Experimental Validation of a Dynamic Linear Model ofPhotovoltaic-Thermal Collector

Lucio Ciabattoni, Student Member, IEEE, Gianluca Ippoliti, Member, IEEE,Sauro Longhi, Senior Member, IEEE

Dipartimento di Ingegneria dell’Informazione, Universita Politecnica delle Marche, 60131 Ancona, Italy, e-mail:{l.ciabattoni, gianluca.ippoliti, sauro.longhi}@univpm.it

Abstract—In this paper a dynamic model of a photovoltaic andwater heating system (PVT) is developed. The model, based onthe energy transfer phenomenon, consists in a set of mathematicalnon linear equations governing the main components of thesystem. The model has been linearized and discretized andsimulations are carried out to compare the non linear continuousmodel and the linear and discrete one. Experimental tests on aprototype of the PVT collector has been used to validate themodel under the effect of water mass flow rate.

Index Terms—photovoltaic thermal collector, non linear mod-eling, experimental validation.

I. INTRODUCTION

As distributed generation (DG) grows in electricity networksthe challenge for electric system operators is how to effectivelyintegrate significant amounts of wind [1]–[4], solar [5]–[9],and other forms of variable generation into the network [10].Photovoltaics (PV), although behind other technologies interms of installed capacity, is currently the most important DGtechnology all around the world. The successful use of solarenergy systems depends on their efficiency or performanceas well as upon other factors such as cost, durability andreliability, and safety characteristics. Researchers are continu-ally striving to improve the efficiency with which solar cellsconvert light energy to electrical energy. The record at themoment stands at an efficiency of around 40%, using multijunction cells (multiple layers of silicon), each layer tuned totrap different frequencies (colors) of light. This type of cellwill however be expensive to produce and in the past has beenmainly used in space where efficiency may be more important.Researchers also strive to increase the amount of light enteringthe cell. Silicon is a naturally shiny substance and cells arecoated with non-reflective layers to ensure that as much lightas possible enters the cell. Cells used in photovoltaic panelsfor electricity production are usually single junction type withan efficiency of somewhere around 15%. Solar panels willnormally have a layer of glass (protecting the cells) throughwhich the light must pass before entering cells. There isscope for reducing light reflected from the glass and also totrap light that is not striking the panel at 90 degrees. Someresearch centers on trapping light from a large area fromvarious angles and directing it onto a relatively small cell.Other two important factors are the angle at which the panelsare mounted and the avoidance of shade. Small areas of shadeon a panel can have a significant effect on the power produced

[11], [12]. The output of a solar cell, and therefore a solarpanel, is affected by its temperature. As a result the poweroutput will be reduced by between 0.25% (amorphous cells)and 0.5% (most crystalline cells) for each ◦C of temperaturerise [13], [14]. Panel temperatures in the summer in warmclimates can easily reach 50 ◦C resulting in a 12% reductionin output compared to the rated output at 25 ◦C. This reductionin efficiency may be important if a high electricity demand inthe summer is required. Hence, a new technology combiningthermal and photovoltaic effects (Photovoltaic/Thermal PV/T)is developed. A photovoltaic/thermal hybrid solar collector (orPV-T collector) is a combination of photovoltaic (PV) panelsand solar thermal components. In fact, a PV-T component isdefined as a device using a PV panel or PV cells as a thermalabsorber. The aims of this technology are: firstly, to cool thePV module and thus improve its electrical performance andsecondly to collect the thermal energy produced, which wouldhave otherwise been lost as heat to the environment. Thisthermal energy available from PV module can be used formany applications namely water and air heating for domestic,agricultural sectors and buildings for thermal heating/cooling.

Over the last few years there has been a growing interest inphotovoltaic-thermal solar collector that generates as statedin [15]–[19]. The collector is attractive for solar energyapplications in which limited space and installation cost areof primary concern. Several designs and non linear modelsof the system have been proposed and analyzed in litera-ture but rarely validated with real prototype data [20]–[24].Simulations and performance analysis had been carried outto optimize and improve them, moreover from the theoreticalpoint of view [25]–[27]. The hybrid PV/T collector prototype,presented in this paper, consists of a glass plate, a photovoltaicpanel, which is composed of polycrystalline silicon cells, anda metal plate in backside is fixed, using an insulant glue, withtubes intended for the circulation of the working fluid (seeFig. 1). The aims of this work are:

• develop a non linear model of the PVT system and derivefrom it a discrete linear model.

• carry on tests of the prototype and compare the resultswith respect a normal PV module under the same condi-tions (ambient temperature, irradiation).

• compare the performances of both models and validatethe model with data carried out in the experimental tests.

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Fig. 1. Section of the PVT panel.

The paper is organized as follows. The proposed non linearmodel is described in section II, the linearization and dis-cretization is discussed in Section III. Simulation tests andresults are reported in Section IV. The validation of the modelwith experimental data is reported in Section V.

II. NON LINEAR EQUATIONS

Several models of a PV/T collector were presented in theliterature. The considered non linear thermal model is based onthe energy transfer phenomenon between the components ofthe collector: outside and inside glass, PV module, outside andinside insulation, plate absorber, tubes and fluid. The presentedmodel derives as a combination of the solutions proposed by[28] and [29] (the nomenclature is in table I).

• External Glass

Tve =2

δvρvCPv

[αv

G2 + hr,v−c · Tc + hventTa

]−Tve

2δvρvCPv

[hr,v−c + hvent + hc,v]

+2hc,v

δvρvCPvTvi

(1)

• Internal Glass

Tvi = αvG2

2δvρvCPv

− Tvi2

δvρvCPv[hr,Pv−v+

hv,Pv−v + hc,v] +2

δvρvCPvTve [hc,v]

+ 2δvρvCPv

TPv [hr,Pv−v + hv,Pv−v]

(2)

• PV cells

TPv =G(ατ)P−E

δPvρPvCPpv+ TP

hc,P−Pv

δPvρPvCPpv−

TPv

δPvρPvCPpv[hc,P−Pv+ hv,Pv−P + hr,Pv−P ]

+ 1δPvρPvCPpv

Tvi [hv,Pv−P + hr,Pv−P ]

(3)

• Plate

TP = − 1δP ρPCPP

TP

[hc,P−Pvc +

APt

APhc,P−t

]+TPv

hc,P−Pvc

δP ρPCPP+ Tt

[APt

AP

hc,P−t

δP ρPCPP

] (4)

• Tubes

Tt = − Tt

δtρtCPt

[APt

APhc,P−t +

Af

Athv,t−f + Ai

Athc,i−t

]+

Tp

δtρtCPt· APt

APhc,P−t +

Tf

δtρtCPt· Af

Athv,t−f

+ 1δtρtCPt

Tii · Ai

Athc,i−t

(5)• Fluid

Tf =m·CPf ·Tf P+At·hv,t−f ·Tt+Aif ·hv,i−f ·Tii

m·CPf+Athv,t−f+Aif ·hv,i−f

(6)

• Internal Insulation

Tii = − 2Tii

δiρiCPiAi[Athc,i−t +Aifhv,i−f +Aihc,i] +

Tf

[2·Aif ·hv,i−f

δiρiCPiAi

]+ Tt

[2·At·hc,i−t

δiρiCPiAi

]+

2·hc,i

δiρiCPiAiTie

(7)• External Insulation

Tie = − 2δiρiCPi

Tie · [hr,i−s + hc,i + hvent] +

Ts

[2hr,i−s

δiρiCPi

]+ Tii

[2hc,i

δiρiCPi

]+ Ta

[2hvent

δiρiCPi

] (8)

The variables hr,x−y, hc,x−y , hv,x−y are the radiation, con-duction and convection heat exchange coefficients respectively,between x and y. The expression of the radiation coefficient isa non linear combination of the temperatures of x and y (eg.the radiation coefficient between sky and the external glassis hr,v−c ∼

(T 2c + T 2

ve

)(Tc + Tve) ). The PV model used to

compute the electrical power production is the single diodeand three parameters model (see [30]). Those aspects give anon linear form to the whole model.

III. LINEAR STATE SPACE MODEL

By gathering the system components sub-models, the globalenergy balance can be written as a state equation:

·x(t) = Ax(t) +Bpu(t) +Bd

y(t) = Cx(t) +Du(t) +Dd(9)

where x is a vector containing the temperatures of the differentsections of the collector, A is the state matrix which containthe heat exchange coefficients between the system elements,Bp is the control matrix which encloses commands applied onthe mass flow rate in the PV/T system, Bd is a disturbancematrix which includes the weather variables, C is the outputmatrix acting on the state vector, y is the output vectorcontaining the electrical power and Dd is the disturbancematrix on the output. Linearization has been performed usingTaylor series technique, considering the equilibrium point tobe the value of the variable previously computed. To linearizethe electrical power production of the module, according to[29], we use the approximate equation:

E = G · P · τv · η0 · [1− ϕc (TPv − 298.15)] (10)

where P is the packing factor, τv the transmittance of glasscover, η0 is the electrical conversion efficiency at 298.15Kand ϕc the temperature coefficient of the cell.

Also the discretization of equations have been performedaccording to the exact Zero-Order hold technique. The formof the matrixes and the state vector is expressed by:

A =

A11 A12 0 0 0 0 0 0A21 A22 A23 0 0 0 0 00 A32 A33 A34 0 0 0 00 0 A43 A44 A45 0 0 00 0 0 A54 A55 A56 A57 00 0 0 0 A65 A66 A67 00 0 0 0 A75 A76 A77 A78

0 0 0 0 0 0 A87 A88

(11)

978-1-4799-3299-3/13/$31.00 ©2013 IEEE 1496

x =

Tve

Tvi

Tpv

Tp

Tt

Tf

Tii

Tie

Bd =

Bd11

Bd21

Bd31

00

Bd61

0Bd81

Bp =

00000

B61

00

(12)

C =[0 0 C13 0 0 0 0 0

]D = 0 Dd = Dd1

(13)

IV. SIMULATION RESULTS

To verify the accuracy of the discrete linear model, a setof simulations has been carried out, in order to compare,under the same inputs (air, water, sky and ground temperature,irradiance, wind speed and flow rate), the output of thetwo models. Simulations shows a MSE% error of 0.095%between the PV temperature output of the models (shown inFig. 2) and a MSE% error of 0.091% for the output flowtemperature.

Fig. 2. Comparison between continuous non linear model (red dotted line)and discrete linear model output (blue continuous line). PV cell temperature.

Fig. 3. Comparison between continuous non linear model (red dotted line)and discrete linear model output (blue continuous line). PV power production.

V. EXPERIMENTAL VALIDATION

In the aim of validating the mathematical model, experimen-tal data have been carried out from the prototype of the hybridPVT collector (see Fig. 5). The experimental setup, as shownin Fig. 4, is composed by the PVT panel and PV panel bothfixed in the same structure, two Enphase micro inverters M215connected to the panels, DC voltage sensors (to compare

the values read from the inverters), an underground watersource, a hydraulic pump, a PLC to control the valve and thepumps. An hydraulic circuit has been designed to maintainthe water temperature in input to the PVT in a range between23.5 ◦C and 24.5 ◦C. Since the underground water has atemperature between 14 ◦C and 15 ◦C, the circuit has beendesigned to simulate the typical operating conditions in aresidential scenario, without underground water sources. Thesystem has been tested during September 2012, at EnergyResources S.p.A., in Jesi (AN), Italy. Temperatures of the

Fig. 4. Experimental setup of the PVT prototype. Red lines are referred tothe input water flow to the module, purple lines are the output water flow,cyan lines are the water source, yellow lines are the control wires.

Fig. 5. Prototype of the hybrid PV-Thermal collector.

different layers of the structures have been measured whilethe fluid was flowing in the tubes with different flow rates.On the PV and PV/T module we used 15 temperature sensorseach, in order to measure the temperature gradient of thewhole panels. Data have been acquired with a cDAQ chassisand 4 analog input modules (model NI9201) of NationalInstruments and processed with the software LabVIEW, thegraphical programming environment of National Instruments.Sampling frequency is 10Hz for each sensor and we averaged50 read values for each one, obtaining the complete tempera-

978-1-4799-3299-3/13/$31.00 ©2013 IEEE 1497

ture measures of the panels every 5sec. The same flow rates,water flow temperature and initial conditions (for what regardsthe cells temperature, the mean of the measures from the 15sensors has been considered) are introduced in the simulationprogram presenting the mathematical model of the collector,and simulation results have been compared to the experimentalones. Figs. 6 and 7 show a comparison between the simulatedand the measured trend of PV cells temperature and the outputflow temperature respectively. The computed MSE% error is0.85% and 0.64% respectively.

Fig. 6. Measured (blue dots) and simulated (red line) temperature evolutionof PV cells.

Fig. 7. Measured (blue dots) and simulated (red line) evolution of the outputfluid temperature.

VI. PERFORMANCE TESTS

The system has been tested during September 2012, atEnergy Resources S.p.A., in Jesi (AN), Italy. In particular thetwo panels (the traditional PV and the PV/T) have been testedunder the same environmental conditions (irradiation, air tem-perature, wind speed) and mounting (tilting and orientation),as shown in Figs. 4 and 5. Experiments are focused on thecomparison of the performances of the two modules under aconstant water flow rate of 1l/min at 24 ◦C for the PV/Tmodule:

• minimum and maximum temperature.• AC power production.• efficiency irradiation - AC production.

The pumping circuit was activated when the temperature of thepanels reached 50 ◦C (in Figs. 8-10, representing tests carriedout on 14 September 2012, it happened at 11 : 30 AM) andstopped when the same temperature was under 40 ◦C.

Fig. 8. 14 September 2012. Maximum and minimum cell temperatures forPV module (red and orange), PV/T module (dark green and green), input andoutput fluid temperature (blue and purple).

Fig. 9. 14 September 2012. Normal Photovoltaic (blue line) and PV/Tmodule (red line) efficiency under the same conditions.

VII. CONCLUSIONS

In this paper we presented the prototype of a novel pho-tovoltaic and water heating system (PVT). A dynamic modelof the collector, based on the energy transfer phenomenon,has been linearized and discretized. Simulations have beenused to proof the validity of the linear and discrete modelwith respect the non linear continuous one. Experimental testshave been performed during September 2012 in Jesi (AN),Italy. Data have been used to validate the model under theeffect of constant water mass flow rate and the results are anMSE% of 0.83% between simulations and experimental tests.We showed also the performance improvements with respecta traditional PV module.

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VIII. NOMENCLATURE

TABLE INOMENCLATURE

Tve External glass Temp Tf Fluid Temp

Tt Tubes Temp Tii Internal Insulant Temp

Tvi Internal glass Temp TPV PV cell Temp

Tie External Insulant Temp Tp Plate Temp

G Irradiance δv Glass thickness

ρv Glass mass density Ts Ground Temp

Ta Ambient Temp ρPv PV mass density

δPv PV module thickness Tc Sky Temp

CPpv Specific heat PV δP Plate thickness

ρP Plate mass density CP Specific heat plate

APt Plate-tubes surface AP Plate surface

δt Tubes thickness ρt Tubes mass density

CPt Specific heat tubes At Tubes surface

Af Fluid-tubes surface Ai Insulation surface

m Flow mass rate CPf Specific heat fluid

Tf P Fluid Previous Temp δi Insulant thickness

ρi Mass density insulant CPi Specific heat insulant

ACKNOWLEDGMENTS

The authors wish to thank Energy Resources S.p.A. for thesupport during experiments.

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