Zvonimir 2007

www.elsevier.com/locate/solener

Solar Energy 81 (2007) 904–916

A model for optimal sizing of photovoltaic irrigationwater pumping systems

Zvonimir Glasnovic a,*, Jure Margeta b

a Faculty of Chemical Engineering and Technology, University of Zagreb, 10000 Zagreb, Marulicev trg 19, Croatiab Faculty of Civil and Architectural Engineering, University of Split, 21000 Split, Matice Hrvatske 15, Croatia

Received 11 July 2005; received in revised form 26 September 2006; accepted 27 November 2006Available online 22 December 2006

Communicated by: Associate Editor Hansjorg Gabler

Abstract

The previous methods for optimal sizing of photovoltaic (PV) irrigation water pumping systems separately considered the demand forhydraulic energy and possibilities of its production from available solar energy with the PV pumping system. Unlike such methods, thiswork approaches the subject problem systematically, meaning that all relevant system elements and their characteristics have been ana-lyzed: PV water pumping system, local climate, boreholes, soil, crops and method of irrigation; therefore, the objective function has beendefined in an entirely new manner. The result of such approach is the new mathematical hybrid simulation optimization model for opti-mal sizing of PV irrigation water pumping systems, that uses dynamic programming for optimizing, while the constraints were defined bythe simulation model. The model was tested on two areas in Croatia, and it has been established that this model successfully takes intoconsideration all characteristic values and their relations in the integrated system. The optimal nominal electric power of PV generator,obtained in the manner presented, are relatively smaller than when the usual method of sizing is used. The presented method for solvingthe problem has paved the way towards the general model for optimal sizing of all stand-alone PV systems that have some type of energystorage, as well as optimal sizing of PV power plant that functions together with the storage hydroelectric power plant.� 2006 Elsevier Ltd. All rights reserved.

Keywords: Photovoltaic pumping; Irrigation; PV generator; Optimal sizing; Dynamic programming

1. Introduction

The previous optimizing of photovoltaic (PV) waterpumping systems, which have been the subject of numerouspapers, mainly dealt with improvement of effectiveness ofvarious system components, as well as their better mutualadjustment, with the aim of total cost reduction of thePV pumping system (AVICENNE Programme, Papersfor distribution, 1997).

On the other hand, optimal sizing of the PV pumpingsystem is basically reduced to calculation of the requiredhydraulic energy at the output of the system and its rela-

0038-092X/$ - see front matter � 2006 Elsevier Ltd. All rights reserved.

doi:10.1016/j.solener.2006.11.003

* Corresponding author. Tel.: +385 1 4597281; fax: +385 1 4597260.E-mail address: [email protected] (Z. Glasnovic).

tion to monthly average daily solar irradiation. Hydraulicenergy for PV pumping systems for irrigation is calculatedbased on required water quantity data, calculated by agri-cultural experts, and total head of water rise, Kenna andGillett (1985).

The equation for nominal electric power of PV genera-tor P el expressed in (W), in referential condition (StandardTest Condition STC – intensity of solar irradiation1000 W/m2, relative air mass AM1.5 and temperature ofPV generator 25 �C), according to Kenna and Gillett(1985) is as follows:

P el ¼1000

fm½1� acðT cell � T 0Þ�gMP

� EH

ES

ð1Þ

where EH ðkW h=dayÞ is output hydraulic energy,ES ðkW h=dayÞ the solar energy at the PV generator input,

mailto:[email protected]

Nomenclature

A area of irrigated location (ha)Dr the most active rooting depth (m)EH hydraulic energy (kW h)Es mean daily radiation on horizontal plane (ter-

restrial radiation) ðkW h=m2dayÞET0 potential evapotranspiration (mm)ETr real evapotranspiration (mm)F objective functionFC field capacity (mm)fiðxiÞ optimal return function per state variable xi in

time stage i

fi�1ðxi�1Þ optimal return function per state variable xi�1

in time stage i � 1fiðW ðiÞÞ optimal return function per state variable W ðiÞ in

time stage i

fði�1ÞðW ði�1ÞÞ optimal return function per state variableW ði�1Þ in time stage i � 1

fm load matching factor to PV generator character-istics

GS intensity of solar irradiance ðW=m2ÞH0T vertical head from the water outlet to the

ground (m)HDT dynamic water level in borehole (m)HF head friction loss (local and linear losses) (m)HST static level (groundwater level) (m)HTE total head (m)i time stage (increment)INF infiltration (mm)Kc crop coefficientL total precipitation losses due to superficial

drainage (mm)N total number of time stages i

nd number of days in time stage i

P el nominal electric power of the PV generator (W)P �el optimal nominal electric power of the PV gener-

ator (W)

P elðiÞðQdðiÞÞ return function from stage i per decision var-iable QdðiÞ

QAP average flow, known also as ‘‘apparent flow’’rate ðm3=hÞ

Qd mean daily water volume at the output of thePV pumping system (decision variable)ðm3=dayÞ

QPVðiÞ water from PV pumping system which is,by irri-gation, added to soil in time stage i (m3)

Qmax maximum discharge capacity of boreholeðm3=hÞ

R total precipitation (mm)Re total effective precipitation (that reach the soil)

(mm)ri return from stage i

tr mean daily insolation (h)T0 referential temperature of PV cell (generator)

(25 �C)Ta temperature of the surroundings (�C)T cell temperature of PV cell (generator) (�C)ui decision variable in stage i

W ðiÞ soil moisture in time stage i (mm)W ði�1Þ soil moisture in time stage i� 1 ðmmÞxi state variable in stage i

xi�1 state variable in stage i� 1ac PV cell (generator) temperature coefficient

(�C�1)goc nominal efficiency of PV generator (%)gI inverter efficiency (%)gMP motor–pump unit efficiency (%)gMPI efficiency of motor–pump unit and inverter (%)gN irrigation efficiency (%)si transformation, i.e. relation between output and

input state variables for every stage of the sys-tem i

# calculation coefficient of average flow

Z. Glasnovic, J. Margeta / Solar Energy 81 (2007) 904–916 905

fm the load matching factor to characteristics of the PVgenerator, ac the PV cell temperature coefficient (�C�1),T0 the referential temperature of the cell (25 �C), gMP mo-tor–pump unit effectiveness, and T cell temperature of thecell (�C) which, according to Markvart and Castaner(2003), can be calculated by the equation:

T cell ¼ T a þNOCT� 20

800GS ð2Þ

where Ta is air temperature (�C), GS the intensity of solarirradiance (W/m2) and NOCT is the Nominal OperatingCell Temperature (�C).

Therefore, the nominal electric power of PV generator iscalculated based on the known monthly average dailydemand for hydraulic energy EH and available monthlyaverage daily solar irradiation Es in the critical month

and the known efficiency of the motor–pump unit gMP inreferential operating conditions, taking into account theeffect of outside temperature on the efficiency of the PVgenerator.

Eq. (1) stands for critical month, i.e. the month in whichthe ratio between hydraulic and radiated solar energyEH=ES is maximum. However, this approach has the fol-lowing flaws:

• Lack of systematic quality – Hydraulic energy valueobtained from an agriculture expert is not in any waycorrelated to the possibilities of its covering with avail-able solar energy in the calculated month. Consequently,the problem is observed particularly and separately, i.e.the agriculture expert first calculates the values of therequired hydraulic energy independently from the

906 Z. Glasnovic, J. Margeta / Solar Energy 81 (2007) 904–916

designer of the PV system. After that, the designerdivides it by available solar energy, and then uses thisvalue as his optimal value.

• Static quality – By monitoring only the critical month, itis not possible to observe properly the demands forhydraulic energy and possibilities of their fulfilling inthe previous and subsequent months. Furthermore, itis completely ignored that the water static level andwater quantity in the borehole can vary from monthto month, thus affecting the determining of critical val-ues. The fact that total head in borehole is dependenton the quantity of the pumped water is also disregarded.Ignored is the possibility of water redistribution regard-ing time and quantity, from water abundant periods(days) into dry period, and thus redistribution of criticalparameters in system sizing (possibilities of water stor-age). Also, possibilities of fulfilling water requirementsfrom available solar energy with the PV pumping systemare not even considered in dynamic sense. Therefore,everything is observed statically.

It is evident that such sizing of the nominal power of thePV generator, whose price is still relatively high, doesn’tyield optimal results. This results in increased investmentcosts and affects possible economic justification of suchsystems.

2. System configuration

Unlike the approach where PV pumping systems areobserved separately from their surroundings (the work of

LOCAL CLES(i)

Ta(i)

Pel(i) ~ EH

η I

HF EH(i)

Eel(i) HTE(i)(Qd(i)) H0T

HST(i) HDT

W(

ηMP

Photovoltaic pumping system

Tcell(i)

INV

MP

Fig. 1. Principle diagram of PV pumping system

Bahaj and Mohammed (1994) is typical in that sense),and in accordance with Glasnovic et al. (1991) (which par-tially unites the main influential elements into one systemand properly chooses the optimizing method that is widelyused in water resources management, but the objectivefunction is inadequately set and in that sense the optimiz-ing problem is solved), in this work the problem is solvedat the level of the system as a technological entirety. Thisentirety equally encloses all components of the system,including natural processes in the system (climate, hydrol-ogy, boreholes, pumping system, irrigation, agriculture andpower supply) during the entire period when the system isin operation (irrigation season), Fig. 1.

This means that throughout the entire operation periodthe system is dynamically analyzed as an integrity, takinginto account all changes that occur in relation to availableresources (capacity and needs). The key components thatdetermine water resources and water requirements are cli-mate and hydrology.

The climate determines the moisture/water input andsolar irradiation, for one part, and water requirementsfor irrigation, for the other part. Land and geological fea-tures determine water storage capacity, both in superficial/pedological layer and in aquifer.

Climate inputs are of stochaic character, therefore needto be treated adequately throughout the entire period.

The result of such systematic approach is the new math-ematical optimizing model of the PV pumping system forirrigation, which considers integrally three main system ele-ments: PV pumping system, boreholes and irrigationsystem.

IMATE

(i)/ES(i) Re(i) ETr(i)

W(i)

INF(i)

i) = W(i–1)+( 10QPV /A) + Re(i)(i) − − ETr(i) INF(i)

i = 1, 2, ....., N

Irrigated area

ηN

QPV(i) A

for irrigation with typical system elements.


2.1. PV pumping system

The PV pumping system consists of PV generator, nom-inal electric power P el, motor–pumping unit which com-bines motor and pump (MP) and inverter (INV) aspower conditioning to the PV generator.

2.2. Borehole

According to Narvarte et al. (2000), the total head fromborehole H TE, can be expressed by the following approxi-mative equation:

H TEðiÞ ¼ H 0T þ H STðiÞ þH DTðiÞ � H STðiÞ

Qmax

QAPðiÞ

þ H FðiÞðQAPðiÞÞ ð3Þ

where

QAPðiÞ ¼ #QdðiÞ ð4Þ

where increment i assumes the values i ¼ 1 to N (N is thetotal number of time stages, decades), HTEðiÞ the total headlift (m), H FðiÞ the head friction loss (local and linear losses)(m), H 0T the vertical head from the water outlet to theground (m), HSTðiÞ the static level (groundwater level)(m), H DTðiÞ the dynamic level of water in borehole (m),Qmax the maximum discharge capacity of borehole (m3/h),QAPðiÞ the average flow rate, known as ‘‘apparent flow’’

(m3/h), QdðiÞ the mean daily quantity of pumped water(m3/day), # calculation coefficient of the average flow rate.

Apart from the aforesaid, it is important to stress theneed for compatibility between discharge capacity of bore-hole and pumped water, i.e. that the power of the PVpumping system should be synchronized in this sense.Due to this, it is necessary to introduce constraint of water

QPV(i) Re(i)

horizontal W(i) water flow = 0

Fig. 2. Superficial soil lay

pumped daily from the borehole in period i, by the follow-ing equation:

QdðiÞ 6 QmaxtrðiÞ ð5Þ

where trðiÞ is actual mean daily insolation (h).

2.3. Irrigation system

Among several available methods of irrigation, in usenowadays, this study the PV pumping system will beapplied only to the trickle irrigation method, because it isthe most cost effective (irrigation efficiency is gN ¼ 85%).

3. Water related system process

3.1. Water balance equation

The irrigated area can be observed as a limited area i.e.as water reservoir (with border and transborder flows), intoand from which water flows. Therefore, observed is onlythe superficial soil layer of a certain depth Dr in whichplants most actively intake water (Doorenbos, 1975). Soilas water reservoir can be observed in Fig. 2. Water flowsinto soil (reservoir) by precipitation ReðiÞ, irrigation QPVðiÞand possibly by capillary lift from deeper layers, and flowsout by infiltration INFðiÞ and evapotranspiration ETrðiÞ.Soil moisture stages in i time period are denoted as W ðiÞ.Considering the water quantities that flow into the soil,as well as those flowing from it, and by observing soil mois-ture as total water quantity in soil on unit area, expressedin ‘‘mm’’ ð1 mm ¼ 10 m3=haÞ, disregarding horizontalflow, at the beginning of time period i, the water balanceequation (soil stages) for soil as water storage (Doorenbos,1975; Fontane and Margeta, 1988) can be expressed asfollows:

ETr(i)

superficial soil layer (e.g. about 30cm deep)

.

INF(i) water in soil (e.g. 35% of volume, or105 mm water, or 1050 m3/ha)

thicker soil layers

er as water reservoir.


W ðiÞ ¼ W ði�1Þ þ ð10QPVðiÞ=AÞ þ ReðiÞ � ETrðiÞ � INFðiÞ ð6Þ

where increment i assumes the values i ¼ 1 to N (N is thetotal number of time stages, decades); W ði�1Þ the soil mois-ture in i � 1 period (mm); QPVðiÞ the water from PV pump-ing system that is, by irrigation, added to soil in i period(m3); Re the total effective precipitation, defined by relationReðiÞ ¼ RðiÞ – LðiÞ, where RðiÞ is total precipitation, LðiÞ are to-tal losses due to superficial drainage (in arid climate areas,Doorenbos, 1975; they can be 80% of total precipitation) intime stage i (mm); ETrðiÞ is quantity of water used for thereal evapotranspiration in i period (mm); A irrigated area(ha); and infiltration INFðiÞ in time stage i (mm) that repre-sents the connection between the observed superficial soillayer and deeper soil layers. In dry period there is mostlyno infiltration.

3.2. Soil and soil moisture

The soil and its characteristics determine water storagecapacity and conditions/processes of water flow from sur-face into deeper layers (infiltration) and from soil into theatmosphere (evapotranspiration).

Soil moisture is the key decision variable that determinesthe conditions within the system and water requirements.Soil moisture is a result of natural processes (precipitation,evapotranspiration) and characteristics of the soil. If mois-ture from natural processes is insufficient, irrigation isapplied.

The soil moisture limits W ðiÞ, in i period, where optimalcrop yield is achieved, can, according to Tomic (1988), beset within the values:

0:6FC 6 W ðiÞ 6 FC ð7Þ

where FC is field capacity.The capacity of such water reservoir is determined by

maximum water quantity that the subject soil layer canintake, i.e. by field capacity FC. For a certain area, suchas that in Osijek, it would have typical value of 35% (of vol-ume percentage, which is, in relation to 1000 mm of soildepth, 350 mm of water). At depth Dr ¼ 0:3 m (depth ofrooting system of, e.g. tomato, as the observed crop),soil moisture value expressed in (mm) is FC ¼ 350�0:3 ¼ 105 mm or, expressed in (m3), on area A ¼ 0:5 ha,maximum water quantity that the observed soil layer canintake is 105� 10 m3=ha� 0:5 ha ¼ 525 m3, Fig. 2. Forthe location in the Split area, the volume of FC of 28%,expressed in (mm) is FC ¼ 84 mm, which is a total of420 m3 on area A ¼ 0:5 ha:

3.3. Crop water requirements

There are potential ET0 and real evapotranspirationETr. Potential evapotranspiration ET0 is the water quan-tity which the plants would use from the soil and the sur-face of plant organs in optimal cultivation conditions.Real evapotranspiration ETr is actual water consumption

of a crop per area unit. As it has been established thatpotential evapotranspiration over a longer period of timeis constant and that it depends only on climate factors ofthe area, it is possible to determine it if climate factors ofthe area are known (Doorenbos, 1975). The realationbetween ETr and ET0 has an important role in definingcrop water requirements, so crop coefficient Kc has beenintroduced. This coefficient depends on the characteristicsof the crop itself, i.e. on growth period (developmentstage), on degree of water supply of crops duringvegetation period and on climate conditions of the area.Therefore, according to Doorenbos (1975), real evapo-transpiration in i period can be expressed as:

ETrðiÞ ¼ KcðiÞET0ðiÞ ð8Þ

For the purpose of testing the optimizing model, (withoutprejudging the correctness and applicability of such calcu-lation on a concrete location, cultivated crops or other rel-evant influential factors), Turc’s formula was applied. Ituses two main parameters, i.e. solar irradiation and tem-perature, which are also input variables of the PV pumpingsystem. The following equation is obtained from the origi-nal form of Turc’s formula for decade calculation (Kos,1987), by inserting solar irradiation expressed in kW h/m2:

ETrðiÞ ¼ 0:13ð86:4ESðiÞ þ 50Þ T aðiÞ

T aðiÞ þ 15ð9Þ

where increment i assumes values i ¼ 1 to N (N is totalnumber of time stages, decades), ETrðiÞ the decade averagevalue of real evapotranspiration (mm/decade), ESðiÞ thedecade average daily value of global solar irradiation ona horizontal surface ðkW h=m2dayÞ, and T aðiÞ is the decademean daily value of air temperature (�C). Irrigation periodis chosen from the first decade in April until the third dec-ade in August ðIV1–VIII3Þ.

4. System management

4.1. Economic estimate

Economic estimates (valorizations) of various alterna-tives generally come down to comparison of PV pumpingsystem with the most effective traditional driving system,i.e. diesel engine (Kenna and Gillett, 1985; McNelis et al.,1989). In that sense their unit water costs are compared.

In the article by Barlow et al. (1993) it is stated that PVpumping systems for potable water supply are more effec-tive than diesel pump unit up to volume head product (prod-uct of total head and flow rate, i.e. Qd � HTEÞ of 800 m4/dayand for irrigation up to 250 m4/day. Namely, the PV waterpumping systems for irrigation operate shorter time, andare characterized by relatively higher demand for pumpedwater. The stated estimates are of a relatively older dateand in the meantime far more efficient PV pumping systemshave appeared at significantly lower prices than in the nine-ties. Therefore it can be concluded that the efficiency oftoday’s PV pumping systems, in relation to diesel pump,


is significantly higher. Also, the prices of diesel fuel haverisen in the meantime. Due to the lack of new relevant anal-ysis, the authors have chosen efficiency limit value of thePV water pumping for irrigation in relation to diesel pump,in the value of volume-head product, in iperiod, expressedby relation:

QdðiÞ � HTEðiÞ 6 800 m4=day ð10Þ

4.2. Determining the nominal electric power of the PV

generator

In the systematic approach to the problem of determin-ing optimal nominal electric power of the PV generator,Eq. (1) has been transformed in this work, in order to showits direct dependency on pumped water, and apart fromcharacteristics of the PV pumping system, to include char-acteristics of the borehole, as well as irrigation method.

In that sense, the initial equation is the one for calculat-ing hydraulic energy at the output of a pumping system in i

time period, which, expressed in kW h, is as follows:

EHðiÞ ¼2:72QdðiÞH TEðiÞ

1000ð11Þ

where EHðiÞ is hydraulic energy (kW h), QdðiÞ the mean dailypumped water ðm3=dayÞ, and H TE is the total head (m).

Furthermore, with modern PV pumping systems, whichare mostly electronically controlled, instead of matchingfactor fm in Eq. (1), it is justified to use inverter efficiencygI, which can include efficiency of the entire electronic sys-tem for matching the load power to the characteristics ofPV generator (INV in Fig. 1). By combining this efficiencywith motor–pump unit efficiency gMP into one efficiencygMPI, and including irrigation efficiency gN (it shows towhat extent the water that enters a certain irrigation systemis exploited), and by inserting them into Eq. (1), and thenby inserting Eq. (11) into Eq. (1) and then first by insertingEq. (4) into Eq. (3) and by inserting such Eq. (3) into Eq.(1), and allocating the index i (multi-stage process) to eachvariable that changes its values during the observed period,the final relation for calculating electric power of the PVpumping system is obtained:

P elðiÞ ¼2:72

½1� acðT cellðiÞ � T 0Þ�gMPIgNESðiÞ

� H 0T þH STðiÞ þH DTðiÞ � H STðiÞ

Qmax

#QdðiÞ þH Fð#QdðiÞÞ� �

QdðiÞ

ð12Þ

Eq. (12) is the basis of calculation of the nominal electricpower of PV pumping system.

It is noted that the nominal electric power is actually acomplex, non-linear function of pumped water QdðiÞ. How-ever, contrary to previous approaches to PV pumping sys-tems, where input elements of solar irradiation ESðiÞ (andtemperature T aðiÞÞ varied and where output water quantitiesfrom the system QdðiÞ were tested and analysed at certaintotal head values for various elements and system configu-

rations and other variable conditions, at the same timeoptimizing the functioning of the PV pumping system itself(internal optimizing); in this system approach it is reversed.In this sense, for set output water quantities QdðiÞ (discreti-zied values of decision variable), the values of nominal elec-tric power are calculated by Eq. (12).

In this manner, through QdðiÞ, which represents the out-put water quantity from PV pumping system, and also theinput water quantity into the irrigation system, waterrequirements and possibilities of their fulfilling are con-nected, i.e. irrigation system with characteristics of PVpumping system. Namely, QPVðiÞ (or QdðiÞ) is by water bal-ance Eq. (6) correlated to soil characteristics (FC, soilmoisture W ðiÞ and W ði�1Þ and infiltration INFðiÞÞ and crops(rooting system depth Dr and crop coefficient KcÞ, as wellas to local climate elements (precipitation ReðiÞ and evapo-transpiration ETrðiÞ), that determine crop water require-ments. On the other side, total head H TEðiÞ is expressed independence of QdðiÞ (Eqs. (3) and (4)), by which boreholecharacteristics are included in the system, whereas throughgN irrigation method is included.

5. Optimization model

5.1. Defining the objectives

In defining the objectives the starting point can be thegeneral aspiration that the PV pumping system meets, inthe best way possible, the requirements for hydraulicenergy at its output, i.e. for irrigation water in the observedperiod.

However, it is not easy to dimension the PV pumpingsystem to produce the hydraulic energy expected from itin every time stage, as the electric energy, that can be pro-duced by PV pumping system, depends on available solarenergy in that period (e.g. this possibility exists in case ofdiesel pump unit, which could be dimensioned to meetthe requirements for hydraulic energy at its output, in everytime stage, because its input energy is not fluctuating).Namely, for a chosen value of the PV pumping system,in certain time periods it would produce too much hydrau-lic energy, i.e. water, and in others too little.

Due to this, in this work, the objective function isdefined in a new manner, i.e. optimizing relation betweenthe output (hydraulic) energy and input (solar) energyEHðiÞ=ESðiÞ, because in that manner the requirements forenergy at the PV pumping system output can best coincidewith the possibilities of its production from the availablesolar energy. As the quotient EHðiÞ=ESðiÞ basically deter-mines the electric power of the PV generator (Eq. 12), theobjective function in this work is reduced to finding thequotient EHðiÞ=ESðiÞ, i.e. the nominal electric power of PVgenerator that will, in the observed period, meet therequirements of the consumer in the best manner possible.

As it is typical for all PV systems that the output energyis dependent on input fluctuating energy of solar irradia-tion, it is not difficult to note that such defining of the scope


function, as a quotient of output energy from a PV systemand input solar energy, could be applied for optimizationof other stand-alone PV systems that have some type ofenergy storage.

In short, the optimization of the PV pumping system forirrigation is possible when for a certain optimal policy ofirrigation water distribution. i.e. for corresponding valuesof hydraulic energy, the certain nominal power of PV gen-erator is found, which could meet all demands in the bestpossible way, throughout the entire observed period.

Considering that investment in the PV pumping system,i.e. PV generator, represents costs from the investors’ pointof view, it is logical to set the objective of achieving theirminimum. On the other hand, as only minimization of elec-tric power would yield relatively significant differencesamong returns from different stages, this paper has setthe objective: minimization of maximum nominal electric

power of PV generator.The extremization of the complex objective function

(Harboe, 1988), which expresses the stated minimizationof maximum nominal electric power of PV generator, foroptimization model based on dynamic programming, canbe written as follows:

MINfMAXF ðW ðiÞ;QdðiÞÞg ð13Þ

where F is the objective function, increment i assumes val-ues i ¼ 1 to N (N represents total time stages, decades), W ðiÞis soil moisture in stage i (mm), QdðiÞ is average daily waterquantity from PV pumping system in stage i ðm3=dayÞ.

5.2. Model formulation

It is evident that the subject problem is multistage. Theimplication is that dynamic programming can be used foroptimal management strategy, according to the model inFig. 3, which is the dominant technique for this kind ofsequential decision problem. The decision variable is therequired irrigation water quantity QdðiÞ, in the observed

ES(i)

Ta(i)

HTE(i)

Qd(i) Re(i) ETr(i) INF(i)

W(i-1) W(i)

Pel(i)

Stage i

PV pumpingsystem

Fig. 3. Mathematical optimization model.

period (stage) i, which is also the mean daily value of waterquantity at the output of the PV pumping system. Statevariable are soil moisture W ðiÞ (total water quantities in soilof an area of location A) that depend on the decision var-iable, state of the system in the previous stage and state ofthe surroundings. Natural variables are effective precipita-tion ReðiÞ, real evapotranspiration ETrðiÞ and infiltrationINFðiÞ. Solar irradiation ESðiÞ, temperature of the surround-ings T aðiÞ and total head H TEðiÞ are influential variables onthe PV pumping system, which appear in calculations ofreturn, i.e. nominal electric power of PV generators.

The system state has already been described by the cor-responding water balance equation.

Given that, according to Eq. (12), the output waterquantity from the PV pumping system QdðiÞ is expressedas daily value, the required water quantity for irrigationQPVðiÞ in given period i (which may be month, decade, week,etc.) should also be expressed as daily value, i.e.

QPVðiÞ ¼ nd � QdðiÞ ð14Þ

where nd is number of days in time stage (day), QdðiÞ is meandaily water output from the PV pumping system in stage iðm3=dayÞ.

By inserting Eq. (14) into Eq. (6) the final form of trans-formation equation of the optimizing process is obtained;i.e. the system state equation:

W ðiÞ ¼ W ði�1Þ þ ð10ndQdðiÞ=AÞ þ ReðiÞ � ETrðiÞ � INFðiÞ

ð15ÞIn this way the connection is established between the re-quired water for irrigation (QdðiÞÞ and possibilities of its ful-filling from the PV pumping system, so this equation andthe equation for calculation of nominal electric power ofPV generator (12) are key relations in formulating the opti-mization model of the PV pumping irrigation system.

5.3. Constraints in the optimization model

In the sense of defining constraint, it is simplest to startfrom the constraint connected with oscillations of soilmoisture level, which has already been defined by Eq. (7).

As the maximum soil moisture in Croatian climate con-ditions occurs in winter and early spring periods (Tomic,1988; analogously it should be established when the maxi-mum soil moisture occurs in other climate areas), the initialvalue of soil moisture W ði¼0Þ can be expressed:

W ði¼0Þ ¼ FC ð16Þ

Considering the average climate and soil characteristics onstandard locations in Croatia that can be irrigated, and thefact that precipitation exceeds evapotranspiration mostlyat the beginning of the observed period (maximum soilmoisture), additional constraint has been introduced.According to this constraint the model will not take intoaccount conditions when total effective precipitation ex-ceeds evapotranspiration, but only conditions when the fol-lowing constraint is fulfilled:


ReðiÞ � ETrðiÞ 6 0 ð17Þ

The constraint of decision variable has been defined basedon economic analysis of cost effectiveness of the PV pump-ing system in relation to diesel pump (Section 4.1, Eq. (10)),according to which it has been established that the PVpumping system is more cost-effective when daily demandfor water is lower than volume-head product which is800 m4=day:

The constraint of water daily pumped from a borehole(functional constraint) has also been included in Eq. (5).

5.4. Recursive formulas of the optimization process

Recursive formulas of the optimization process bydynamic programming, for the purpose of minimizing themaximum objective function, calculating in advance,because initial conditions are known (Harboe, 1988; Nem-hauser, 1966), can be expressed as follows:

fiðxiÞ ¼ min maxui

½riðuiÞ; fi�1ðxi�1Þ��

ð18Þ

in stage transformation conditions:

xi�1 ¼ siðxi; uiÞ ð19Þwhere fiðxiÞ is the function of optimal return by state vari-able xi in time stage i, riðuiÞ is the return function from stagei, fi�1ðxi�1Þ is return function per state variable xi�1 in timestage i� 1, ui is decision variable in time stage i and si aretransformations that express the relation between the out-put and input state variables for each system stage.

In this case of optimization of nominal electric power ofPV generator, where state variables are expressed by soilmoisture W ðiÞ in stage i, i.e. W ði�1Þ in stage i� 1, and deci-sion variables QdðiÞ by mean daily water quantity from thePV pumping system in stage i, these formulas can be writ-ten as follows:

fðiÞðW ðiÞÞ ¼MIN MAXQdðiÞ½P elðiÞðQdðiÞÞ; fði�1ÞðW ði�1ÞÞ�

� �ð20Þ

in conditions of state transformation Eq. (15), calculationof nominal electric power P elðiÞ by Eq. (12) which representsreturn, and with all mentioned constraints and defined timestage i, i.e.

W ðiÞ ¼ W ði�1Þ þ ð10ndQdðiÞ=AÞ þ ReðiÞ � ETrðiÞ � INFðiÞ

P elðiÞ ¼ 2:72½1�acðT cellðiÞ�T 0Þ�gMPIgNESðiÞ

fH 0T þ H STðiÞ

þ HDTðiÞ�HSTðiÞQmax

#QdðiÞ þ HFð#QdðiÞÞgQdðiÞ

0:6FC 6 W ðiÞ 6 FC

ReðiÞ � ETrðiÞ 6 0

0 6 QdðiÞ 6800

HTEðiÞ

QdðiÞ 6 QmaxtrðiÞ

W ði¼0Þ ¼ FC; i ¼ 1; 2; . . . ;N

1CCCCCCCCCCCCCCCA

ð21Þwhere ETrðiÞ is calculated by Eq. (9), and T cellðiÞ by Eq. (2).

State variable values in observed time stages W ðiÞ and inprevious time stages W ði�1Þ, and values of decision variableQdðiÞ, as well as returns per various stages P elðiÞ, are calcu-lated in the course of the process.

Optimal electric power of the PV generator, which isobtained as an output result, should meet the demands ofthe consumers throughout the whole observed period.

5.5. Computer programme of PVPS-Irrigation

In order to solve the subject problem, the computer pro-gramme PVPS-Irrigation has been developed in pro-gramme language Matlab, Version 6.1. (Matlab, 2002). Ituses hybrid simulation-optimization model where optimi-zation is conducted by dynamic programming, which canadjust well to problem characteristics. This model is sup-plemented by connection to smaller simulation models, inorder to define functional constraints. In computing stepsand combinations of state variable and decision variable,in accordance with dynamic programming method, theconnection with simulation models is performed in orderto test the validity of the combination. The flow diagramis shown in Fig. 4.

6. Verification of the optimization model

For the purpose of verification of the proposed optimi-zation model, to outline to what extent it describes the sys-tem, and for the purpose of obtaining the appropriateresearch results, two locations in corresponding climateregions of Osijek and Split have been selected, as patternsof continental and coastal Croatia.

Considering that all system parameters are variable, it isprimarily required to determine their reference values,upon which certain variables changed, so their effect on cal-culation of optimal value of nominal electric power of PVgenerator was observed, as well as their effect on otherparameters in the system.

6.1. Area in the Osijek region

6.1.1. Reference parameters

The following reference parameters for the area in theOsijek region have been selected:

(a) PV pumping system:
– nominal efficiency of PV generator: goc ¼ 10%,– nominal efficiency of motor–pump unit and inver-
ter: gMPI ¼ 35%.

(b) Climate region:
– location: Osijek,– observation period: decades IV1–VIII3.
(c) Borehole – irrigation well of typical values:
– static level: HST ¼ 7:5 m,– dynamic level: HDT ¼ 10 m,

LOOP OVER TIME

CONSTRAINT

LOOP OVER

STATE TRANSITION EQUATION

DOES IT FIT?

FEASIBLE VARIABLES COMBIN.

DOES IT FIT?

CALCULATION

DOES IT FIT?

RETURN CALCULATION

VARIABLES STORING

OBJECTIVE FUNCTION- MAX return search- MIN return search

STORING OF BEST SOLUTION

LAST ?

LAST STEP?

TRACE BACK

PRINT OUT

EXIT

INPUT

DATA INPUT

DOES IT FIT?

REPEATED DATA INPUT

CREATING FIELD STRUCTURES

1

2

3

4

5

6

7

8

9

10

12

13

14

15

16

17

18

19

20

22

23

24

25

26

11

CONSTRAINT R - ET 0−e(i) r(i)

Q Q t−d(i) max r(i)

W(i)

LOOP OVER Qd(i)

0.6 FC W F; W = FC− −(i) (i=0)

HTE(i)

0 Q 800 / H− −d(i)T E(i)

Pel(i)

Qd(i)

LAST ?W(i)

21

Fig. 4. Flow diagram of programme PVPS-Irrigation.


– head lift from ground surface: H 0T ¼ 1 m,– maximum discharge capacity:Qmax ¼ 20 m3=h.

(d) Irrigation area:
– area size: A ¼ 0:5 ha,– type of soil: medium mechanical composition
(medium clay),– field capacity: FC ¼ 35% or FC ¼ 105 mm,– depth of observed soil: Dr ¼ 30 cm,– optimal moisture limits: 60–100% FC.

(e) Crops:
– crops: tomatoes,– evapotranspiration calculation method: Turc’s
method,– crop coefficient Kc ¼ 1.

(f) Irrigation:
– irrigation method: trickle,– trickle irrigation effectiveness: gN ¼ 85%.
(g) Other elements
– cost effectiveness limit of PV system: 800 m4=day,– variable discretization stage: 10 ðm3Þ.
In the daily operating cycle there are no significantchanges in the aquifier and moisture of the active soil layerfrom which crops intake water. Hour values of aquifierdrawdown and borehole are connected solely to theinstalled pump capacity, not to the available water in theaquifier. Because of that, this work selects decade (10 days)for time period, which is in practice more realistic, in viewof water consumption.

6.1.2. Input data for the computer programme

A great number of input data in the computer pro-gramme PVPS-Irrigation, has conditioned the applicationof table in Excel programme, shown in Table 1. It containsdata on climate, data on PV pumping system, well andcrops.

The first three rows contain decade average daily valuesof solar irradiation Es, insolation tr and air temperature Ta.These data were obtained from the Meteorological andHydrological Service of Croatia for measurements fromthe year 1981–2000 (Hrabak-Tumpa, 2003). Rows 4 and5 contain cell temperatures T cell (that are recalculatedaccording to Eq. (2)) and average efficiency of the motorpump unit and inverter gMPI. Rows 6, 7 and 8 contain liftheight from the ground H 0T and static HST and dynamiclevel H DT of water in borehole. Row 9 contains total pre-cipitation R.

Apart from the data contained in the said table, it isrequired to directly input to the programme the data onirrigation period (decade range), area size, FC size, soildepth, limits of optimal soil moisture, evapotranspirationcalculation method, if needed (if it differs from Turc’smethod) or to input finished values, irrigation efficiencyand value of variable discretization.

Table 1Input data for area in Osijek

STAGE i 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

DECADE IV1 IV2 IV3 V1 V2 V3 VI1 VI2 VI3 VII1 VII2 VII3 VIII1 VIII2 VIII3

1 Es (kW h/m2d) 4.3 4.2 4.6 5.3 5.5 5.6 6.0 5.5 5.7 5.8 5.7 6.0 5.4 5.6 5.02 tr (h) 5.5 5.3 6.4 6.6 7.2 7.4 8.4 7.3 7.4 8.5 8.2 8.9 8.6 9.1 7.53 T a (�C) 11.1 10.1 13.5 15.4 17.6 17.6 19.9 19.1 20.4 21.6 21.4 22 22.3 21.9 19.64 T cell (�C) 22.7 21.4 26 29.7 32.4 32.7 36 33.9 35.9 37.3 36.8 38.2 36.9 37 33.25 gMPI (%) 35 35 35 35 35 35 35 35 35 35 35 35 35 35 356 H0T (m) 1 1 1 1 1 1 1 1 1 1 1 1 1 1 17 HST (m) 7.5 7.5 7.5 7.5 7.5 7.5 7.5 7.5 7.5 7.5 7.5 7.5 7.5 7.5 7.58 HDT (m) 10 10 10 10 10 10 10 10 10 10 10 10 10 10 109 R (mm/dc) 11.8 19.1 19.3 20 19.8 21.5 28.5 30.8 21.7 16.3 19.4 20 20.1 13.9 24.8


6.1.3. Optimization results for reference parametersThe optimization results of the PV pumping irrigation

system are shown in Table 2. The first column containsthe sequence of decades (stage i), in the second are refer-ence soil moisture values at the beginning of certain stages(e.g. W ði�1Þ is soil moisture at the beginning of stage i). Col-umns 3 and 4 contain mean decadal required water quan-tities for irrigation QPVðiÞ ðm3=decÞ and decadal meandaily values QdðiÞ ðm3=dayÞ. Column 5 contains effectiveprecipitation and column 6 contains real evapotranspira-tion (calculated by Turc’s method). Column 7 contains soilmoisture W ðiÞ ðm3Þ at the end of stage i. Water balanceequation for decade 10 is: 500 ¼ 525þ 44þ 47:2� 116:3.The last column contains optimal returns fðiÞðW ðiÞÞ, repre-senting optimal nominal electric power of PV generatorper various time stages. Therefore, the optimal nominalelectric power, obtained by MIN–MAX process, for refer-ence parameters in the Osijek area, that would satisfy theconsumer demands throughout the entire observed period,is P �el ¼ 254 W.

It can be deducted that the optimal power that would besatisfactory only in the 10th decade, would be 136 W.

Table 2Optimization model results for reference values of Osijek

1 2 3 4

Stage i (decade) W ði�1Þ (m3) QPVðiÞ (m3/dec) QdðiÞ (m3/day)

10 525 44 4.411 500 50 5.012 520 60 6.013 520 80 8.014 520 90 9.015 510 90 9.016 500 90 9.017 500 80 8.018 520 100 1019 510 100 1020 470 90 9.021 440 10 1022 410 90 9.023 380 90 9.024 320 70 7.0

However, in sequence of decades, optimal returns increase.From 18th to 24th decades return values are 254 W, repre-senting optimal return values (minimized maximum nomi-nal electric power), that should meet the consumers’demands throughout the entire observed period.

6.2. Location in the Split area

For the area in the Split location the same referenceparameters have been selected as for the area in the Osijeklocation, except climate elements and types of soil. There-fore, the soil of lighter mechanical composition is observed(sandy or lighter clay) of field capacity FC ¼ 28% (of vol-ume percentage) or FC ¼ 28� 0:3� 1000 mm ¼ 84 mm(which is 420 m3 at the location of 0.5 ha).

The optimization results are shown in Table 3. The opti-mal nominal electric power that would meet the demandsthroughout the entire observed period, for reference values,at the location in Split, is P �el ¼ 482 W:

It can be noted that for reference conditions, which areequal for areas in Split and Osijek, except climate valuesand type of soil, big differences in the required nominal

5 6 7 8

ReðiÞ (m3) ETrðiÞ (m3) W ðiÞ (m3) fðiÞ(W ðiÞ) (Watt)

47.2 116.3 500 13676.4 107.8 520 15777.2 138.6 520 17580.0 166.7 520 21179.2 184.0 510 23586.0 187.0 500 235

114.0 209.4 500 235123.2 190.6 520 23586.8 204.5 510 25465.2 212.1 470 25477.6 207.3 440 25480.0 219.3 410 25480.4 201.4 380 25455.6 205.9 320 25499.2 178.1 320 254

Table 3Optimization model results for reference values of Split

1 2 3 4 5 6 7 8

Stage i or decade W ði�1Þ (m3) QPVðiÞ (m3/dec) QdðiÞ (m3/day) ReðiÞ (m3) ETrðiÞ (m3) W ðiÞ (m3) f �ðiÞ(W ðiÞ) (Watt)

10 420 73 7.3 58.0 140.51 410 20611 410 0 0 – – 420 –12 420 80 8 83.6 160.07 420 21513 420 100 10 84.0 193.79 420 24114 420 110 11 88.4 207.01 420 26515 420 140 14 83.6 225.69 420 32516 420 170 17 78.4 254.19 420 37617 420 190 19 51.2 251.19 420 43718 420 210 21 48.8 267.92 420 47219 420 210 21 22.4 272.33 390 48220 390 200 20 26.0 268.92 350 48221 350 210 21 22.4 277.09 310 48222 310 190 19 23.2 249.23 280 48223 280 180 18 28.4 243.57 250 48224 250 100 10 109.2 212.83 250 482


electric power of the PV generator are obtained. This canbe explained by the following facts:

• precipitation at the Split location is lower in theobserved period than in Osijek (in the warmest summermonths effective precipitation in Split is only 4–5 mm),

• the values of real evapotranspiration in Split exceed sig-nificantly those in Osijek, due to very big quantities ofsolar iradiation (in the most critical decades they canreach 54–55 mm),

• due to big evapotranspiration and low precipitation, bigwater deficits in soil occur, meaning that relatively bigquantities of water are needed for irrigation, soil oflower capacity for water, as is most frequently the soilat the Split location, cause greater nominal electricpower of the PV generator.

6.3. Variation of typical elements in the system

By variation of different elements in the system it can beconcluded that the model describes the subject system verywell, and in that sense the following has been establishedand confirmed:

• In relation to reference parameters, where variable dis-cretization is 10 m3, by selecting the stage of 1 m3 verysmall increases of accuracy are obtained (relative erroris only 0.3%), but calculation time on standard comput-ers is significantly longer. Due to this, great discretiza-tions don’t have significant practical value.

• By decreasing the static level of water in borehole, opti-mal nominal power of the PV pumping system increases,and by increase of borehole abundance (intake) itdecreases.

• The increase of irrigated area results in increase of cropwater requirements, which will result in increase of opti-mal nominal electric power of the PV generator. How-

ever, in that way it has been established that thedependence of optimal nominal electric power is not alinear function of the irrigated area.

• Soil of lower capacity requires greater nominal electricpower, because the PV pumping system must compen-sate greater water deficits that occur in them.

• By increase of soil depth where crops most activelyintake water, water resources in it increase, thereforethe required nominal electric power of the PV generatordecreases.

• By decreasing the bottom limit of soil optimal moisture,the required nominal electric power of the PV generatordecreases. This is due to greater intake of water from thesoil, so less water quantities are expected from the PVpumping system in the observed period.

• As the amount of potential evapotranspiration has a sig-nificant effect on the required water quantity for irriga-tion, and therefore on the nominal electric power ofthe PV generator, it is best that the agriculture expertinputs its calculation mode into the optimization modeldirectly, as well as other elements regarding irrigation(crop coefficient, depth of the most active part of therooting system, etc.).

• In addition to reference parameters of an area, the big-gest areas that can be cost-effectively irrigated by the PVpumping system, directly depend on volume-head prod-uct, expressed in value of 800 m4/day, where static waterlevel in borehole can be observed as a dominant element.However, the output results of the optimization modelare interesting: the biggest areas that can be cost-effec-tively irrigated at locations of lower insolation (Osijek)are significantly bigger than those at locations of higherinsolation (Split). This is due to the fact that at locationsof higher insolation the PV pumping irrigation systemmust compensate relatively bigger water requirementsthat occur due to bigger evapotranspirations, lower pre-cipitation and smaller soil capacities for water.


7. Comparison of results of the usual and new optimization

method

Based on the data on required water for irrigation,obtained from the agriculture expert and on identicalremaining data (climate data, method of evapotranspira-tion calculation, total head, total effectiveness of the pump-ing unit, etc.), the usual optimising method by Eq. (1) givesthe following results:

• The critical period for the Osijek location area is decadeVIII2 (the second decade in August), with decade aver-age daily water requirements of 15.6 m3/day, which,with total head of 9.35 m gives 0.392 kW h of hydraulicenergy, so nominal electric power is P el ¼ 418 W.

• The critical period for the Split location area is decadeVII3 (the third decade in July), with decade averagedaily water requirements of 26.2 m3/day, which, withtotal head of 9.35 m gives 0.7 kW h of hydraulic energy,so nominal electric power is P el ¼ 572 W:

By comparison of these data with the ones obtained insubchapter 6.1.3. (Osijek: P �el ¼ 254 WÞ and 6.2. (Split:P �el ¼ 482 WÞ, it can be observed that optimal values ofthe nominal electric power of the PV generator, obtainedby the new optimizing method, are lower than thoseobtained by the usual sizing method.

8. Comments regarding the optimization model

The accuracy of the obtained results will depend mostlyon the input data quality, where climate data are of partic-ular importance.

As the produced electric energy from the PV generator isgreater on the inclined array plane, for the observed periodis necessary to find the optimal angle of inclination of thearray, and reduce the obtained optimal nominal electricpower proportionally to greater solar irradiation on theinclined plane.

However, although is not the subject of this paper, dueto insufficient relevant analysis, in the future the cost-effec-tiveness of the PV pumping system for irrigation shouldalso be calculated in view of connection to power network.Up to now the PV pumping systems were applied mostly atsmall geographic latitudes, where there was practically nopower network. However, in the last years they have beenmore widely used within greater geographic latitudes (loca-tions in Europe, even in Finland; Ecofys Energy and Envi-ronment, 2001) with developed power network.

In the described model new elements and system con-straints can be changed and added, which indicates itsfavourable adjustability to characteristics of the problem.It is possible, e.g. to add the constraint of competitivenessof the PV pumping system in relation to the wind-genera-tor, supply from power network or other available energysource, and analyze the application of the PV pumping sys-tem in irrigation by infiltration method, take into account

other well types (river, lake or sink hole), various crops,water storage, different calculation of potential evapotrans-piration and/or other elements in the system, etc.

As the proposed model for dimensioning the optimal PVpumping system for irrigation functions well in the subjectconditions, good results are expected for even more com-plex conditions, where more variables could be introduced,as well as stochastic process elements and optimizing byneural dynamic programming. Also, with the same objec-tive function (that the PV system meets the consumers’demands in the observed period) and equal methods forproblem solving (dynamic programming), with corre-sponding improvements and adjustments to specific char-acteristics of the problem, it is possible to solve theproblem of optimizing all other stand-alone PV systemswith any form of energy storage (electric energy in batteriesor its transformation into hydrogen), as well as optimizingof photovoltaic power plant in operation with hydro elec-tric power plant.

9. Conclusion

The performed observations show that the previousapproach to sizing the PV pumping irrigation system,which separately views the demands for hydraulic energyand possibilities of its production by PV pumping systemfrom the available solar energy, is basically non-systematicand static, therefore is not optimal.

Because of that, this paper presents a systematicapproach to the problem, taking into account all relevantelements, from PV pumping system, boreholes, local cli-mate, soil, crops, to irrigation system.

The result of such approach is a new mathematicalmodel for optimal sizing of the nominal electric power ofthe PV generator.

As the possibilities of meeting the hydraulic energydemands at the output of the PV pumping system dependon solar energy in the observed stage, this work optimizestheir relation, which basically defines the electric power ofthe PV generator. In that sense a complex objective func-tion of minimization the maximum electric power of thePV generator, has been defined.

By verification of this model on two locations in Croa-tia, it has been established that it describes the subject sys-tem very well, where a number of useful research resultswere obtained.

The work proves that electrical power of the PV gener-ator, obtained by the new optimization method, is rela-tively smaller than that obtained by the usual method.

As the proposed model for dimensioning the optimal PVpumping system for irrigation functions well in the subjectconditions, good results are expected for even more com-plex conditions, where more variables could be introduced,as well as stochastic process elements and optimizing byneural dynamic programming. Also, with the same objec-tive function (that the PV system meets the consumers’demands in the observed period) and equal methods for


problem solving (dynamic programming), with corre-sponding improvements and adjustments to specific char-acteristics of the problem, it is possible to solve theproblem of optimizing all other stand-alone PV systemswith any form of energy storage (electric energy in batteriesor its transformation into hydrogen etc.), as well as opti-mizing of photovoltaic power plant in operation withhydro electric power plant.

References

AVICENNE Programme, Papers for distribution, 1997. Prepared by ITPower as part of the Concerted Action for the Testing and Costreduction of PV Water Pumping Systems, Contract AVI-CT094-0004,IT Power Ref. 95501.

Bahaj, A., Mohammed, A., 1994. Sizing of a photovoltaic pumping systemand its storage capacity to meet crop water requirements in remoteareas. In: Proceedings of 12th European Photovoiltaic Solar EnergyConference, Amsterdam, pp. 1969–1972.

Barlow, R., McNelis, B., Derrick, A., 1993. Solar Pumping, AnIntroduction and Update on the Technology, Performance, Costs,and Economics. Intermediate Technology Publications and The WorldBank, Washington, DC.

Doorenbos, J., 1975. Crop Water Requirements. Food and AgricultureOrganization of the United Nations (FAO), Rome.

Ecofys Energy and Environment, 2001. Autonomous PV-generators inAgriculture and Water Managment in The Netherlands, Germany,

Spain, France and Finland. Final Report. Project number SE/218/95NL/DE/ES. Postbus 8408, NL-3503 RK Utrecht, Netherland.

Fontane, D.G., Margeta, J., 1988. Introduction to System Engineering inDesigning and Management of Reservoir, Faculty of Civil Engineer-ing. University of Split, Croatia (on Croatian).

Glasnovic, Z., Sesartic, M., Fancovic, M., 1991. Optimization photovol-taicly-powered water pumping system for irrigation. In: Proceedings ofSolar World Congress of The International Solar Energy Society,August 17–21, Denver, Colorado.

Harboe, R.C., 1988. Complex Objective Functions in Dynamic Program-ming, Lecture #4. Denver, Colorado (unpublished).

Hrabak-Tumpa, G., 2003. Climate Data for Osijek and Split. Meteoro-logical and Hydrological Service of Croatia.

Kenna, J., Gillett, B., 1985. Solar Water Pumping – Handbook.Intermediate Technology Publications.

Kos, Z., 1987. Hydrotechnical Melioration of Soil – Irrigation, Skolskaknjiga, Zagreb, Croatia (on Croatian).

Matlab, 2002. The Language of Tehnical Computing, Version 6.5, TheMathworks Inc.

Markvart, T., Castaner, L., 2003. Practical Handbook of Photovoltaics:Fundamentals and Applications. Elsevier Science Ltd., Oxford, UK.

McNelis, B., Derrick, A., Starr, M., 1989. Water Pumping, RegionalWorkshop on Photovoltaic Systems. UNDP, Belgrade.

Narvarte, L., Lorenzo, E., Caamano, E., 2000. PV pumping analyticaldesign and characteristics of boreholes. Solar Energy 68 (1), 49–56.

Nemhauser, G.L., 1966. Introduction to Dynamic Programming. JohnWiley and Sons Inc., New York.

Tomic, F., 1988. Irrigation, Croatian Association of Agriculture Engineersand Technicians and Faculty of Agriculture Zagreb (on Croatian).

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