Real-time analytical model for predicting the cell temperature modules of photovoltaic water pumping systems

Sustainable Energy Technologies and Assessments 6 (2014) 93–104

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Original Research Article

Real-time analytical model for predicting the cell temperature modulesof photovoltaic water pumping systems

http://dx.doi.org/10.1016/j.seta.2014.01.0092213-1388/� 2014 Elsevier Ltd. All rights reserved.

⇑ Corresponding author. Tel.: +216 97862751; fax: +216 75600632.E-mail address: [email protected] (A. Mahjoubi).

Ammar Mahjoubi ⇑, Ridha Fethi Mechlouch, Belgacem Mahdhaoui, Ammar Ben BrahimApplied Thermodynamic Research Unit (UR11ES80), National School of Engineering of Gabes, Gabes University, Avenue of Omar Ibn Elkhattab, 6029 Gabes, Tunisia

a r t i c l e i n f o a b s t r a c t

Article history:Received 11 June 2013Revised 6 January 2014Accepted 24 January 2014

Keywords:Temperature of a photovoltaic modulePredicted modelsPV water pumping systems efficiencySimulationDesert of southern Tunisia

In this paper an energy balance model to predict the cell temperature modules of photovoltaic waterpumping system was developed. Results from this model are experimentally validated and also com-pared to results from published cell temperature models. The models were used to predict real-time per-formance from a PV water pumping systems in the desert of Medenine, south of Tunisia using 60-minintervals of measured performance data during one complete year. Statistical analysis of the predictedresults and measured data highlight possible sources of errors and the limitations and/or adequacy ofexisting models, to describe the temperature and efficiency of PV-cells and consequently, the accuracyof performance of PV water pumping systems prediction models.

� 2014 Elsevier Ltd. All rights reserved.

Introduction losses, temperature of reference, and moto-pump and inverter

A photovoltaic water pumping system is a type of installationwhere three major components are used: the PV-generator (com-prising a number of PV-modules connected in series or parallelon a mounting structure); the DC or AC inverter and a moto-pump[1,2]. Inverters play a key role in energy efficiency and reliabilitysince they operate the PV-array at the Maximum Power Point(MPP). Moreover, inverters convert the DC power generated byPV-modules into alternating current (AC) of the desired voltageand frequency (e.g., 230 V/50 Hz). Installations of this type do notinclude batteries.

During the operation of the PV cell, only around 15% of solarradiation is converted to electricity with the rest converted to heat.The electrical efficiency will decrease when the operating temper-ature of the PV module increases.

Many research efforts have focused on the development ofempirical models that are able to predict photovoltaic waterpumping system performance for any climatic condition using var-ious parameters. These parameters include global solar radiation,ambient temperature, temperature of a PV module, latitude, longi-tude, albedo, orientation and inclination of PV array, nominal PVmodule efficiency, NOCT, PV array area, PV temperature coefficient,miscellaneous power conditioning losses, miscellaneous PV array

efficiency.In Algeria, Hamidat and Benyoucef [3] proposed two mathemat-

ical models that directly link the operating electric power to thepump’s output water flow rate. The first is a simple model that al-lows the calculation of water flow only with the experimentationtotal heights. The second model is more general and allows the cal-culation of water flow for any total height of pumping. These mod-els are based principally on the analysis of the experimentationresults for different types of motor-pump set. Ghoneim [4] adopteda computer simulation program to assess the performance of PVpumping systems in the Kuwait climate, this program was com-pared with the manufacturers PVCAD program. The PVCAD pro-gram is used by Solar Jack Company (Solar Jack Solar PumpingProducts) to simulate the performance of PV pumping systems.The program consists of component subroutines for each systemcomponent.

This program takes into account the effect of solar source vari-ations on the system performance.

These models typically require the temperature of the photo-voltaic component of the module, known as the cell, as an input.It is well documented that the cell temperature affects module per-formance, so accurately characterizing the cell’s operating temper-ature is critical to achieving accurate estimates of systemperformance. Some investigations stated that the electrical effi-ciency of PV cells decreases with temperature increase.

Many cell temperature models are available in literature. De-tails of a sampling of these models are included in the next section.

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Nomenclature

Ee energy delivered by the PV array (Wh)Ehydr hourly hydraulic energy demand (Wh)Et equation of timef fraction diffuseg acceleration of gravity (m s�2)H hourly global solar irradiation on a horizontal surface

(Wh/m2)H0 extraterrestrial solar irradiance (Wh/m2)Hd diffuse solar radiation (Wh/m2)Hb direct radiation (Wh/m2)HGreenwich world time (h)HMT total head (m)Ht hourly irradiance in the plane of the PV array (Wh/m2)hcon convective heat transfer coefficient (Wm�2 K�1)hrad radiative heat transfer coefficient (Wm�2 K�1)KT hourly clearness indexQ hourly water flow rate (m3/h)_QH absorbed solar radiation (W)_Q conv thermal losses by convection (W)_Q rad thermal losses by radiation (W)

q quantum of the day (i.e., q = 1 for January 1 and q = 365for 31 December)

NOCT nominal operating cell temperature (�C)Rb ratio of beam radiation on the PV array to that on the

horizontalRinv inverter efficiency (%)Rp pump system efficiency (%)RMSE root mean square errorS area of the array (m2)TSV real solar time (h)t hour of the day (h)

Ta hourly ambient temperature (�C)Tc average module temperature (�C)Tr reference temperature (= 25 �C)Tsky effective temperature of the sky (�C)Up overall heat transfer coefficient (Wm�2 K�1)Vw wind speed (m s�1)s transmissivitya absorptivitya0 ground albedob slope of the PV array (�)bopt optimum tilt angle (�)q density of water (kg/m3)u latitude (�)k longitude (�)kc power conditioning losses (%)kp miscellaneous PV array losses (%)gf factor accounting for friction losses in the piping (m)gp array average efficiency (%)gr PV module efficiency (%)bp temperature coefficient for module efficiency (�C)d solar declination (�)x sunrise hour angle (�)x0 sunset hour angle (�)h incidence angle of beam irradiance on the array (�)c orientation of the array (�)hz zenith angle of the sun (�)e emissivityr Stefane Boltzman’s constant

94 A. Mahjoubi et al. / Sustainable Energy Technologies and Assessments 6 (2014) 93–104

These models can be generally grouped into three categories:empirical models, semi-empirical models, and theoretical models.

In this paper, a general energy balance model was developed topredict the cell temperature of a PV module. This model was sim-plified and validated using field measurements into a steady statemodel.

The objective of this paper therefore, is, to develop a validatedreal-time mathematical model that predicts the cell temperatureof a PV module. The model can be used to generate the perfor-mance of PV water pumping systems.

Modeling of the performances of a PV water pumping systems

PV-cell temperature

Many PV cell temperature (Tc) models are available in literature.Details of a sampling of these models are included below. Thesemodels can be generally grouped into three categories: empiricalmodels, semi-empirical models, and theoretical models.

Empirical models are developed purely from experimental data.Empirical models have the possibility to be very accurate sincethey are developed from actual data and do not need to makeassumptions. The downside is the fact that, since the model re-quires measured data, it is difficult to predict the performance ofa module before installation.

King et al. [5] have developed an empirical model for threemounting types that is dependent on irradiance, wind speed, andambient temperature. Shrestha et al. [6] have developed a cell tem-perature model for the interior modules in a gap mounted rooftoparray.

Theoretical models apply established heat transfer correlationsto predict the temperature of the module. As opposed to empiricalmodels, theoretical models do not require measured performance.Fuentes [7] developed an extensive transient model that is adapt-able to different mounting using equations manipulating the re-ported Nominal Operating Cell Temperature (NOCT). Jones andUnderwood [8] developed a transient model for a rack mountedmodule, while Davis et al. [9] developed a transient model for abuilding integrated module. Duffie and Beckman [10] presented asimplified steady state model that adapts to the mounting of themodule using the reported NOCT.

Semi-empirical models are defined as theoretical models thatare slightly modified by information from experimental data. Forexample, Del Cueto [11] empirically calculates coefficients to im-prove a theoretical steady state model. Krauter et al. [12] devel-oped a semi-empirical transient model. Skoplaki et al. [13]developed a simplified steady state model and modified the tem-perature for mounting conditions based on an empirical mountingcoefficient.

In addition to previous models, Skoplaki and Palyvos [14] listadditional cell temperature models. They include explicit and im-plicit models.

The explicit correlation models express the PV-cell temperature(Tc) as a function of ambient temperature, solar radiation, windspeed and other system parameters ignoring heat exchangedynamics between the module and its environment. Several equa-tions for Tc found in the literature are listed in Table 1.

The implicit correlations, i.e., involve variables such as cell effi-ciency or heat transfer coefficients, which themselves depend onTc. Some implicit correlations are shown in Table 2.

Table 1Explicit correlations of PV-cell temperature.

Models N�

Tc = Ta + [(a � g)GT + (a + bTa)]/(17.8 + 2.1Vw) (E.1)Tc = Ta + aGT(1 + bTa)(1 � cVw)(1 � 1.053gc) (E.2)

Tc ¼ Ta þ GTsaUL

� �1� gc

sa� �� (E.3)

Tc ¼ Ta þ GTðsa�gcÞ

UL(E.4)

Tc ¼ Ta þ GTC1þC1 Vw

ð1� qÞð1� gÞ � r eT4c�T4

skyGT

�(E.5)

Tc ¼ Ta þ GTGNOCT

ðTc;NOCT � TaÞ 1� gcsa

� � (E.6)

Tc ¼ Ta þ GTGNOCT

ðTc;NOCT � Ta;NOCTÞ 1� gcsa

� � (E.7)

Tc ¼ Ta þ GTa

UL

� �1� gc

a� �� (E.8)

Tc ¼ Ta þ GTGNOCT

� �UL;NOTC

UL

� �ðTNOCT � Ta;NOCTÞ 1� gc

sa� �� (E.9)

Tc ¼ Ta þ 9:55:7þ3:8Vw

� �GT

GNOCT

� �ðTNOCT � Ta;NOCTÞ 1� gc

sa� �� (E.10)

Tbs¼hp1ðsaÞeff GTþUtb Taþhb Tw

Utbþhb

(E.11)

Tc¼s½acpþaTð1�bcÞ�GT�gcbcGTþUtTaþUtTbsUtþUTa

(E.12)

T 0c¼s½acpþaTð1�bcÞ�GTþðUtþUTaÞTaUtþUTa

(E.13)

Table 2Implicit correlations of PV-cell temperature.

Models N�

Tc = 3.12 + 0.025GT + 0.899Ta � 1.30Vw) (E.14)Tc = 3.81 + 0.0282GT + 1.31Ta � 1.65Vw (E.15)

Tc ¼ ðU0=FpÞðm0=FpÞðTaþTcOÞ2ðm0=FpÞ

h i½ð1þ YÞ1=2 � 1� (E.16)

Tc = Ta + 0.028GT � 1.0 (E.17)Tc = Ta + 0.035GT (E.18)Tc = Ta + aGT(1 + bTa)(1 � cVw) (E.19)

Tc ¼Taþ½acell�gref ð1þbTref Þ�

GTh

� �1�gref b

GTh

� � (E.20)

Tc ¼½sa�gref �GTþTa hcfþ2recsIR ð1þcos bÞTskyT2

aþhcbþ4rT3a Fe Fb½ �

hcfþ2recsIR T3a ð1þcos bÞþhcbþ4rT3

a Fe Fb

(E.21)

Tc = Ta + 0.0155GT + 0.7 (E.22)Tc = 30 + 0.0175(GT � 150) + 1.14(Ta � 25) (E.23)

Tc ¼ Ta þ GTGNOCT

0:0712V2w � 2:411Vw þ 32:96

h i(E.24)

Tc ¼ Ta þ GTGNOCT

� �ðTc;NOCT � Ta;NOCTÞ ¼ Ta þ GT

800 ðNOCT� 20Þ (E.25)

Tc ¼ Ta þ ð219þ 832KtÞ NOCT�20800

(E.26)

Tc = 30 + 0.0175(GT � 150) + 1.14(Ta � 25) (E.27)

Tc ¼ T i þ KgTGT ;K ¼ 1�FRFR UL

(E.28)

Tc ¼ Ta þ GTGTref½T1ebVw þ T2 þ DTref � (E.29)

Tc ¼ Ta þ GT eðaþbVF Þ þ DTrefGTref

h i(E.30)

Tc = Ta + 0.031GT (E.31)Tc = Ta + 0.031GT � 0.058 (E.32)

Fig. 1. Simple schematic of the thermal processes in PV module.

A. Mahjoubi et al. / Sustainable Energy Technologies and Assessments 6 (2014) 93–104 95

Steady-state analysis

A general energy balance model was developed to predict thecell temperature of PV module. This model was simplified and val-idated using field measurements into a steady state model thatcreates a thermal resistance network between the cell and the endsof the module. Fig. 1 gives a schematic of the thermal energy ex-changes between PV module and the environment, which involvesuch variables and parameters.

The equation for the PV-cell temperature operating under stea-dy state is derived assuming that the incident energy on a solar cellis equal to the electrical energy output of the cell plus the sum ofthe energy losses due to convection and radiation. The resultingenergy balance equation is given as [15,16]:

_Q H ¼ Ee þ _Q rad þ _Q conv ð1Þ

where:

_Q H ¼ saHtA ð2Þ

where: a: absorptivity and s: transmissivity.Ee is the electrical energy produced by the module, where

Ee ¼ gpHtA ð3Þ

_Q ray þ _Q conv ¼ UpAðTc � TaÞ ð4Þ

Up = hrad + hconv: Thermal losses by radiation and thermal losses byconvection.where:

hrad ¼ erðTsky þ TcÞðT2sky þ T2

c Þ ð5Þ

hconv ¼ 5:82þ 4:07Vw ð6Þ

where: e and r are, respectively emissivity and Stefane Boltzman’sconstant, Vw is the wind speed.

The effective temperature of the sky (Tsky) is calculated from thefollowing empirical relation:

Tsky ¼ Ta � 6 ð7Þ

Substituting the relevant terms in Eq. (1) results in Eq. (8) given as:

saHt ¼ gpHt þ UpðTc � TaÞ ð8Þ

Finally, the PV-module temperature is given as:

Tc ¼Ht

erðTsky þ TcÞðT2sky þ T2

c Þ þ ð2:8þ 3VwÞ1�

gp

sa

� �þ Ta ð9Þ

Description of the algorithm

The performance of PV water pumping system is simulatedusing MATLAB 7 software with an hourly time step. The programis based on the proposed models for calculation of hourly waterflow rate. This simulation program uses the hourly global solarradiation, the hourly ambient temperature and the hourly windspeed as the input, moreover the characteristics of region (Lati-tude, Longitude, ground albedo) and characteristics of photovoltaicwater pumping system (orientation, inclination, NOCT, PV arrayarea, temperature of reference, HMT,...).

The first step consists of the determination of the various astro-nomical parameters, and more precisely the exact position of thesun at any moment of the year.

The first parameter to be determined is the declination; it is theangular position of the sun at solar noon, with respect to the plane


of the equator. Its value in degrees is given by Cooper’s equation[17]:

d ¼ �23:45Sin360ð284þ qÞ

365:25

�ð10Þ

where q is the day of year (i.e., q = 1 for January 1, q = 32 for Febru-ary 1, etc.). Declination varies between �23.45� on December 21and +23.45� on June 21.

The stage which comes then is the calculation of the true solartime (TSV):

TSV ¼ HGreenwich þk

15þ Et ð11Þ

where k is the longitude of the site and Et is the equation of time;

Et ¼ ½9:87Sinð2BÞ � 7:5CosB� 1:5SinB� 160

ð12Þ

B ¼ 360ðq� 81Þ365

�ð13Þ

The solar hour angle is the angular displacement of the sun east orwest of the local meridian; morning negative, afternoon positive.

x ¼ 15ðTSV� 12Þ p180

ð14Þ

The sunset hour angle x0 is the solar hour angle corresponding tothe time when the sun sets.

It is given by the following equation:

x0 ¼ arc cosð�tanu � tandÞ ð15Þ

where d is the declination, calculated through Eq. (10), and u is thelatitude of the site, specified by the user.

The zenith angle of the sun hz is calculated through thisequation:

Coshz ¼ Sind � Sinuþ Cosd � Cosu � Cosx ð16Þ

The hourly extraterrestrial radiation on a horizontal surface, H0, canbe computed for hour from the following equation [17]:

H0 ¼24p� I0 1þ 0:033� cos

q� 360365

� ��

Cosu � Cosd � Sinx0 þ2p �x0

360� Sinu � Sind

�ð17Þ

The ratio of global solar radiation at the surface of the earth toextraterrestrial radiation is called the clearness index. Thus thehourly clearness index, KT, is defined as:

KT ¼HH0

ð18Þ

The fraction diffuse is the ratio of the diffuse solar radiation to theglobal solar radiation [18]:

fd ¼Hd

Hð19Þ

fd ¼ 1� 0:09KT for 0 � KT � 0:22 ð20Þ

fd ¼ 0:9511� 0:1604KT þ 4:388K2T � 16:638K3

T

þ 12:336K4T for 0:22

< KT � 0:8 ð21Þ

fd ¼ 0:1 for KT > 0:8 ð22Þ

where the diffuse solar radiation Hd, is defined as:

Hd ¼ fd � H ð23Þ

After calculation of diffuse radiation, the direct radiation is obtainedjust by difference with global solar radiation which can be ex-pressed as:

Hb ¼ H � Hd ð24Þ

Calculation of hourly irradiance in the plane of the PV array, Ht [17]:

Ht ¼ Hb � Rb þ Hd �1þ cos b

2

� �þ H � a � 1� cos b

2

� �ð25Þ

where a represents the diffuse reflectance of the ground (also calledground albedo is set to 0.2) and b represents the slope of the PV ar-ray (inclination).

Rb is the ratio of beam radiation on the PV array to that on thehorizontal, which can be expressed as:

Rb ¼cos hcos hz

ð26Þ

where h is the incidence angle of beam irradiance on the array and cis the orientation of the PV array.

Cosh ¼ Sind � Sinu � Cosb� Cosu � Sind � Sinb � Coscþ Cosd

� Cosu � Cosb � Cosxþ Sinu � Cosd � Sinb � Cosc

� Cosxþ Cosd � Sinb � Sinc � Sinx ð27Þ

The energy delivered by the PV array, Ee, is simply [19]:

Ee ¼ SgpHtð1� kpÞð1� kcÞ ð28Þ

where S is the area of the array. It has to be reduced by miscella-neous PV array losses kp and other power conditioning losses kc.

The array is characterized by its average efficiency, gp, which isa function of average module temperature Tc:

gp ¼ gr½1� bpðTc � TrÞ� ð29Þ

where gr is the PV module efficiency at reference temperatureTr (= 25 �C), and bp is the temperature coefficient for moduleefficiency.

Tc is related to the hourly ambient temperature Ta through for-mula (9):

Tc ¼Ht

erðTsky þ TcÞðT2sky þ T2

c Þ þ ð2:8þ 3VwÞ1�

gp

sa

� �þ Ta

where NOCT is the Nominal Operating Cell Temperature, KT thehourly clearness index and bopt is the optimum tilt angle and b isthe actual tilt angle.

The water pumping model is based on the simple equations [20].The hourly hydraulic energy demand Ehydr (Wh), corresponding tolifting water to a height h (m) with a hourly volume Q (m3/h) is:

Ehydr ¼ q � g � Qð1þ gf Þ ¼ q � g � Q �HMT ð30Þ

where g is the acceleration of gravity (9.81 m s�2), q is the densityof water (1000 kg m�3), and gf is a factor accounting for frictionlosses in the piping. This hydraulic energy translates into an electri-cal energy requirement Ee:

Ee ¼Ehydr

Rp¼ q � g � Q �HMT

Rpð31Þ

where Rp is the pump system efficiency. If the pump is AC, thisequation has to be modified to take into account the inverter effi-ciency Rinv:

Ee ¼q � g � Q � HMT

Rp � Rinvð32Þ

Equalizing the two Eqs. (28) and (32),

Ee ¼q � g � Q � HMT

Rp � Rc¼ SgpHtð1� kpÞð1� kcÞ ð33Þ

Fig. 2. Geographic localization of two solar water pumping stations Bel Khchab and Om Chraket.


The hourly volume pumping (flow rate) Q (m3/h), which can be ex-pressed as:

Q ¼SgpHtð1� kpÞð1� kcÞ � Rp � Rc

q � g � HMTð34Þ

Description of the PV water pumping systems

In this work, two solar water pumping stations are selected.These stations are located in the isolated site in Medenine city(south of Tunisia); with a Saharan climate. The first station named

Fig. 3. Photographs of station Bel Khchab (a) and station Om Chraket (b).

Table 3Characteristics of the PV pumping stations.

Station Bel Khchab Om Chraket

Modules Monocrystal (100 Wc,24 V)

Monocrystal (65 Wc,12 V)

PV array area (m2) 85 21Inclination angle of the

modules (�)35 35

Nominal PV moduleefficiency (%)

14 14

Miscellaneous PV arraylosses (%)

5 5

Miscell. power conditioninglosses (%)

2 2

Moto-Pump Grundfos SP 8A-44(7.5 kWh)

Grundfos SP 3A-10(1.2 kWh)

Pump system efficiency (%) 45 45Inverter Telemecanical DC/AC Grundfos SA-1500

DC/ACInverter efficiency (%) 90 90HMT (m) 100 40Capacity storage tank (m3) 500 20PV temperature coefficient

(%/�C)0.4 0.4

NOCT (�C) 47 47Temperature of reference

(�C)25 25

Fig. 4. Schema of data a


Bel Khchab (33.10 N, 10.05 E) and the second station named OmChraket (32.55 N, 11.33 E). Both stations are located at isolatedsites with a Saharian climate near the city of Medenine in SouthernTunisia. Fig. 2 shows the geographic localization of the twostations.

The two systems are used for potable water. A PV water pump-ing system consists of a PV array, inverter, submersible pump, stor-age tank, and auxiliary system of measuring devices and weathermonitoring sensors, the two station work without storagebatteries.

The Photographs and characteristics of the PV pumping stationsare shown, respectively in Fig. 3 and Table 3.

The modules in two stations are mounted on a supportingstructure so that the surface azimuth angle and the inclination an-gle of the modules are 0 and 35�, respectively.

The proposed system consists of a set of sensors (can record upto 15 different sensors) for measuring both meteorological (e.g., so-lar irradiance, temperature, etc.) and photovoltaic parameters(water flow rate, photovoltaic voltage and current, etc.).

The experimental data is registered in a data logger locatedabout 150 km from the CRDA of Medenine. The data acquisitionsystem used in these stations is designed and made by us[21,22]. The information from the data logger will be transmittedto the PC at anytime via GSM modem, where they are processedusing the data acquisition software. The data acquisition software

cquisition system.


is used to further process, display and store the collected data inthe PC disk. Fig. 4 presents a schematic of our data acquisitionsystem.

The pumped water flow rate is the principal measuring param-eter, the ambient temperature, global irradiation on an inclinedand horizontal surface (Ht, H), voltage and current of photovoltaicfield (V, I) are also measured in order to construct a complete database of the site and the system. The following transducers havebeen used for the various measurements. Turbine flow meters havebeen used for the flow rate measurements. The flow meters workin the range of 0–10 m3/h with an accuracy of ±0.05%. The flow

Fig. 5. Variation of mean monthly irradiance measured i

Fig. 6. Variation of mean monthly irradiance measured in th

meters give 4:20 mA analog signal calibrated to the measured flowrate. Silicon pyranometers have been used to measure the solarradiation in the horizontal plane and in the plane of the PV arrays.The accuracy of the pyranometers is 10 W/m2 or ±5%. Ambienttemperature has been measured using an RTD sensor shieldedfrom the sun. The uncertainty in the measurement of temperatureis ±0.5 �C. The DC voltage and current generated by the PV arraysand the AC voltage and current drawn by the pump motors havebeen measured using Hall Effect transducers of appropriate ranges.The accuracy of the voltage and current transmitters is ±0.25% offull scale.

n horizontal surface of the year 2010 for Medenine.

e plane of the PV array of the year 2010 for Medenine.

Table 4Root mean square error (RMSE) for PV-cell temperature predictions.

PV-cell temperature models RMSE (%)

Present model/measured 2.6Model of Durisch et al./measured 5.4Model of Kou et al./measured 5.7


The experimental data registered in a data logger during onecomplete year (2010) at a time interval of one hour and transmit-ted from remote photovoltaic water pumping systems in the desertof Tunisia to the PC are analyzed.

The mean monthly global solar radiations measured in thehorizontal surface of Medenine city are plotted in Fig. 5. From thisfigure, we find that the maximum monthly mean of globalradiation occurs in summer 289 kWh/m2 per month and in spring261 kWh/m2 per month. In autumn and winter we find,respectively 197 and 153 kWh/m2 per month.

The mean monthly irradiance measured in the plane of the PVarray of Medenine city is plotted in Fig. 6. We find that the maxi-mum of global solar radiation is high throughout the summermonths reaching 281 kWh/m2 per month, in spring month’s274 kWh/m2 per month. In autumn and winter we find, respec-tively 208 and 171 kWh/m2 per month. Another factor which isprejudicial to the good behavior and the efficiency of the PV arrayis the ambient temperature. We can see in Fig. 7, the monthly aver-age variations of the ambient temperature of the year 2010. Tem-peratures can be very high and reach 44 �C, especially in the periodstretching from May to September.

The areas of the desert of Medenine, in the South of Tunisia,benefit from a very significant irradiation for which the use of PVwater pumping system can be considered with strong chances ofsuccess.

Results and discussion

In order to quantify variations between predicted and measuredvalues, root mean square error (RMSE) was used. It evaluates thepercentage mean of the sum of absolute deviations arising due toboth over-estimation and under-estimation of individual observa-tions. RMSE is given as:

RMSE ¼PðXmC � XmDÞ

N

2 !1

2

ð35Þ

N is the total number of observations while XmC and XmD are the ithcalculated and measured values, respectively.

Fig. 7. Measured monthly average ambient tem

PV-cell temperature

Table 4 presents RMSE for PV-cell temperature predictionsusing the present model, Model of Durisch et al. and the Modelof Kou et al. (respectively the models in Eqs. (9), (E.6) and (E.18).The results show that the empirical models in Eqs. (9), (E.6) and(E.18) produce the least RMSE of 2.6%, 5.4% and 5.7%, respectively.Where field trial data is not available to derive the empirical coef-ficients, Eq. (9) can be used to predict the PV-cell temperature witha higher RMSE of 2.6%.

Fig. 8 shows plots of measured and modeled PV-cell tempera-ture using Eqs. (9), (E.6) and (E.18). It can be seen from Fig. 7 thatthe predicted PV-cell temperatures show good correlation with themeasured data.

PV water pumping system performance

Tables 5–8 show the predict results obtained for the PV waterpumping system PVWPS (Station Bel Khchab) for four days duringthe month of January, March, July and October, respectively.

Table 9 shows the predict results obtained for the PV waterpumping system (Station Om Chraket) for four days during themonth of January, March, July, and October, respectively.

Fig. 9a–d shows the comparison between the measured and thepredicted hourly flow rate one using a conventional model duringfour specific days. We note that these specific days correspond tothe summer and winter solstice days and the vernal and autumnalequinox days. It is found that the conventional model consistentlyover-predicts the measurements at all times.

This is expected since the model does not account for localweather conditions such as the presence of cloud and dust.

perature of the year 2010 for Medenine.

Fig. 8. Measured and modeled PV-cell temperature.

Table 5Predicted performance using simulation model in January1th, 2010 (Station BelKhchab).

Hours KT Ht (kWh/m2) Ee (kWh) Q (m3/h)

6 0.00 0.000 0.00 0.007 0.00 0.000 0.00 0.008 0.44 0.030 0.02 0.039 0.27 0.101 0.39 0.58

10 0.74 0.452 4.88 4.0511 0.46 0.469 2.08 4.0912 0.23 0.309 0.86 2.2813 0.22 0.287 0.85 1.9814 0.22 0.258 0.81 1.6915 0.29 0.221 0.95 1.4116 0.35 0.190 0.90 1.3317 0.38 0.103 0.65 0.9718 0.00 0.020 0.01 0.0219 0.00 0.000 0.00 0.0020 0.00 0.000 0.00 0.00Daily water flow rate m3/day 18.43

Table 6Predicted performance using simulation model in March 16th, 2010 (Station BelKhchab).


6 0.00 0.00 0.00 0.007 0.00 0.00 0.08 0.128 0.44 0.00 1.12 1.679 0.27 0.05 2.79 4.14


Table 7Predicted performance using simulation model in July 15th, 2010 (Station BelKhchab).


6 0.00 0.00 0.29 0.437 0.00 0.00 0.63 0.948 0.44 0.00 1.09 1.629 0.27 0.05 2.24 3.32


Table 8Predicted performance using simulation model in October 15th, 2010 (Station BelKhchab).


6 0.00 0.00 0.16 0.237 0.00 0.00 1.10 1.648 0.44 0.00 2.27 3.379 0.27 0.05 3.26 4.85



Table 9Predicted performance in January 1th, March 16th, July 15th, October 15th, 2010(Station Om Chraket).

Hours January 1th March 16th July 15th October 15th

Q (m3/h)

6 0.00 0.00 0.00 0.147 0.00 0.07 0.25 0.968 0.02 0.98 0.55 1.989 0.44 2.44 0.95 2.85

10 1.78 3.36 1.96 3.4911 1.89 4.02 2.82 3.8212 1.68 4.52 3.57 4.5913 1.42 4.66 3.99 4.9514 1.19 4.50 4.10 4.4115 0.98 3.89 3.90 3.0216 0.86 2.65 3.48 1.2417 0.57 1.50 2.72 0.0818 0.01 0.50 1.75 0.0019 0.00 0.05 0.78 0.0020 0.00 0.00 0.13 0.00Daily flow rate

m3/day10.84 33.12 30.95 31.52

Fig. 9. Comparison between the predicted and the measured daily flow rate for two statequinox (c) and autumnal equinox (d) days.


Moreover, one can observe that there are visible discrepanciesbetween the measurements for the 3 years. The performance ofthe conventional model was evaluated in terms of the root meansquare error (RMSE). Table 10 summarizes the RMSE of one year(2010) as well as the global error for the whole samples duringthe four above mentioned days. It is found that the highest globalRMSE is occurred in the autumnal equinox, winter solstice and ver-nal equinox days with values of 4.26%, 3.7% and 4.34%, respec-tively. This is generally attributed to the local cloud formationsin these wintry, autumnally and vernally days which obviouslyvary from year-to-year.

In contrast, the results presented for the summer solstice day inFig. 9b show that both the measurements and the conventionalmodel predictions differ only slightly. In fact, there is a practicallyclose agreement between the results of the conventional modeland the mean values of the measurements with a slight over-pre-diction of the hourly flow rate by the conventional model with aglobal RMSE equal to 1.87%. Results for the other days of the years,which are not presented here follow a similar trend and agree withthe above conclusion, i.e., some discrepancies occur among themeasurements for different years and with the predictions for

ions of PV water pumping during the winter solstice (a), summer solstice (b), vernal

Fig. 9 (continued)

Table 10Root mean square error (RMSE) age between the predicted and the measured dailyflow rate.

Predicted/measured dailyflow rate

RMSE (%)station 1

RMSE (%)station 2

Global RMSE(%)

Winter solstice day 3.57 3.83 3.7Summer solstice day 1.78 1.96 1.87Vernal equinox day 4.09 4.43 4.26Autumnal equinox day 4.19 4.49 4.34


the cloudy periods, namely, October–May. These differences de-crease obviously for the months of June–September (summermonths) for which the sky is relatively clearer.

The promising similarities between the measured and predictedhourly flow rate variations during the clear-sky dates suggest thathourly flow rate have estimated basing on this conventional model.The deviation, though small, is attributed to fluctuations in the so-lar irradiance and unsteady module temperatures during themeasurements.

Conclusions

An energy balance model to predict the cell temperature mod-ules of photovoltaic water pumping system was developed andvalidated using field measurements. The energy balance modelconsiders convection and radiative exchange with the ambientenvironment as well as conduction through the cover, cell, andthe back sheet substrate.

The convection heat loss is predicted using established free andforced convection heat transfer correlations and the radiative heattransfer calculations are based on first principles rather thanempirical relations. Both a transient model that accounts for thethermal capacitance and temperature of each material in the celland a steady energy balance model that creates a thermal resis-tance network between the cell and ends of the module weredeveloped. In order to quantify variations between predicted andmeasured values, root mean square error (RMSE) was used. Thepredicted PV-cell temperatures show good correlation with themeasured data with RMSE of 2.6%.

The predicted PV-cell temperatures models were used to pre-dict real-time performance from a PV water pumping systems.

The method has been validated by predicting the performanceof two PV water pumping systems in the desert of Medenine (southof Tunisia) during one complete year. The daily flow rates pre-dicted by the method in two stations (Bel Khchab and Om Chraket)have been compared with experimental measurements; the pre-dicted water flow rate shows good agreement with the measureddata. A daily comparison basing on the root mean square errorshowed that this model predicted with good accuracy the real dataunder the clear sky conditions with an error of 1.87% while its per-formance decreases in the cloudy conditions with an error reachingthe value of 4.34%.

Acknowledgments

The corresponding author thanks the CRDA of Medenine (Tunisia)for their cooperation. We would like to extend our sincere thanksto Mrs. Fadhel Leffet, Mongi Arabi and Cherif Mhemdi for his


constructive advice and recommendations given during the execu-tion of the work.

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Real-time analytical model for predicting the cell temperature modules of photovoltaic water pumping systems