaire à cause d'une sédimentation permanente au sein du réservoir.
INTRODUCTION
The optimal operation of a reservoir which is used to supply an irrigated area in an
arid or semiarid region is as optimal as the forecasts are. Because the inflows to a
reservoir are produced by irregular rainstorms, the forecasts consist of expected values
based on the application of an optimization technique, on the use of generated inflows
and on the size of the area to be irrigated. At the beginning of a growing period, the
area to be planted has to be decided. It depends on the local situation whether the
whole developed area or only part of it should be planted. The whole area which is
to be developed may be determined by the proposed model of Plate and Treiber
(1979). Once the area to be planted has been decided, the initial storage content
together with the future inflows have to be allocated, so that no crop failure occurs
and no areas have to be abandoned.
The main concepts of the simulation model are given by Plate and Treiber (1979)
and are only briefly repeated here. However, the derivation of the forecasting model
is described in detail. The hydrological simulation model and all the submodels are
shown schematically below. It is the model of a single purpose reservoir of which the
491
492 Otto Schmidt and Erich J. Plate
input is a mixture of water and sediment. The water is stored in, or released from, the
reservoir according to demand, water availability, and forecast. Because of the gradual
filling up of the reservoir as a result of the continuous sediment deposit, the process
of reservoir operation is a nonstationary one. The model as a whole consists of three
main parts:
(1 ) Input model, (a) Generation model of daily discharges, and (b) rating curve
for sediment deposit.
(2) Reservoir, (a) Evaporation model, and (b) stage—area and stage—volume curves
for the reservoir.
(3) The operating rule, (a) Determination of the irrigation area, and (b) forecast of
the reservoir contents.
In this paper the main stress is on the development of forecasts, and the purpose of
the paper is to discuss the method itself rather than to present detailed results.
THE OPERATION RULE
Determination of the irrigation area
The number of hectares Nk for a crop in year k can be calculated either by
Nk = SA VAIL/DEMAND
(1 )
where SA VAIL is the total quantity of water which should be available for irrigation
purposes and DEMAND is the product of the cumulative potential évapotranspiration
of the n stages of the growing period and an assumed project efficiency, or is given as
a fixed value as a result of the model of Plate and Treiber (1979). The value of
SA VAIL is determined by:
SAVAIL = S-mi+Zx
(2)
where £„„ is the initial storage content at the beginning of the irrigation period and
Zx is the sum of the expected total monthly inflows during the forecasting range.
Investigations have shown that the values of Zx are gamma-distributed and x is therefore the level of the chosen probability which should exceed x per cent in all cases. A
better alternative would be to calculate the area to be planted every year on the basis
of a decision analysis, but such a calculation is beyond the scope of this paper.
THE FORECASTING METHOD
Once the size of the irrigation area has been decided, the question arises of how to
operate the reservoir to get the highest crop yield from the irrigated area. Because of
the irregular nature of the inflows, no traditional forecasting technique can be used.
Therefore the method described in this paper approaches the problem in a different
way consisting of the following three steps:
(1) Development of a data generation model for arid rivers.
(2) Development of a mathematical model of a water resources system and
simulation of its operation over a long trace of synthetic inflows (simulation coupled
with a mathematical programming technique).
(3) Statistical analysis of the results of the simulation—optimization computations
and identification of the optimal operation rule for the reservoir system.
Optimal scheduling of reservoir releases
493
Qtcm/i]
FIGURE 1.
Schematical runoff record.
THE RUNOFF GENERATION MODEL
Since the measured data series of discharges were too short to be taken as representative of conditions during the lifetime of the reservoir, a model was designed to
generate daily values. A model known from literature is that of Yakowitz (1973),
specially developed for arid and semiarid areas. However, to build the data generation
model a considerable amount of measured data are required for estimation of the
model parameters, but such data were not available. Based on an investigation of the
available data, a schematic record of daily flows of a wadi is shown in Fig. 1. The
flow in larger wadis consists of a continuous low flow on all days (baseflow), plus a
flood peak added to this baseflow on days with rainfall. In smaller wadis, the flow
comprises flood events only, without baseflow. It is possible, therefore, t o construct
a model which consists of four parts:
(1) A generation model for the sequence of wet and dry days (days with and
without floods).
(2) A generation model for the magnitude of floods on wet days.
(3) A model for the magnitude of the baseflow for the larger wadis.
(4) For preservation of the standard deviation of the total annual inflows a process
is incorporated in the model which determines if a year is a wet or a dry one and
related to the model for the generation for the magnitude of the floods on wet days.
Using the model 10 series of runoff data each of 100 years duration were generated,
whose statistics are compared with those of the measured data. In Fig. 2 a visual
comparison of observed and generated discharges is shown, which shows that the
records appear to be from the same family of curves, in particular with regard to peak
flows. Because of the relatively short length of the measured data only the measured
data can serve for comparison. A comparison made between observed and calculated
annual flows shows that the mean annual flows are identical and the standard
deviation differs less than two per cent. Measured and generated daily flows are
compared in Figs 3 and 4, measured and generated monthly flows in Figs 5 and 6. In
these figures the measured values, the mean values of the 10 runoff series (100 years
each) and the smallest and the greatest values of one of the 100-year data series are
plotted. Good agreement is considered to exist. For a detailed description of the data
generation model see Schmidt and Treiber (1980).
SIMULATION-OPTIMIZATION MODEL
Several researches have been conducted to develop an optimal irrigation strategy
(Flinn and Musgrave, 1967; Hall and Butcher, 1968; Dudley, 1969). The main scope
of these models, however, was irrigation management, and in the case of Dudley
494 Otto Schmidt and Erich J. Plate
Qlcm/sl
neosured daily flows
I li- XkJL4I.
-1
1
1000
1
2000
1
1
1
1
r-"
3000
days
[cm/sj ,
,
g e n e r a t e d daily f l o w s
(0_
1.
J
1000
FIGURE 2.
2000
days
Comparison between measured and generated data.
Q [cm/s]
-_ Mean of 10 — 100 years sequences
— Largest value
2.5
.._ Smallest value
—. Measured value
2.0
1.5-
0.5
0.25~1
N
FIGURE 3.
1
D
1
J
1
1
F
M
1
A
Means of daily flows.
1
M
1
J
1
J A
1
I
S O
I
w
Month
3000
Optimal scheduling of reservoir releases 495
o [cm/s]
Mean of 10 - 100 years sequences
Largest value
5
Smallest value
. Measured value
D
J
*i[10bcm]
7.0-
F
M
A
M
J
——
1
J
0
Month
A
Standard deviations of daily flows.
FIGURE 4.
Mean of 10 — 100 years sequences
Largest value
_.„.
__,
„ . . Smallest value
.
Measured value
f\
i \
~~1
N
FIGURE 5.
1
D
1
J
1
F
Î
M
!
A
I
M
I
J
I
J
1
A
T
S
1
O
W
Month
Means of monthly flows.
(1969) the integration of the future inflows into the reservoirs was done using mean
monthly inflows. In addition the yield model used was very rough. To obtain better
results a simulation model should be used. This should also account for reservoir
sedimentation (the sediment yield is a result of the daily inflows) and reservoir
evaporation. It should also be capable of calculating the soil moisture, and as a
consequence a dated production function which allows for the susceptibility of the
plant to stress during the growing period.
496 Otto Schmidt and Erich J. Plate
oHO b cm]
8.0-
Mean of 10— 100 years sequences
Largest value
Smallest value
.
,
Measured value
N
FIGURE 6.
O
J
F
M
A
M
J
J
A
S
O
Month
Standard deviations of monthly flows.
THE OPTIMIZATION MODEL
The optimization technique applied to solve the problem is backward dynamic
programming. Dividing the irrigation season into n stages (which need not all have
the same length), two state variables, reservoir content and the soil moisture available
at the beginning of each stage, and one decision variable, the quantity of irrigation
water which should be applied, may be calculated. For this it is necessary to describe
the reservoir system and the soil regime.
Reservoir system
The state transition of the reservoir system is given through the following equation:
Sj = S/_ t + Qj - Ej - Dj - SPILLj
(3)
subject to 0 < Sj « 5 m a x — SStP where
5)_i
= reservoir content at the beginning of stage/;
= inflow during stage /";
Qi
= evaporation losses of the reservoir during stage/ (function of the reservoir
sedimentation);
Dj
= decision variable, water release during stage/, depending on the available sou
moisture of the irrigation area;
SPILLj= spill volume during stage/;
^max = maximum storage capacity;
^s,p = volume of reservoir filled with sediment at the beginning of year p (the
assumption is made, that the reservoir sedimentation during any year is
negligible and can be added as a whole at the end of the year);
/ = ! , . . . , « = number of stages.
Optimal scheduling of reservoir releases
497
The soil regime
The state transition for the sou moisture content yields equation (4) (values per
hectare irrigated area):
BFj = BFj^ ! + IRj - Wdj
(4)
subject to
PWP < BFj < FC
where
BFj_ i
IRj
Waj
PWP
FC
= available soil moisture at the beginning of stage /;
= irrigation quantity during stage /';
= water use of the crop during stage/ (calculated on a daily basis);
= permanent wilting point;
= field capacity.
The relation between the two branches of the system is given through the following
equation:
D/ = IRj AREA
G
(5)
subject to
IRmin « IRj < (Sj_l + Qj -Et -SPILLj)l(AREA
• G)
where AREA = size of the irrigation area;/i? min = irrigation quantity which should be
given within a stage to avoid plant wilting; and G = total project efficiency of water
use. The return of a stage is expressed through Bellman's recursion formula:
Yj (Sj,BFj) = max(Dj o Y*__ 1 (Sj_. 1, BF^
))
(6)
where Yj = return obtained from a / stage process when starting with specific values
S/_i and BFj_t; o = mathematical operator (e.g. +or x). The variable which is
unknown until now is the soil moisture available in the root zone BF, a knowledge of
which allows the calculation of both the irrigation quantity D to be applied and the
actual daily évapotranspiration ETA. Hence the resulting yield Y of the irrigated area
can be established.
THE SOIL MOISTURE MODEL
Much research has been done to explain the relationship between the atmospheric
demand, plant behaviour, available soil moisture and yield. The main problem seems
to be to identify a relationship between potential évapotranspiration ETPOT, actual
évapotranspiration ETA and available soil moisture SMin the root zone. Because no
measured data are available the following procedure proposed by Minhas etal. (1974)
was adopted.
(1) Establish the functional relationship between these values
ETA
= f(SM)
ETPOT
(2) Solve the following differential equation
dSM
= ETPOT (t) • {(ISM)
df
(7)
(8)
498 Otto Schmidt and Erich J. Plate
Knowing ETPOT and SM at time t0, SM and ETA at time ty (tt > t0 ) can be calculated
and with the knowledge of ETA, the harvestable yield can be determined. Because there
are two or more possible harvests, a grain yield model and a model which calculates
the dry matter yield as a number for fodder crops have to be used.
THE GOAL FUNCTION
Provided that water is the only limited factor, the goal function used consists of two
parts: the first is the model proposed by Jensen (1968) for grain yield, and the second
is based on results given by Doorenbos and Pruitt (1975) for the estimation of the dry
matter yield of grasses.
Y
m
/Wa\Xi
P2
^n
Pj and P2 are the unit values for the net return on water for grain and dry matter
yield, to relate the two crops to one another. Where Y/Y0 is the ratio of actual yield to
maximum yield if the soil moisture is not limited, X;- the relative sensitivity of the crop
to water stress during the/th stage and Wa/W0 is the ratio of net water use to use of
water if the soil moisture is not limited, that means Waj is the sum of the daily values
of ETA within the stage /, m is number of stages of the grain producing period and
Wmax is the cumulative potential évapotranspiration of the dry matter period.
With all the models described above it is possible to calculate the reservoir releases
dynamically according to water demand and the inflow situation. The resulting yield
of the irrigation area is optimal per unit of water, the limiting factor. If we do the
optimization in a long-term run (r years) we get r optimal yields. As a second result of
the optimization computations, a set of (r x «) optimal control vectors S;-, final storage
volumes, are obtained for each of these stages.
Establishing the operational rules of the system
In the method presented above the decision maker has a perfect knowledge of the
future inflows, that means he has the perfect forecast. To receive a reliable forecast of
the inflows, and furthermore to establish the operation rule of the reservoir, a linear
regression analysis based on the optimal control vectors Sy was used. That means for
each stage/ of the annual growing period, a multiple regression was assumed to exist
between the following parameters:
S,- = « 0 + ^ ! 5/_i + a2 Qj
0°)
where a0,ax and a2 are the regression coefficients. As Qj normally is not predictable,
the mean monthly inflows were replaced in the operation system. The set of
regression equations constitutes the operation that secures long-term optimality of the
reservoir operation. With the known reservoir content at the beginning of a stage
Sj^i, the regression coefficients and the mean monthly inflow,Sj can be calculated.
The difference between Sj and S/_ i is the quantity of water which should be applied
(or not) for irrigation purposes.
The application of the model
For purpose of demonstration, some assumptions have to be introduced in this paper,
but without limiting the demonstration of the practicability of the method itself. (A
more detailed description of the implementation of the proposed method for the
operation of a single purpose reservoir will be given in the future.) The assumptions
Optimal scheduling of reservoir releases 499
/
Soil-plantatmosphere
parameters
Historical
Run-off data
QSYE
Flow generation
soil moisture
estimation
SOPLAT
Genera ted
run-of f data
Soil mois
ture values
Simulation
optimization
DAS SA
MULREG
Regression
analysis
Forecasting
Values
FIGURE 7.
The computer models for the determination of the operation rules.
are (1) the only investigated crop is grain sorghum, (2) two crops can be harvested
(grain and fodder), (3) the growing cycle consists of six monthly stages (three
months for grain producing, and three for fodder crops). The susceptibility factors for
the grain yield were used with X = 0.5, X = 1.5 and X = 0.5 according to the threemonth period. The estimation of the regression coefficients (forecasting values) means
the determination of the operation rule was done according to the three steps in Fig. 7.
The flow generation and the estimation of the soil moisture is done by two computer
programs QSYE and SOPLAT, the simulation coupled with optimization by the
program DASSA and the regression analysis by MULREG. The forecasting values will
be used by another model which simulates the operation of a single purpose reservoir
according to the model described in the introduction of this paper. For demonstration
purposes the reservoir simulation model was run over 50 years. Every year the planted
area was determined by the first method described previously. The probability level x
was chosen arbitrarily as 20 per cent. With the same sequence of the estimated areas
the simulation optimization model DASSA was run. The cumulative total yield was
computed according to the following equation:
A = i \Y0)i
•AREAi
(H)
where (Y/Y0)i is the relative yield of a year and AREAt is the irrigated area of this
year. For i = r and for the results of DASSA the maximum possible yield J m a x is
given. In a graphical comparison the relatively cumulative total yields y/ymax are
plotted in Fig. 8. Variant (1) is the relative yield if DASSA has been run. That means
the system operator has a perfect knowledge of inflows. Variant (2) shows the
cumulative relative yield as a result of the simulation run by the reservoir simulation
model. That means the system operator uses the coefficients given above by the
implicit stochastic approach.
500 Otto Schmidt and Erich J. Plate
FIGURE 8.
Cumulative relative total yield (simulation for 50 years).
CONCLUSIONS
A set of mathematical models and computer programs was developed to determine
the optimum operation rule of a storage reservoir supplying an irrigation area. The
implicit stochastic optimization by combining data generation of the inflows,
simulation with optimization and regression analysis seems to be a valuable
technique for the solution of such problems. Research should continue in estimating
the annual cropping area based on a decision analysis, and cost and benefit values
derived from this work should be integrated in the model. Furthermore, the effects
of different project efficiencies on the total yield should be investigated.
REFERENCES
Doorenbos, J. and Pruitt, W. O. (1975) Crop water requirement. FAO Irrigation and Drainage
Paper no. 24.
Dudley, N. J. (1969) A simulation and dynamic programming approach to irrigation decision
making in a variable environment. PhD Thesis, University of New England, Australia.
Flinn, J. C. and Musgrave, W. F. (1967) Development and analysis of input-output relations for
irrigation water. Austral. J. Agric. Economy 11, 1-19.
Hall, W. A. and Butcher, W. S. (1968) Optimal timing of irrigation. / . Irrig. Drain. Div., Amer. Soc.
Civ. Engrs 94, 487-492.
Jensen, H. E. (1968) Water consumption by agricultural plants. In: Water Deficits and Plant
Growth, chapter 1 (edited by T. T. Kozlowski): Academic Press, New York, USA.
Minnas, B. S. et al. (1974) Toward the structure of a production function of wheat yields with
dated inputs of irrigation water. Wat. Resour. Res. 10, no. 3, 383-393.
Plate, E. J. and Treiber, B. (1979) A simulation model for determining the optimum area to be
irrigated from a reservoir in arid countries. Proceedings of the III World Congress on Water
Resources (Mexico), vol. 1, 1-15.
Schmidt, O. and Treiber, B. (1979) A data-generation model for daily flows in arid regions.
Die Wasserwirtschaft 70, no. 1, 5-9.
Yakowitz, S. J. (1973) A stochastic model for daily river flows in an arid region. Wat. Resour. Res.
9, no. 5, 1271-1285.