optimization of reservoir operation for irrigation and...

Scientific Procedures Applied to the Planning, Design and Management of Water Resources Systems (Proceedings of the Hamburg Symposium, August 1983). IAHSPubl.no. 147.

Optimization of reservoir operation for irrigation and determination of the optimum size of the irrigation area

OTTO SCHMIDT Landesanstalt fur Vmweltschutz, Baden-Wurttemberg, Griesbachstr. 3, 7500 Karlsruhe, FR Germany ERICH J. PLATE Institut fur Hydrologie und Wasserv/irtschaft, Universitat Karlsruhe, Kaiserstr. 12, 7500 Karlsruhe, FR Germany

ABSTRACT Since in arid zones water is the limiting factor for irrigation, it is very important to establish the relationship between the size of the irrigation area and the operation schedule of a reservoir delivering the irrigation water. The objective for operation of an irrigation system should be the optimum use of all available water in the sense of maximization of the crop production for the whole irrigation area. The solution of this planning problem is obtained by means of a stochastic simulation method. It required to develop a computer model consisting of several sub-models by means of which an irrigation system, supplied by a single purpose reservoir, can be simulated. The operation rule results from an improved implicit-stochastic-optimization. The advantages and improvements of the water utilization by use of the developed optimum models are demonstrated, and for a case study on the Arabian Peninsula the results are compared with those from a simulation model which operates without the optimum operation schedule (so-called conservative model).

Optimisation d'une règle opérationelle pour un réservoir destiné a 1'irrigation et évaluation de la superficie irrigée optimale RESUME Etant donné que, dans les zones arides l'eau d'irrigation représente le facteur limitant pour l'agriculture, une grande importance doit être attribuée à la mise au point d'une relation entre la superficie ayant besoin d'être irriguée et une règle opérationelle pour le réservoir qui fournit l'eau d'irrigation. Le but de l'aménagement du système d'irrigation devrait être d'utiliser une certaine quantité d'eau de manière que la production agricole soit la plus élevée possible. On obtient la solution de ce problème à l'aide des modèles de simulation aléatoire. A ce propos, il a été nécessaire de développer un modèle d'ordinateur qui est composé de plusieurs sub-modèles adaptés à la simulation d'irrigation, ces derniers étant approvisionnes en eau d'irrigation pour

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452 Otto Schmidt s Erich J.Plate

un réservoir uniquement destiné à l'irrigation à l'aide de l'optimisation basée sur la simulation aléatoire il a été possible d'améliorer la règle opérationelle. Pour la presque-île d'Arabie les avantages ainsi que les améliorations dans l'utilisation de l'eau obtenus par le modèle ainsi mis au point sont montrés et comparés avec un modèle de simulation où un procédé d'optimisation n'est pas appliqué.

NOTATION

AO, Al, A2 c C CETa

CETp

CIR

D ETa ETp FC IR

NWY PWP QM QZ S SA S T

X Y/Y0

regression coefficients parameter numerical factor cumulative actual évapotranspiration within a time interval (mm) cumulative potential évapotranspiration within a time interval (mm) cumulative quantity of water for the replenishment of the soil profile during a time interval (mm)

3

estimate for the draft within a time interval (m ) actual évapotranspiration (mm day ) potential évapotranspiration (mm day- ) field capacity (mm) quantity of water for the replenishment of the soil profile (mm) net water yield (ha) permanent wilting point (mm) mean inflow (m3) inflow during a time interval (m~)

3

storage content (m ) constant draft every time interval (m")

3

estimate for the storage content (m ) irrigation quantity, required for avoiding yield reductions (m ) available soil moisture (mm day L) relative yield = ratio of actual to maximum yield project efficiency soil suction (bars)

INTRODUCTION

Establishing an irrigation system, supplied by a reservoir, one has to determine the size of the crop area which has to be developed (design area). As in arid or semiarid zones in irrigated farming often more arable land is available than can be irrigated with the available water, the free parameter crop area has to be considered in design and operation. To illustrate this effect: if a small design area is irrigated almost no losses due to lack of water can be expected. However, not all available water will have been used for plant production. If on the other hand a large design area is irrigated with the same supply of water during the same time, then one has to expect many seasons in which the water does not suffice

Optimization of reservoir operation for irrigation 453

to water the crops without corresponding losses. Unfortunately in these regions the water budget varies considerably from season to season and from year to year. If the little water is stored in a reservoir, it will be important to establish an optimum schedule for the operation of the whole system. The condition for operation should be to use all available water in order to maximize the agricultural return. The area to be used is assumed either fixed or variable, i.e. the crop area could be less than the design area (especially in very dry years). The relationship between the size of the design area and the operation of the system is established through that of the water supply and the consumptive use of the cropping pattern. In addition one has to take account of a very strong reservoir sedimentation and a high rate of evaporation from the stored water. A schematic representation of the system is given in Fig.l. The aim of the paper is not to give a detailed description

water supply operation irrigation area

sediment evaporation transport

t water discharge

évapotranspiration

soil moisture

FIG.1 Schematic representation of the system.

of the developed models, but to answer fundamental questions in the context of design and operation of irrigation systems and to demonstrate the advantages of the developed models in comparison with conservative design methods.

PROBLEM IDENTIFICATION AND STRATEGY OF SOLliïlON

The above problem can be broken down into two main problems each of which comprise several subsystems. The first main problem is the "design-problem", i.e. the determination of the size of the irrigation area and the sizing and design of the associated systems (channels, weirs, roads etc.). The size of the design area must be larger than the crop area which is adjusted during the irrigation season. We assume the following rules for adjusting the size of the area:

(a) the crop area is constant during the lifetime of the system (crop area = design area);

454 Otto Schmidt S Erich J.Plate

(b) the initial crop area is determined according to reservoir content and expected inflows at the beginning of every year (crop area é design area).

The second problem is the "operation problem", i.e. the determination of the optimum operation rule for a single purpose reservoir used to irrigate a monocrop under consideration of the stochastic behaviour of the water supply, évapotranspiration and the size of the irrigation area. In this context we have distinguished two irrigation strategies:

(a) The first tries always to deliver enough water to the plants, to irrigate to field capacity, so that no yield reductions occur. If there is not enough water, parts of the fields are abandoned for the rest of the season.

(b) The second permits temporary water stress and therefore leads to yield reductions. The objective is to obtain from a larger design area with temporarily stressed plants a total higher yield than that from a smaller area where the plants are always sufficiently irrigated. Additionally one has to investigate if the reservoir should be operated in a year or over-year-mode in order to maximize the return.

THE DEVELOPED MODELS

The solution to the above-described design and operation problems is obtained by means of a stochastic simulation model. An overview of the model as a whole is given in Fig.2. It is solved in two steps. First, for defined constraints, like the fixed size of the design area and storage capacity, the optimal operation of the system is determined with known reservoir inflows on the basis of historical or generated river discharges. Then the model is extended to manipulate the parameter crop area (constant over all years, yearly adjustment and reduction of size within the irrigation season). The operation schedule is derived using a modified implicit-stochastic-optimization. The size of the design area is determined by use of simulation runs. The main submodels are described below. For purpose of demonstration and transparency the models which are described in detail elsewhere (Schmidt & Plate, 1980; Schmidt, 1981) are only briefly described.

The runoff generation model

Often measured data series of discharges are too short to be taken as representative of conditions during the lifetime of the system. Especially in the case of arid or semiarid zones. Therefore a model was designed to generate daily discharges. The model is based on investigations of available data from basins on the Arabian Peninsula. With this model one can generate runoff records of any desired length. Details of the model are given by Schmidt & Plate (1980) and Schmidt &. Treiber (1980).

The soil-plant-atmosphere system

The plant production process depends mainly on three factors - soil,


water supply

data generation model for daily runoff - values

1 reservoir model

deterministic models

evaporation

sedimentation

continuity equation

r—S—v { a g r i c u l t u r e J

o p e r a t i o n

forecast : control curve

s imulat ion and opt imizat ion

- year - s torage

- over - year' - storage

I feed - back

size of the irr ig, area - f i xed

- year ly var iable

- reduct ion dur ing i r r igat ion season

i r r iga t ion area _ZL

s o i l - mo is tu re - m o d e l

so i l - mo is tu re -equa t ion act. évapotranspirat ion

monocrop

y ie ld est imat ion

water product ion

func t i on

consumpt ive use

pot. evapotrans -

p i ra t ion

soil -

p lan t -

atmosphere

sys tem

eff ic iency of

wa te r use

FIG.2 Flow chart of the developed model.

plant and atmospheric conditions. Each contributes to and must be considered in the calculation of the consumptive use. In a mathematical model the effect of these factors is included by the following steps:

(a) calculation of the potential évapotranspiration ETp, (according to the FAO Guide (Doorenbos & Pruitt, 1975), if no field measurements are available);

(b) estimation of the reduction of the actual évapotranspiration ETa in the case of decreasing soil moisture (évapotranspiration concepts).

(c) use of a soil water balance equation for the calculation of actual available soil moisture (soil moisture strategies).

(d) calculation of the expected plant production.

Evapotranspiration concepts After application of irrigation water, the rate of potential évapotranspiration gradually decreases, i.e. some time after a water application the actual évapotranspiration will be less than the potential evaporation:

456 Otto Schmidt S Erich J.Plate

ETa = c-ETp (1 )

where c is a function of atmospheric demand, plant behaviour and available soil moisture x or soil suction i|i (x and tji are interrelated by the so-called desorption curve). Quantitative and qualitative concepts for modelling the c-function are given in detail by Yaron (1973). For incorporation of these results in a computer model, the best known approaches, based on experimental results, have been used:

(a) c = f (x): only the soil moisture between two fixed limits can be used by plants. Most researchers believe that only the soil moisture between the upper limit, field capacity (FC), and the lower limit, permanent wilting point (PWP), is available. Yet there are differences between the functional behaviour of the c-function.

(b) c = f(x,ETp): according to Denmead & Shaw (1962) the different c-functions are special cases of a general function describing the dependence of ETa on soil , plant and atmospheric conditions. Therefore the c-function depends on ETp. To find out how important the specifics of the soil moisture are for the results, two different models were adopted, which are able to reproduce the different approaches above:

(a) the model of Minhas et al. (1974) which is representative of the first approach;

(b) the model of Norero (1972) which is able to reproduce the second approach.

The consequences of the use of the different approaches on the results obtained by the developed models are given later. Details of the models and their adaptation are given in Schmidt (1981) and Schmidt & Plate (1980).

The soil moisture balance and yield estimation With the models described above, it is possible to calculate x and ETa at time 112 using the soil water balance equation if ETp and x at time III (112 > III) and the replenishment of the soil profile by a water application IR(II) is known. Supposing ETp is constant during a time interval J (e.g. a month), we are able to calculate irrigation strategies as a function of the soil moisture at the beginning of the interval. The allowed level of soil moisture deficit is a scale for the yield reduction.

Provided that water is the only limited factor, the yield estimation is calculated by an equation of the following type:

. . dry matter Y/Y^ = f(CETa/CETp) = g1*"" y l e + production (2)

o model , model

with Y/Y = ratio of actual to maximum yield = relative yield, CETa = cumulative actual évapotranspiration within a time interval, CET„ = cumulative potential évapotranspiration within a time interval.

The first part is the model proposed by Jensen (1968) for grain yield and the second is based on results given by Doorenbos & Pruitt (1975) for the estimation of dry matter yield of gras. The model is valid for one or several harvests from the same plant during one year. The detailed description of the objective function is given by Schmidt (1981) and Schmidt & Plate (1980).


Operation rule

The operation rule results from an improved implicit stochastic optimization. This proceeds by first generating runoff hydrographs which are used as perfect forecasts from which optimum water releases are calculated by means of dynamic programming optimization. Then the optimum releases are correlated with parameters known at the time at which the releases have to be decided. For this, multi-regression analyses are used and the results improved by means of control curves.

The optimization of the water resource system on the basis of perfect forecasts is done by backward dynamic programming. For purpose of demonstration it is assumed that the system is operated on a one-year storage mode and that the size of the irrigation area is always constant. The system can be described by two submodels which are linked: a storage model and a soil moisture model. The linkage is received through the size of the crop area and the project efficiency of water use <)>. A schematic representation is given in Fig.3. (A detailed description of the equation system is given by Schmidt & Plate (1980) and (1983)).

QZ(J)

SU) •SIJ-I)

CETaU)

FIG.3 Dynamic programming scheme of the one-year storage.

The on-line operation rule is obtained by use of implicit stochastic optimization which is modified as follows:

(a) If the model with the known inflows is operated for a fixed area over a time horizon of N years, a set of optimum values for reservoir contents and drafts for every time interval is obtained. By use of regression analysis an estimate for the storage content S(J) at the end of an interval J is calculated:

S(J) = A0(J) + A1(J) * S(K,J 1) A2 * QZ(K,J) (3)

with A0,A1,A2 = regression coefficients, S(K,J - 1) = storage content at the beginning of J in the K-th year and QZ(K,J) = total inflow during J in the K-th year.

(b) By use of the continuity equation (spill and evaporation losses neglected) and replacing QZ(K,J) through the mean inflow

458 Otto Schmidt & Erich J.Plate

QM(J) an e s t i m a t e for the d r a f t D(J) for r e a l t ime o p e r a t i o n during J i s r e c e i v e d

D(J) = S(K,J - 1 ) * ( 1 - A1(J ) ) + C*QM(J)*(1 - A 2 ( J ) ) -

AO(J) + SA(K) (4 )

and

D(J) S T(J)

w i th T(J) = the irrigation quantity, required for avoiding yield

reduct ions; C = numerical factor, obtained by simulation runs; SA(K) = constant draft for every interval J

= storage at the beginning of the irrigation season divided by the number of intervals of the season.

The draft within J is only a function of the storage at the beginning of J although a statistical forecast is integrated. The irrigation system itself is operated by employing the simulated soil moisture strategies.

(c) If this procedure - optimization coupled with regression analysis - is repeated for different sizes of the irrigation area a set of regression values is obtained which secures longterm optimality of the reservoir operation.

Model modifications

To take into consideration the possibilities of the reduction of the crop area during the irrigation season and the operation of the storage in an over-year mode the optimization model was modified (more details are given by Schmidt, 1981). The on-line operation rule was calculated again with the implicit-stochastic-optimization approach.

The on-line operation models

To give a solution of the design problem and to demonstrate the advantages of the developed operation rule two models for the online operation were developed. The first operates with the new operation rule, and is henceforth called the optimum model, the second is based only on the actual storage content at the beginning of a time interval without any forecast of future inflows and is called conservative model. Both work on a daily base and account for soil moisture strategies, reservoir sedimentation, evaporation losses and spillway actions . The models were modified for handling the parameter crop area (for more details see Schmidt, 1981).

THE RESULTS OF THE SIMULATION RUNS

With the developed models simulation runs over 5 to 50 years were done for a basin located on the Arabian Peninsula. The size of the


2

Wadi is 870 km . The climate can be characterized as relatively-arid. The dominant soil in the irrigation area is sandy loam. The irrigation water is not saline and there exists no connection to a saline aquifer. No rainfall occurs in the irrigated area.

With the soil moisture model after Minhas et al. (1974) simulations were run according to size of the irrigation area. As a number of comparison the net-water-yield NWY is used. The NWY is the sum of the products of the relative yield and the size of the irrigation area of every year over the lifetime of the system. In the following tables the mean value of the 5 to 50 years simulations is used.

TABLE 1 Results of the simulation runs in case of a fixed irrigation area

DAREA NWY.IOOO (ha)

Optimum Conservative

model model

NO WATER lOOO

WITH

1100 1200

1300

1400

1500

DEFICIT

WATER DEFICITS

168.

179. 183.

190.

192. 194.

.7

.3

.4

.7

.1

.0

165.6

176.3

181.3

Constant area during the lifetime of the system

For an assumed value of é = 0.7 (a high value, but very often used by consultants) the results are seen in Table 1. Comparison of the results show:

(a) the design area (DAREA) with the best results is in the case of the optimum mode 1500 ha respectively with the conservative model 1200 ha;

(b) the NWY is improved by about 7%; (c) furthermore with both types of model it is more economical

to allow water deficits of the crops because a total higher yield can be obtained.

New areas at the beginning of a year

In Table 2 the different results are listed. As a function of different <f>-values the following interpretations can be given:

(a) independently of the used efficiency of water consumption there exists a remarkable increase in NWY if during the irrigation season, water deficits are allowed.

(b) in this case an improvement of the NWY by aid of the optimum model of 6% is obtained.

460 Otto Schmidt & Erich J.Plate

TABLE 2 Results of the simulation runs in the case of every gear new fixed irrigation area

NWY.IOOO

<j> DAREA Optimum Conservative (ha) model model

0.7 1500* 198.7* 187.1 0.7 WOO 168.3 167.4

0.5 llOO* 142.4* 134.2*

0.5 800 129.3 124.9

Results, if water deficits are allowed.

Area reduction during the irrigation season

Based on the simulation runs it is found, that if once at the beginning of the year the size of the irrigation area is fixed, no reduction during the season is necessary.

Comparison of over-year and one-year storage operation

For different sizes of constant irrigation areas during the lifetime no significant differences in yield are obtained, if the system is operated on an over-year or an one-year storage concept. Furthermore the same optimum design area is obtained.

Effect of the soil moisture models

To investigate the importance of the used soil moisture concept , simulations were done with the model of Norero et al. (1972) for the case of a constant irrigation area during the lifetime of the system. The results showed that:

(a) the advantage of operating the system with the optimum model in comparison to the conservative model is reproduced;

(b) there also exists a remarkable increase in NWY, if during the irrigation season water deficits are allowed;

(c ) if the actual values of the ÎWY-calculations are compared, then a significant difference between the two models is found to exist.

CONCLUSIONS

The models described above are valuable tools in the design process for irrigation systems in arid or semiarid lands. Based on the results of the simulation runs it is expected that research in this field will help to contribute to solve the problem of feeding the world's population. Limitations of the models are, that they are only able to model a monocrop and that the processes at the farm-level are only considered in a very approximate fashion. Furthermore


it would be very interesting to extend the models to include monetary objectives instead of yield only.

REFERENCES

Denmead, O.T. & Shaw, R.H. (1962) Availability of soil water to plants as affected by soil moisture content and meteorological conditions. Agron. J. 54, 385-390.

Doorenbos, J. & Pruitt, W.O. (1975) revised (1977) Croo water requirements. Irrigation and Drainage Paper no. 24, FRO, Rome.

Jensen, M.E. (1968) Water consumption by agricultural plants. In: Water Deficit and Plant Growth (ed. by T.T.Koslowski) . Academic Press, New York.

Minhas, B.S. et al. (1974) Towards the structure of a dated production function for wheat yield with dated inputs of irrigation water. Wat. Resour. Res. 10 (3), 383-393.

Norero, A.L. et al. (1972) Effect of irrigation frequency on the average évapotranspiration for various crop-climate-soil-systems. Trans. Am. Soc. Agric. Engrs 15, 662-666.

Schmidt, 0. (1981) Die Optimierung des Speicherbetriebs fiir die Bewasserung mittels Simulation. Mitteilungen des Instituts Wasserhau III, Universitat Karlsruhe, Heft 18.

Schmidt, 0. & Plate, E.J. (1980) A forecasting model for the optimal scheduling of a reservoir supplying an irrigated area in an arid environment. In: Hydrological Forecasting (Proc. Oxford Symposium, April 1980), 491-506. IAHS Publ. no. 129.

Schmidt, 0. & Plate, E.J. (1983) Optimaler wasserwirtschaftlicher Speicherbetrieb. Z. Operations Research 27, B1-B19.

Schmidt, 0. & Treiber, B. (1980) A data generation model for daily flows in arid zones. Die Wasserwirtschaft 70 (1), 5-9.

Yaron, B. et al. (1973) Arid Zone Irrigation. Springer Verlag, Berlin.

Young, G.K. (1967) Finding reservoir operating rules. J. Hydraul. Div. ASCE 93, 297-321.

optimization of reservoir operation for irrigation and...

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