Electrical, thermal and structural performance of a cooled PV module: Transient analysis using a multiphysics model

Applied Energy 112 (2013) 300–312

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Electrical, thermal and structural performance of a cooled PV module: Transient analysis using a multiphysics model M.U. Siddiqui, A.F.M. Arif ⇑ Department of Mechanical Engineering, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia

h i g h l i g h t s  Multiphysics model for electrical, thermal and structural performance prediction.  Structural performance of PV modules is important under thermal cycling and cooling.  Methodology for coupling thermal and electrical model of PV modules.  Simulations done for four different full days representing different timings of a year.  Cooling effectiveness is most strongly dependent on incident solar radiation.

a r t i c l e

i n f o

Article history: Received 22 November 2012 Received in revised form 11 June 2013 Accepted 15 June 2013 Available online 9 July 2013 Keywords: Photovoltaic module Multiphysics model Thermal model Electrical model Structural model Thermal stress

a b s t r a c t The main performance metric of any PV device is its electrical power output. But the ability to predict its thermal and structural response under different environmental conditions are also important in order to estimate its overall performance and for useful life prediction and reliability analysis. In the current work, the development of a multiphysics model is presented which is capable of estimating the three dimensional thermal and structural performance as well as the electrical performance of a PV module under given meteorological conditions. The model is also capable of including the effect of module cooling. The thermal modeling has been carried out in ANSYS CFX CFD environment, the structural modeling has been done in ANSYS Mechanical FEA code and the electrical modeling has been developed in MATLAB environment. The electrical model used is an improved seven-parameter electric circuit model which is capable of better simulating the electrical performance of the module at low irradiance and high temperature conditions. A coupling methodology to include the effect of electrical performance of the PV module in the thermal model inside the CFD environment has also been presented in the paper. Using the developed model, the electrical, thermal and structural performance of a PV module with and without cooling has been analyzed for four different days representing different environmental conditions at Jeddah, Saudi Arabia. Ó 2013 Elsevier Ltd. All rights reserved.

1. Introduction Photovoltaic technology (PV) provides a direct method to convert solar energy into electricity. In recent years, the use of PV systems has increased greatly with many applications of PV devices in systems as small as battery chargers to large scale electricity generation systems and satellite power systems. Commercially available PV modules convert around 13–20% of incident solar radiation into electricity. The remaining solar energy absorbed into the panel is converted to heat and increases the panel temperature. This increase in temperature causes the development of thermal stresses in the panel and also causes the module efficiency to decrease [1]. To reduce the panel temperature, cooling of the PV pan⇑ Corresponding author. E-mail addresses: [email protected], [email protected] (A.F.M. Arif). 0306-2619/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.apenergy.2013.06.030

els is usually done which improves the electrical performance of the module and reduces the thermal stresses developing in the module. Several types of thermal models have been developed for calculating the temperature of PV cells as a function of solar radiation and the environmental conditions. These include models for PV panels without cooling [2–4] as well as with cooling [5–10]. For PV module cooling two different approaches have been adopted. Either custom-made photovoltaic–thermal (PV/T) collectors are developed with the objective of PV module cooling and collecting hot water or air or commercially available PV modules are cooled using auxiliary thermal collectors attached to the back surface of the module. Thermal modeling of custom-made PV/T collectors is extensively reported in literature [5–8]. Tiwari and Sodha [5] developed a one-dimensional analytical model for the Integrated

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Nomenclature a modified diode ideality factor aref modified diode ideality factor at STC Ai anisotropy index Apanel area of PV panel (m2) [B] displacement gradient matrix Cp specific heat capacity (J/kg K) Cl, Ce1, Ce2 constants for turbulence model [D] material properties matrix (Pa) Eg band-gap energy of PV cell material (eV) Eg,ref band-gap energy of PV cell material at STC (eV) {Fth} thermal force vector (N) Ffront/rear-ground geometric factor for front glass or back sheet to ground radiation Ffront/rear-sky geometric factor for front glass or back sheet to sky radiation Gb horizontal beam radiation (W/m2) Gd horizontal diffuse radiation (W/m2) Gref reference condition (STC) incident radiation (W/m2) h heat loss coefficient (W/m2 K) hconv,forced forced convection coefficient (W/m2 K) hconv,free free convection coefficient (W/m2 K) hradiation radiation heat loss coefficient (W/m2 K) I PV panel output current (A) IL light generated current (A) IL,ref light generated current at STC (A) IMP PV panel maximum power current (A) Io diode reverse saturation current (A) Io,ref diode reverse saturation current at STC (A) kcond thermal conductivity (W/m K) k turbulent kinetic energy (m2/s2) K extinction coefficient [K] stiffness matrix (N/m) Ksa,b incidence angle modifier for beam radiation Ksa,d incidence angle modifier for diffuse radiation Ksa,g incidence angle modifier for ground reflected radiation Lglass thickness of front glass layer (m) m irradiance dependence parameter for IL M air mass modifier n temperature dependence parameter for a n surface normal NCS number of PV cell in series in PV panel Nu Nusselt’s number Pk production term Pr Prandl’s number q heat conduction (W) Q volumetric heat generation (W/m3) Qvh viscous energy dissipation (W/m3) Rbeam geometric factor for beam radiation

photovoltaic/thermal system (IPVTS) collector of Huang et al. [11] and validated its performance against experimental data. An improvement to this model was suggested by Sarhaddi et al. [6]. They suggested that the accuracy of the model could be improved by using the equivalent electric circuit model to determine the electrical performance of the system and by using more detailed expressions for determining the thermal resistances within the system. A similar model was developed by Amori and Taqi Al-Najjar [7] who applied their model to predict the performance of a PV/ T collector for two different environmental conditions in Iraq. They considered the temperature variation across the thickness only and used an equivalent electric circuit model for electrical output prediction. Amrizal et al. [8] carried out dynamic modeling of a PV/T

Ra Re Rs Rs,ref Rsh Rsh.ref S Sref t T(x,y,z) Tamb Tcell Tcell,ref Tf,in Tfront/rear Tsky u {u} V VMP Vf,in Vpv,cell

Raleigh’s number Reynolds’s number series resistance (X) series resistance at STC (X) shunt resistance (X) shunt resistance at STC (X) plane-of-array absorbed solar radiation at operating conditions (W/m2) absorbed solar radiation at STC (W/m2) time (s) temperature field (K) ambient temperature (K) PV cells temperature (K) PV cells temperature at STC (K) inlet water temperature (K) temperature of front or back surface (K) sky temperature (K) fluid velocity (m/s) displacement vector (m) PV panel output voltage (V) PV panel maximum power voltage (V) inlet water velocity (m/s) volume of PV cells inside the module (m3)

Greek symbols a absorptivity of PV cells b slope of PV panel gpv electrical efficiency of PV panel gpv,ref PV electrical efficiency at reference condition {e} elastic strain vector {eth} thermal strain vector {eT} total strain vector e turbulent dissipation rate (m2/s3) efront.rear emissivity of front glass or back sheet eground emissivity of ground qground reflectivity of ground q density (kg/m3) h incidence angle of solar radiation hr refracted angle of solar radiation re turbulent Prandtl number for e rk turbulent Prandtl number for k s transmitivity of top cover (sa) transmitivity–absorptivity product l dynamic viscosity (Pa s) lisc temperature coefficient of short circuit current lT turbulent viscosity (Pa s)

collector. They used a very simplified thermal model which calculated the thermal power output and the PV cell temperature and their model required experimental data for estimating the model parameters. For electrical performance prediction, they used an equivalent electric circuit model. For a commercially available PV module cooled by auxiliary thermal collectors, a relatively smaller amount of work has been reported. In one work, Teo et al. [9] designed an air cooled PV/T system using a commercially available PV panel and a custom made air collector attached to it. They developed a one-dimensional thermal model for their PV/T collector and used it to analyze its performance. In a previous work done by the authors, they developed a three-dimensional thermal model for commercial PV

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Fig. 1. Performance modeling of PV systems [10].

Table 1 Material properties used in thermal model.

Fig. 2. Equivalent circuit of an actual PV cell [10].

Material

Layer

Thermal conductivity (W/m K)

Specific heat capacity (J/kg K)

Density (kg/m3)

Silicon Tempered glass Polyester

Solar cell Top cover Bottom cover Encapsulent Heat exchanger

130 2 0.15

677 500 1250

2330 2450 1200

0.311 237

2090 903

950 2702

Ethyl vinyl acetate Aluminum

Table 2 Material properties used in the structural model. Material

Layer

Modulus of elasticity (GPa)

Poisson’s ratio

Coefficient of thermal expansion (K1)

Silicon Tempered glass Polyester

Solar cell Top cover Bottom cover Encapsulant

150 70 4

0.17 0.22 0.37

2.616  106 9  106 60  106

7.8 72

0.3 0.32

90  106 22  106

Fig. 3. Modes of energy transfer in a PV panel [10]. EVA Aluminum

modules cooled using an auxiliary thermal collector. They used the model to carry out parametric studies to see the effect of various environmental and operating parameters include the contact resistance between the PV module and thermal collector on the electrical and thermal performance of the PV module [10]. For modeling the electrical performance of PV modules, various electrical models have been developed. These include models based on the analytical knowledge of PV cell behavior, models based on empirical correlations, as well as models which combine the two approaches. King et al. [3] developed an empirical model for simulation of PV systems called the Sandia Labs PV Model. Hishikawa et al. [12] and Marion et al. [13] used current–voltage curve

interpolation for estimating the module electrical performance. Another approach adapted for PV electrical performance prediction was to represent the PV device by an equivalent electric circuit in which five model parameters represent the specific characteristics of a PV device. The model parameters can be modified for different input conditions and the model can then be solved to find the PV electrical characteristics [14–16]. The inputs to the electrical model are the absorbed solar radiation in the PV cells and the PV cells temperature. An improvement to the five parameters model, the seven parameters model, was suggested by the authors [17] in

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Fig. 4. Coupling methodology for thermal and electrical models.

Fig. 6. Maximum power prediction errors. (Module 1 = AP-110, Module 2 = S-36, Module 3 = KC-40T, Module 4 = MST-43LV, Module 5 = ST-35, Module 6 = PVL-124).

Fig. 5. Model validation for thermal model without cooling using experimental data [10].

Table 3 Model validation for thermal model with cooling [10]. Inlet velocity (m/s)

Cell temperature (°C)

Outlet fluid temperature (°C)

Current model

Using Sarhaddi et al. [6]

Current model

Using Sarhaddi et al. [6]

0.1 0.05 0.01

30.6 32.5 41.1

29.1 31.5 39.8

25.7 26.5 30.7

25.6 26.1 26.3

which the equations to modify the model parameters were adjusted to improve the electrical model accuracy at low irradiances and high temperatures. As far as structural performance of PV panels is concerned, the main approach used is using finite element methods [18–21]. In an effort to optimize the design of PV cell interconnects to reduce thermal stresses, Eitner et al. [18] developed a structural model for a string to PV cells laminate using finite element method. The model was exposed to uniform temperature loads from 150 °C to 40 °C and the thermal stresses were calculated in a static timeindependent analysis. Gonzalez et al. [19] used an FEA based model to study the effect of encapsulent and adhesive materials and PV cell size on the thermal stresses developing in the cells and the interconnects. Meuwissen et al. [20] developed a finite element model for a single PV cell laminate to study the structural response of adhesive cell interconnects. Using the developed FE model, they studied the effect of adhesive interconnect on the stresses

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Fig. 7. Input meteorological conditions for (a) January 17, 2000 (b) July 17, 2000 (c) October 15, 2000 and (d) December 10, 2000.

Table 4 Electrical model reference parameters for the selected PV module. Parameter

Value

IL,ref Io,ref aref Rs,ref Rsh,ref m n

7.5084 A 3.4686  106 A 1.4249 V1 0.0527 X 46.8713 X 1.0959 1.1368

developing in the cell interconnects by reducing the laminate temperature uniformly from 150 °C to 20 °C. Dietrich et al. [21] presented a finite element based modeling methodology which consisted of an overall model of the entire panel and several submodels for studying the effects of thermal and mechanical loads on the PV module. They studied the stresses developing in the module during the lamination process and during thermal cycling between 40 °C and 150 °C. In all of these previous works, the focus was to study the stresses developing in the module during the lamination process and/or the IEC 61215 standard thermal cycle. Additionally, the structural models were not combined with a thermal model to first determine the temperature distribution inside the module which would have extended the model capabilities to simulate the effect of real-life environmental conditions on the module performance. Instead, uniform temperature load was applied to the entire model. In a previous work, the authors used

the structural model in combination with a thermal model in order to simulate the structural performance of the PV module under a variety of environmental and operating conditions [22]. The objective of the current work was to develop a multiphysics model capable of predicting the thermal, electrical and structural performance of a PV module under real-life conditions with and without cooling. The developed model is capable of calculating the three-dimensional distribution of temperatures in the PV module and the auxiliary thermal collector, the three-dimensional stress distribution inside the PV module as well as its electrical power output and electrical efficiency. The three-dimensional thermal model, developed in CFD environment, allows the use of complex thermal collector geometries which cannot be modeled by one-dimensional analytical models. For the electrical performance prediction, the improved seven parameters model is used. Moreover, a scheme to incorporate the electrical model effects in the CFD model is also presented. The developed model was applied for simulating the performance of a commercially available PV module for four different days representing different temperatures and cloud conditions at Jeddah, Saudi Arabia. 2. PV performance prediction model Fig. 1 shows the flow chart of the performance prediction model used in this work. It has four sub-models: radiation and optical, electrical, thermal and structural which are sequentially coupled together. When the meteorological data and the PV site information are input into the model, the temperature distribution in the PV module and its electrical and structural performance can be

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predicted. The details of the sub-models have been discussed in detail in the following sections. 2.1. Electrical model The electrical model used in the present work is a newly developed 7 parameters model [23] in which a PV device is represented by an equivalent electric circuit [24] of Fig. 2 and the current–voltage relationship for the PV devices is governed by the following equation:



   V þ I  Rs V þ I  Rs 1  I ¼ IL  Io  exp a Rsh

ð1Þ

To use the model, the parameters IL, Io, a, Rs and Rsh are first determined at a reference condition and then translated to the operating condition using the translation Eqs. (2)–(6). The parameters m and n are determined using two additional maximum power values at a higher temperature and a lower irradiance.

a ¼ aref ðT cell =T cell;ref Þn IL ¼ ðS=Sref Þm ðIL;ref þ lisc ðT cell  T cell;ref ÞÞ  3

Io ¼ Io;ref ðT cell =T cell;ref Þ e Sref Rsh;ref Rsh ¼ S Rs ¼ Rs;ref

NCS:T cell;ref aref

ðEg;ref =T cell;ref Eg =T cell Þ



ð2Þ ð3Þ ð4Þ ð5Þ

305

the authors in [10]. The model is capable of calculating PV panel temperature distribution with and without considering PV panel cooling. A PV panel consists of several solid domains for which the energy equation need to be solved. In case of cooling, the energy equations for the heat exchanger body and working fluid and the continuity and momentum equations for the working fluid also need to be solved. Eqs. (7) and (8) are the heat transfer equations for solid and fluid domains respectively [25].

@T i ðx; y; zÞ ¼ r  ðqi Þ þ Q i i ¼ 1; 2; . . . ; n @t @T qC p þ qC p u  rTðx; y; zÞ ¼ r  ðqÞ þ Q v h @t

qi C p;i

ð7Þ ð8Þ

where

q ¼ kcond rT

ð9Þ

and q is the density, Cp is the specific heat capacity, T(x,y,z) is the temperature, t is the time, kcond is the thermal conductivity, q is the heat transferred by conduction, Q is the internal heat generation, u is the fluid velocity and Qvh is the viscous dissipation. The momentum and continuity equations governing the fluid flow inside the heat exchanger are given by Eqs. (10) and (11) [26].

q

  @u 2 ð10Þ þ qðu  rÞu ¼ r  pI þ ðl þ lT Þðru þ ruT Þ  qkI @t 3

qr  u ¼ 0

ð11Þ

ð6Þ

2.2. Thermal model A thermal model to calculate the three-dimensional temperature distribution in PV panels was developed and validated by

where p is the pressure, l is the viscosity, lT is the turbulent viscosity and k is the turbulent kinetic energy [26]. For a PV panel, the various modes of energy transfer in the PV panel are shown in Fig. 3. The panel gains energy by absorbing incoming solar radiation while energy is lost from it by convection and radiation to the environment, by energy transfer to the

Fig. 8. Temporal variation of cell temperature in PV panel for (a) January 17, 2000 (b) July 17, 2000 (c) October 15, 2000 and (d) December 10, 2000.

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working fluid in the heat exchanger and in the form of electrical energy delivered to the electrical load. In case of PV panel without cooling, the energy transfer to the heat exchanger through the PV panel back surface is replaced by convection and radiation losses to the environment. The absorbed solar radiation was calculated using HDKR model [27,28] and applied to the heat transfer equation of the PV cell layer as an internal heat generation, Q, which was calculated using the following equation:

Q¼

ð1  gpv Þ  S  Apanel V pv cell

ð12Þ

where gpv is the electrical efficiency of the PV panel, Apanel is the front area of the PV panel and the Vpv cell is the volume of the PV cells in the panel. The PV panel loses heat from its top and bottom surfaces through convection and radiation. The equivalent heat loss coefficient for the PV panel is given by the following equation:

h ¼ hconv ;forced þ hconv ;free þ hradiation

( Nu ¼

0:76Ra1=4 0:15Ra1=3

ð14Þ for 104 < Ra < 107 for 107 < Ra < 3  1010

þ

rF front=rearground ðT 4front=rear T 4ground Þ 1

1

efront=rear þeground 1

ð16Þ

where Nu is the Nusselt’s number, Re is the Reynolds’s number, Pr is the Prandl’s number, Ra is the Raleigh’s number, e is the emissivity, F is the geometric factor and r is the Stefan–Boltzmann’s constant. Tsky was taken as Tamb  20 K [2] and the ground temperature was assumed to be equal to the ambient temperature [31]. In the present study, the boundary conditions were calculated for a PV cell temperature of 70 °C and an ambient temperature of 25 °C and assumed independent of temperature for numerical solution simplicity. The corresponding boundary conditions applied to the heat transfer equations of top and bottom layers of the PV panel is given by Eq. (17).

ð13Þ

where h is the equivalent heat loss coefficient. Eq. (14) by Sparrow et al. [29], Eq. (15) by Lloyd and Moran [30] and Eq. (16) by Duffie and Beckman [28] can be used to calculate hconv,forced, hconv,free and hradiation iteratively for each time step.

Nu ¼ 0:86Re1=2 Pr1=3

hradiation ðT front=rear  T amb Þ ¼ efront=rear F front=rearsky rðT 4front=rear  T 4sky Þ

ð15Þ

n  q ¼ hðT amb  T front=rear Þ

ð17Þ

where n is the surface normal and Tamb and Ts are the ambient and surface temperatures. In the current work, the thermal model for cooled PV module was implemented in ANSYS CFX code while the model for PV module without cooling was implemented in ANSYS Mechanical code. The material properties used in the thermal models are listed in Table 1.

Fig. 9. Temporal variation of stress in PV cells for (a) January 17, 2000 (b) July 17, 2000 (c) October 15, 2000 and (d) December 10, 2000.

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Since {du}T is an arbitrary displacement vector, the above equation reduces to,

½Kfug ¼ fF th g R

ð19Þ T

th

where [K] = vol[B] [D][B]dv is the element stiffness matrix; {F } = R T th th vol[B] [D]{e }dv is the element thermal load vector; {e } = {a}DT is the thermal strain vector and {a} is the vector of coefficient of thermal expansion, [B] is the displacement gradient matrix, [D] is the material properties matrix and DT is the temperature difference from the reference condition.To calculate the thermal stresses developing in the body, the Hooke’s law, given by Eq. (20), is used. Fig. 10. Paths for studying spatial distribution of temperature and stresses.

frg ¼ ½Dfeg

where {r} is the stress vector and {e} is the elastic strain vector equal to the difference of the total and thermal strains.

2.3. Structural model

feg ¼ feT g  feth g Finite element method was used to calculate the thermal stresses developing in the PV panel and two structural models were implemented [22,32]. Applying principle of virtual work under only temperature body load gives,

fdugT

Z v ol

½BT ½D½Bdv fug ¼ fdugT

ð20Þ

Z v ol

½BT ½Dfeth gdv

ð18Þ

ð21Þ

The structural model for PV panel without cooling was implemented in ANSYS Mechanical using SHELL181 layered structural elements. The structural model for cooled PV panel was implemented in ANSYS Mechanical environment. Only the PV panel was modeled for structural analysis using the same geometry and mesh as the thermal model. The finite element used was SOLID185 which is a brick element defined by eight nodes having

Fig. 11. PV cell temperature (a) with cooling, and (b) without cooling.

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was then used to calculate the electrical efficiency at each iteration using Eq. (23).

  @ gpv 1 @V mp @Imp ¼ Imp þ V mp @T cell Gref Apanel @T cell @T cell   @ gpv gpv ¼ gpv ;ref 1  ðT cell;av g  T cell;ref Þ @T cell

ð22Þ ð23Þ

where Tcell is PV cell temperature, Tcell,avg is the average cell temperature in the PV panel, G is the incident solar radiation, Apanel is the front surface area of the PV panel and Imp and Vmp are the maximum power point current and voltage. Different methodologies were adopted to include this coupling in the thermal model for panels with and without cooling. The thermal model for panel with cooling was developed in ANSYS CFX environment which employs an iterative solver to solve momentum, continuity and energy equations. For the model of panel without cooling, the above methodology cannot be employed. This is because ANSYS Mechanical FEA code, in which the model without cooling has been implemented, does not use iterative solver. Therefore, instead of using the PV cell temperature from the same time step in Eq. (23), the cell temperature from the previous time step was used. The flowchart of the coupling methodology is shown in Fig. 4. 3. Model validation 3.1. Validation of thermal model without cooling using experimental data The thermal model for PV panel without cooling was validated using experimentally measured data for a 5940 Watts PV site using Schott Solar SAPC-165 multi-crystalline silicon PV panel located at Tallahassee, Florida, USA. The meteorological data and measured PV back surface temperature for the PV site for 1 day (May 15, 2005) was taken from the PV performance database maintained by Florida Solar Energy Center. A comparison of the predicted PV back surface temperature by the thermal model and the measured temperature is shown in Fig. 5. The root mean square error in the model prediction of panel back surface temperature was 4.9 °C. The correlation factor (r) was calculated to be 0.95. Fig. 12. PV cell von Misses stress (a) with cooling, and (b) without cooling.

three degrees of freedom at each node. Temperature distribution calculated in the thermal analysis was applied to the model as body load. To restrain the model structurally, the four corners of the model were fixed in all degrees of freedom. The structural properties of all panel materials were assumed to be temperature-independent linear-elastic and are shown in Table 2. The strain-free temperature was assumed to be 25 °C which is the standard test conditions (STC) cell temperature used in PV module datasheets.

3.2. Validation of thermal model with cooling using analytical model The thermal mode with cooling was validated by comparing the average cell temperature and outlet fluid temperature predicted by the thermal model at various fluid velocities with the temperatures predicted by the analytical model for parallel channel heat exchanger PV/T collector presented by Sarhaddi et al. [6]. The results of the comparison are shown in Table 3. The root mean square errors in the prediction of PV cells and outlet fluid temperatures were 0.74 °C and 1.47 °C respectively.

2.4. Coupling thermal and electrical models

3.3. Validation of electrical model using manufacturer supplied experimental data

As shown by Eq. (12), the thermal and electrical performance of PV panels are coupled together. As the PV cell temperature increases, efficiency decreases which then modifies the amount of absorbed energy heating the PV panel. During a transient analysis, this two-way coupling needs to be modeled in order to accurately calculate the correct temperature distribution in the panel. To establish a two-way coupling between electrical efficiency and cell temperatures, first an expression for temperature dependence of electrical efficiency, given by Eq. (22), was developed by differentiating the equation for efficiency with respect to temperature. It

The validation of the electrical model was carried out by comparing the model predictions for the maximum power of PV modules of various technologies with the values reported in the module datasheets. For six different modules and considering three different operating conditions, the root mean square errors (RMSE) and the mean bias errors (MBE) in the maximum power prediction were calculated and are reported in Fig. 6. As can be seen from the figure, all errors for 5 out of the six modules are less than 1.5%. For the sixth module, the RMSE and MBE are 3% and 4% respectively.

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Fig. 13. PV cell temperature and thermal stress along (a and b) path P1 (c and d) path P2 and (e and f) path T1.

4. Results and discussion Using the developed multi-physics model, the thermal, structural and electrical performance of a crystalline silicon PV module (AstroPower AP-110) was simulated for 4 days with different meteorological conditions at Jeddah, Saudi Arabia. The ambient temperature and total irradiance on a horizontal plane for the four selected days is presented in Fig. 7. The total horizontal irradiance was used to calculate absorbed solar radiation in the PV cells using HDKR model. The absorbed radiation and the ambient temperature were used in the thermal model to calculate the temperature distribution in the PV panel which was then used to calculate the stress distribution inside the panel. The electrical model used the

absorbed radiation and average PV cell temperature to calculate the electrical power output and the electrical efficiency throughout the day. The electrical model parameters for the reference condition are given in Table 4. 4.1. Temporal variation of PV cell temperature and thermal stresses For all 4 days, the temporal variation of PV cell temperature and thermal stresses are shown in Figs. 8 and 9 respectively. Fig. 8a–d show temporal variation of PV cell temperature at point T1 for the four simulated days. For January 17, 2000, the maximum temperature occurred at 12:30 pm and was 61 °C for the panel without cooling and 31 °C for the cooled panel. For July 17, the maximum

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Fig. 14. Variation of electrical power with time in PV panel with cooling for (a) January 17, 2000, (b) July 17, 2000, (c) October 15, 2000 and (d) December 10, 2000.

cell temperature occurred at 12:30 and was 74.7 °C for panel without cooling and 33.1 °C for panel with cooling. For October 15, the maximum PV cell temperature occurred at 3:45 pm and was 84 °C for the panel without cooling and 35 °C for the cooled panel. For December 10, the panel without cooling showed lower cell temperature through most part of the day. This is because the inlet water temperature for the heat exchanger cooling the PV panel was set to 25 °C throughout the day. The maximum PV cell temperature occurred was 29 °C at 12:00 pm while it was 25.5 °C for the cooled panel. The temporal variation of thermal stresses for the 4 days is shown in Fig. 9a–d. Because the strain-free temperature was assumed to be 25 °C, the von Misses stress showed zero magnitude when the PV cell temperature equaled 25 °C. For January 17, the maximum von Misses stress for the panel without cooling was 27 MPa and for the cooled panel was 3.5 MPa. For July 17, the corresponding maximum von Misses stress values for the panel without cooling and with cooling were 38 MPa and 4.6 MPa respectively. For October 15, the maximum stress in the PV cells was 44 MPa for the panel without cooling and 5 MPa for the cooled panel. Finally, for December 10, the maximum stress was 5.5 MPa for the panel without cooling and 0.7 MPa for the cooled panel. 4.2. Spatial distribution of PV panel temperatures and thermal stress For studying the spatial temperatures and stresses, initially 1 day was studied in detail. Three paths were defined along the PV panel. Paths P1 was along the length of the panel, path P2 was along the panel’s width and path T1 was across the thickness of the panel starting from the bottom. All paths are shown in Fig. 10. The day selected for the detailed study was July 17, 2000 because of its high and uniformly distributed solar irradiance and high ambient temperatures. For the time 12:30 when the PV cell

temperature and stresses were maximum, the PV cell temperature and von Misses stress in the PV panels with and without cooling are shown in Figs. 11 and 12. The spatial variation of temperature and stresses were plotted along the three paths shown in Fig. 10 and are shown in Fig. 13. Fig. 13a and c show the PV cell temperature variation along paths P1 and P2 in the panel with and without cooling. The figures show that the PV cell temperature for PV panel without cooling showed almost constant values along the paths with cyclic variation within 0.2 °C. On the other hand, the cell temperature for cooled PV panel showed almost 5 °C rise along paths P1 and 2 °C rise along paths P2. The non-uniformity of cell temperatures in the cooled panel was due to non-uniform flow in the heat exchanger. Fig. 13b and d shows the variation of von Misses stress in PV cells along paths P1 and P2. In the figures, stress values for the first and the last cells include the effect of the applied boundary condition and the effect will increase as the path is moved closer to the boundary. Along, the paths, the panel without cooling showed symmetric variation in stress with von Misses stresses around 35 MPa. The cooled panel showed one order of magnitude lower stress values (around 1 MPa) with a gradual increase in the stress along the path due to the increase in the cell temperature. In the panel without cooling there was also a strong stress gradient between the cells but the stress gradient dropped significantly for the cooled panel. Fig. 13e shows the temperature variation across the thickness of the PV panel at location T1. It shows that PV panels with and without cooling have different types of variation along the thickness of the panel. For the panel without cooling, the temperature varied between 73.6 °C and 74.7 °C with the minimum temperature being at the top where the convection losses were maximum. For the panel with cooling, the temperature varied between 30 °C and 33.1 °C with the minimum temperature being at the bottom where the

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heat exchanger is attached. The von Misses stress variation across the thickness is shown in Fig. 13f. The figure shows no significant stresses developing in the encapsulent layer due to the very low elastic modulus of the encapsulent material. For panels with as well as without cooling, the maximum stresses occurred in the PV cell layer (37.5 MPa for the panel without cooling and 4.6 MPa for the cooled panel).

4.3. Temporal variation of electrical performance Using the measured irradiance and the average predicted PV cell temperature, the electrical performance of a crystalline silicon module with and without cooling was simulated for the conditions of the 4 days considered in this study. The electrical power output of the PV module for the 4 days is shown in Fig. 14 and the variation of electrical efficiency is shown in Fig. 15. For the 3 days (January 17, 2000, July 17, 2000 and October 15, 2000) in which the difference between the PV cell temperature of panel with and without cooling was high, there is marked improved in the electrical performance of the PV panel as shown by Fig. 14a–c. For December 10, 2000, the maximum difference in PV cell temperatures of panels with and without cooling at any time during the day was 6 °C. Within this temperature difference range, the electrical performance of the PV panel shows no appreciable variation. Fig. 15 shows variation of electrical efficiency with time during the 4 days considered in this study. As expected, the cooled PV panels showed higher efficiencies than the panels without cooling for the 3 days with higher difference between cooled and uncooled cell temperatures (January 17, 2000, July 17, 2000 and October 15, 2000) as shown in Fig. 15a–c. For December 10, 2000, the two panels show same efficiency throughout the day.

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Fig. 15 also sheds light on the dependence of electrical efficiency upon irradiance and cell temperature. In general, increasing irradiance increases efficiency while increasing PV cell temperature decreases it. At low irradiance, the influence of irradiance is higher but as the cell temperature increases, the positive effect of increasing irradiance decreases until eventually the efficiency starts to drop with increasing cell temperature. This phenomenon is visible in Fig. 15a–c. During morning and evening, the efficiency increased with increasing irradiance. During the peak sunshine hours, the panel with cooling showed slight decrease in efficiency despite the increasing irradiance while the efficiency of the cooled panel completely followed the trend of irradiance. 5. Conclusions In this paper, the implementation of a multiphysics model for the thermal, electrical and structural performance prediction of PV modules is presented. The developed multiphysics model was used to simulate the thermal, structural and electrical performance of a PV module for 4 days representing four different types of environmental conditions at Jeddah, Saudi Arabia. From the study, the following conclusions were drawn.  The effectiveness of cooling in improving the electrical conversion efficiency is more strongly dependent on irradiance than ambient temperature. For example, January 17, 2000 and December 10, 2000 showed similar ambient temperatures (around 20–25 °C) but the incident radiation was much lower for December 10, 2000 (less than 350 W). Therefore, the cooled panel provided around 20 W more power than the uncooled panel on January 17, 2000 while there was no performance improvement for December 10, 2000.

Fig. 15. Variation of electrical efficiency with time in PV panel with cooling (a) January 17, 2000, (b) July 17, 2000, (c) October 15, 2000 and (d) December 10, 2000.

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 The cooled panel showed lower cell temperatures than the uncooled module as expected but the temperature gradients in the cooled panel were also higher (4.5 °C along path P1).  The cooled panel showed one order of magnitude lower stress level than the uncooled panel. The maximum stress along path P1 was 1 MPa for the cooled panel while it was 35 MPa for the uncooled panel.  The results shows that the modeling methodology presented in the current work may be used to develop simple correlations to decide correct heat exchanger operating conditions as a function of irradiance, ambient temperature, wind speed, etc.

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