*Article* **A Novel Method for Thermal Modelling of Photovoltaic Modules/Cells under Varying Environmental Conditions**

#### **Ali Kareem Abdulrazzaq \*, Balázs Plesz and György Bognár**

Department of Electron Devices, Budapest University of Technology and Economics,

Hungarian Scientists Tour 2, H-1117 Budapest, Hungary; plesz@eet.bme.hu (B.P.); bognar@eet.bme.hu (G.B.)

**\*** Correspondence: kareem@eet.bme.hu; Tel.: +36-1-463-3073

Received: 12 May 2020; Accepted: 20 June 2020; Published: 29 June 2020

**Abstract:** Temperature has a significant effect on the photovoltaic module output power and mechanical properties. Measuring the temperature for such a stacked layers structure is impractical to be carried out, especially when we talk about a high number of modules in power plants. This paper introduces a novel thermal model to estimate the temperature of the embedded electronic junction in modules/cells as well as their front and back surface temperatures. The novelty of this paper can be realized through different aspects. First, the model includes a novel coefficient, which we define as the forced convection adjustment coefficient to imitate the module tilt angle effect on the forced convection heat transfer mechanism. Second, the new combination of effective sub-models found in literature producing a unique and reliable method for estimating the temperature of the PV modules/cells by incorporating the new coefficient. In addition, the paper presents a comprehensive review of the existing PV thermal sub-models and the determination expressions of the related parameters, which all have been tested to find the best combination. The heat balance equation has been employed to construct the thermal model. The validation phase shows that the estimation of the module temperature has significantly improved by introducing the novel forced convection adjustment coefficient. Measurements of polycrystalline and amorphous modules have been used to verify the proposed model. Multiple error indication parameters have been used to validate the model and verify it by comparing the obtained results to those reported in recent and most accurate literature.

**Keywords:** module temperature; solar energy; thermal modelling; heat transfer mechanisms

#### **1. Introduction**

The increasing need for electricity and the risks of environmental pollution and global warming are the main problems increasing the interest in renewable and clean energy sources [1]. Solar energy sources using photovoltaic (PV) modules recently have the main focus among other renewable sources. This is due to several reasons such as the abundance of the solar irradiance, the photovoltaic (PV) phenomenon, by which a direct conversion is achieved from solar radiation to electricity, employable at both small and large scale, non-polluting, clean and reliable energy sources. The increase in the temperature of the silicon-based technology PV modules has direct effect on the current–voltage (I–V) characteristics of the device, that is, adversely affecting the power production and causes a significant drop in efficiency [2,3]. Therefore, it is insufficient to rely only on the rated efficiency to estimate the output power. One has to consider the operating temperature of the PV module as well as other environmental conditions and structural parameters [4]. The temperature of the PV module is affected by the module material compositions, mounting structure and the environmental conditions [5,6]. Multiple heat sources are physically contributing to the increment of the module temperature [5]. The first is the incoming short-wave solar irradiance, where only up to 20% will be converted to electrical energy, and the rest will be converted to thermal energy [4,7]. The second heat source is the long-wave infrared radiation. Accurate temperature prediction is not only needed for a precise prediction of the output power, but is also essential for estimating lifetime and quantifying the degradation of PV modules [7–10].

The heat generated in the PV module is conducted through the stacked layers of the PV module to the external surfaces (front and back surface). Radiation, forced convection and free convection heat transfer mechanisms are involved in dissipating the generated thermal energy from the surfaces to the surrounding environment. Therefore, a robust PV thermal modelling is required to estimate the operating temperature of the PV module under the given environmental, physical and structural conditions. These conditions are represented by physical parameters, which act as an input for the model.

The main objective of this work is to propose a novel thermal model to estimate the PV module temperatures at three different planes: The semiconductor p-n junction (electronic junction temperature), the front and the back surface of the PV module. The proposed model is constructed by new combination of effective sub-models found in the literature and including a novel solution for considering the effect of the module tilt angle on the forced convection heat transfer mechanism.

#### **2. Thermal Modelling General Considerations**

This section discusses the main environmental, physical and structural parameters that determine the thermal behaviour of a PV module.

#### *2.1. Physical Structures of Pv Modules*

An accurate description of the PV module is fundamental to achieve precise estimation for the operating temperature as well as its profile through different layers. Although the photovoltaic technologies are advancing rapidly with higher efficiency and lower cost, the basic solar module physical structure has not changed much over the years [11]. Figure 1 shows the basic structure of a typical PV module.

**Figure 1.** Schematic structure of a basic photovoltaic (PV) module.

The active semiconductor layer is consisting of several photovoltaic cells interconnected in series and parallel depending on the required output current and voltage levels.The active layer is encapsulated between two layers of, as a most often used material, ethylene-vinyl acetate (EVA) to bind the PV cells to the top and bottom layers and provide moisture resistance and electrical insulation [6,12]. Fundamentally, the glass layer is tempered (to increase the mechanical strength of the module), highly transparent, low iron content and has a textured upper-surface (to reduce the solar irradiance reflection and absorption losses). The back layer is usually made of tedlar polymer that is

functioning as irradiance blocker and also providing moisture resistance [13]. Anti-reflection coating (ARC) layer is typically added to the PV layer for efficient light trapping (not shown in Figure 1) [14]. The active semiconductor layer may consist of materials like mono-crystalline, polycrystalline or amorphous silicon.

#### *2.2. Parameters That Affect the Pv Module Thermal Behaviour*

A well-known fact is that the temperature has a direct effect on the output power of the PV module. The maximum output power is decreased by 0.3 to 0.5% per Kelvin of temperature increase [15,16]. This is because the open-circuit voltage decreases significantly with increasing temperature while the short-circuit current increases only slightly [17]. However, several parameters affect the PV module temperature. These parameters have a different impact on the temperature value; therefore, some of them are essential to be considered when constructing the thermal model. Following is a list of these parameters [4,5,13,18–23].


Some of these parameters are strongly influencing the module temperature; however, other parameters effect the thermal properties of the module only to a smaller extent [24]. For example, the module temperature is highly sensitive to the wind speed and much less to the wind direction [25,26]. Some of these parameters are not easy to be included in a general approach for estimating the module temperature since the module thermal behaviour is changing for different technologies [17].

#### **3. Thermal Modelling Concepts**

This section will review different thermal modelling techniques and focuses mainly on the energy balance and heat transfer mechanisms.

#### *3.1. Classification of Pv Modules Thermal Modelling Concepts*

The wide range of parameters that affect the PV module temperature (material and environmental parameters) as well as different heat transfer mechanisms that take place through the module or on its surfaces give rise to the need of complex models for estimating the junction temperature. However, for commercial products, the manufacturers do not provide all of the required information. Generally speaking, the module temperature is a dynamic, nonlinear and implicit function incorporating the controlling parameters [9]. Factors like the required level of the accuracy, details of the temperature changing profile and the model complexity produce different types of modelling approaches. Many researchers treated temperature variation as a static function; hence, it is abruptly changing to reach a steady state. That is, neglecting the material thermal capacity effect and discarding the lag in temperature variation with respect to one or more of the affecting parameters [6]. Based on this concept, the main classification of the PV temperature modelling is whether it is a static (steady-state) [12,15–17,22,25,27,28] or dynamic model [2,5–7,18,29–33]. Although the static model

requires lower computational cost, its accuracy level could be affected in case of rapid changing of the controlling parameters. Temperature requires time between 4 and 10 minutes to reach the steady-state from its initial value, depending on the difference between the initial and the final temperatures and the PV module technology [5,34]. When the model input parameters are available with a frequency below this range, the dynamic model will be applicable for an accurate evaluation of the temperature [12].

The various existing thermal models in the literature, which are different in accuracy and complexity, can be grouped depending on their nature to be represented by the following.


The latter approach, recently, attracts the researchers focus and interest because of its applicability and high level of accuracy for estimating the module temperature. However, different researchers consider different methodologies when building up their thermal models. Based on this approach, this paper aims to propose a new thermal model. Sections 3.2 and 4 will discuss its physical translation and review the existing methods in the literature, respectively.

#### *3.2. Energy Balance and Heat Transfer Mechanisms*

It is a well-known fact that electronic junction temperature is not accessible from outside and cannot be directly measured using normal methods. Instead, models are used to estimate its value. One of the widely used methods considers the PV module as a single block of material and employ a single thermal heat balance equation (HBE), in which the absorbed energy (*qabsorbed*) should equal to the sum of the converted (*qconverted*) and the lost (*qlost*) energies.

$$q\_{absorbed} = q\_{convted} + q\_{lost} \tag{1}$$

The absorbed energy results from collecting the irradiance by the front surface of the PV module represent the overall input energy for the PV system. The converted energy includes the output produced electrical energy and the heat energy generated within the PV module. The last term in Equation (1) is related to the heat losses to the surrounding environment by different heat transfer mechanisms. The heat losses can be classified in two groups. The first group is mainly driven by the temperature difference between the module and its surrounding environment. The second group involves different effects such as joule heat in the wire contacts, diode losses and dirt accumulation. Considering all types of heat loss mechanisms gives rise to a complex modelling design that requires various parameters, which are related to the material properties and also the surrounding environment. Such a detailed model is impractical to be used for commercial products. Therefore, some of these losses are neglected due to their minor effects [9]. These minor losses include the energy initiated due to partial shading, low irradiance, dirt accumulation, joule heat through the wire contacts and diodes losses. Conduction heat transfer between the PV module and the holding structure is also neglected because of the small contact area and relatively small temperature difference [3,13,41].

Each of the heat balance equation terms (including their different components) need to be modelled to estimate their values individually because their effects cannot be directly measured [7,29]. Such an analysis should be based on the instantaneous temperature of the PV module. Therefore, an iterative process is needed. Conduction (between the PV module layers), convection and radiation (from front and back surfaces) are the three main heat transfer mechanisms that have to be evaluated to calculate the amount of losses.

Even in case it is not explicitly mentioned, the majority of the existing models share common assumptions, which are listed as follows [3,4,6–9,18,22,31,34,41].


Some researchers are going further in simplifying their models by eliminating other effects or parameters as a result of dealing with specific environmental conditions, materials properties or mounting structures. Table 1 shows some of these simplifications and assumptions.


**Table 1.** Special case assumptions found in the literature.

#### **Table 1.** *Cont*.


#### **4. Reviewing the Existing Sub-Models**

As previously mentioned, each of the HBE terms (including their different components) need to be modelled and individually estimated. This section is dedicated to briefly discussing each term of the HBE with scanning the literature to review the typical methods adopted to estimate their values.

#### *4.1. Absorbed Energy*

The absorbed energy represents the amount of energy received by the PV module due to the total captured short wave irradiance. Different parameters are affecting the amount of absorbed energy, such as [2,4,6,7] the following.


A widely used equation to determine the short wave absorbed energy given as [2,3,29–33]

$$
\mathfrak{q}\_{absobed} = \mathfrak{a} \cdot \Phi \cdot A,\tag{2}
$$

where *α* is the absorptivity of the front surface of the PV module, Φ is the total received irradiance and *A* is the surface area.

#### *4.2. Converted Energy*

The energy is converted into two forms: electrical energy (*qelec*) and thermal energy (*qtherm*).

$$q\_{convted} = q\_{therm} + q\_{clcc} \,\text{.}\tag{3}$$

To determine the amount of produced electrical energy, the values of the current and voltage at the maximum power point (*Im*, *Vm*) are required. Therefore, the fill factor (*FF*) and the efficiency of the PV module (*η*) play a major role as shown below [3,13],

$$q\_{elcc} = I\_m V\_m = (FF) I\_{sc} V\_{oc} = \eta \,\tau q\_{absorb} \,\text{s} \tag{4}$$

where *τ* is the front layer transmittance, and *Isc* and *Voc* are the short-circuit current and open-circuit voltage, respectively. Many authors, for the sake of a higher level of accuracy, tend to consider the environmental effects on the electrical performance of the PV module. Therefore, they include a dedicated electrical model for estimating the instantaneous value of the generated electric power [19,40]. These details are out of the scope of this paper.

The portion of the absorbed energy not converted to electrical energy is converted to heat, causing higher PV module temperature. With time, the thermal energy will be lost to the surrounding environment, mainly due to the temperature difference. However, this process requires some time before reaching a steady-state depending on the thermal properties of the PV module, represented

by its thermal capacity and resistivity. In case of temperature evaluation is required within small periods, as described in Section 3.1, then the dynamic analysis is required to include different layers' thermal capacity. The module heat capacity (*Cmodule*) is determined as the sum of each layer's capacity [2,13,29–31] from the following formula,

$$\mathbb{C}\_{modul\varepsilon} = \sum\_{i=1}^{n} A \cdot d\_i \cdot \rho\_i \cdot \mathbb{c}\_{i\prime} \tag{5}$$

where *n* is the number of PV module physical layers and *i* is the layer index. Moreover, in Equation (5) we see, for each layer, *A* is the area, *d* is the layer thickness, *ρ* is the material density and *c* is the specific heat.

Another modelling approach adopts the concept of assuming that the temperature is abruptly following the changes in the absorbed energy. These methods are applicable in case the module temperature and its output power is required to be estimated with time resolution large enough to reach a steady thermal state, higher than its thermal time constant.

#### *4.3. Heat Transfer Mechanisms*

As indicated previously, different heat transfer mechanisms are involved in this context, including conduction (within the PV module), radiation and convection. The rest of this section presents a brief description of each one of these mechanisms.

#### 4.3.1. Conduction Heat Transfer Mechanism

Typically, conduction is only considered between the structural layers of the PV module. Conduction to the holding structure is neglected due to the small contact area between the module and the holding structure and the low-temperature difference. The conduction heat transfer between the different layers is analysed based on the thermal resistivity and the thermal capacity of each layer of the PV module [9]. In such models, the HBE is derived for each layer [18].

#### 4.3.2. Convection Heat Transfer Mechanism

Convection is a heat transfer mechanism between the surfaces of the PV module and the surrounding air based on Newton's law of cooling [43]. It is modelled by the corresponding heat transfer coefficients (*hc*). The amount of heat convection per unit area (*qconv*) is evaluated using the following equation:

$$\mathfrak{q}\_{conv} = -\mathfrak{h}\_{\mathfrak{c}} \cdot \mathcal{A} \cdot (T\_{module} - T\_{ambient}) \, . \tag{6}$$

where *Tmodule* and *Tambient* are module and ambient temperatures, respectively. The convection heat transfer involves two mechanisms—the forced convection mechanism and the free convection mechanism—which are characterised by forced convection coefficient (*hc*, *f orced*) and the free convection coefficient (*hc*, *f ree*), respectively. Different researchers deal with their overall effect to be substituted in Equation (6) in different ways, as shown in Table 2.

**Table 2.** Determining the overall convection coefficient.


The significance of both types of convection is differs under different environmental conditions [5,30]. The authors of [33] considered only forced convection for the front surface of the PV module and only free convection for the back surface. Other authors consider only free convection for both surfaces [42]. However, most of the recent literature work that aims for high accuracy is incorporating both free and forced mechanisms [2,6,31]. Modelling and estimating the value of each of the heat transfer coefficients is performed using various techniques [2,6]. Tables 3 and 4 summarise the well-known equations for estimating the free and forced convection, respectively. The following points are common between the different expressions listed in both tables. If any sub-model in the mentioned table use a different expression or parameter definition it will be explicitly mentioned.





**Table 4.** Forced convection equations.

#### 4.3.3. Radiation Heat Transfer Mechanism

The heat exchange by radiation heat transfer mechanism involves the long-wave irradiance [9]. The amount of radiative energy per unit time per unit area (*qrad*) is determined based on the Stefan–Boltzmann law as follows

$$q\_{rad} = \epsilon \cdot F \cdot \sigma \cdot (T\_{ob}^4 - T\_{sur}^4)\_\prime \tag{7}$$

where *σ* is the Stefan–Boltzmann constant, *Tob* is the radiating object temperature, *Tsur* is the surrounding temperature,  is the emissivity of a surface and *F* is the view factor. Table 5 summarises various existing methods from the literature for estimating the amount of thermal radiation. The following notes are common between the expression listed in Table 5 unless explicitly defined again: If any sub-model in Table 5 uses a different expression or parameter definition it will be explicitly mentioned.


$$10. \quad F\_{mf\xi\eta} = \frac{(1+\cos(\beta\_{surface}))}{2}, \; F\_{mf\xi\eta} = \frac{(1-\cos(\beta\_{surface}))}{2}, \; F\_{m\text{bdy}} = \frac{(1+\cos(\pi-\beta\_{surface}))}{2}, \; F\_{m\text{bdy}} = \frac{(1-\cos(\pi-\beta\_{surface}))}{2}, \; F\_{m\text{bdy}} = \frac{(1-\cos(\pi-\beta\_{surface}))}{2}, \; F\_{m\text{bdy}} = \frac{(1-\cos(\pi-\beta\_{surface}))}{2}$$

11. The ground temperature (*Tground*) is assumed to be equal to the ambient temperature (*Tambient*). 12. Some authors define the radiative heat transfer coefficient (*hrad*) as: *hrad* = *<sup>σ</sup>* · *Fxy* ·  *<sup>x</sup>* · (*T*<sup>2</sup> *<sup>x</sup>* + *T*2 *<sup>y</sup>* )(*Tx* + *Ty*); therefore, the heat energy per unit time per unit area is *qrad* = *hrad*(*Tx* − *Ty*)

**Table 5.** Radiation thermal energy losses equations.


#### **5. Detailed Construction of Thermal Model**

Constructing the thermal model in this research work is based on the approach of treating the PV module as a single block of material and employing the HBE, including different heat transfer mechanisms. Figure 2 shows the thermal behaviour of a PV module that described by the HBE.

The model will provide the PV module junction temperature as well as the temperature difference to both surfaces. Therefore, both front and back surface temperatures will be estimated. This result will be useful in the validation phase because then we can compare the back surface estimated temperature to the measured one by a thermometer attached to the backside of the PV module. The model was constructed based on different, already existing models from the literature (see Section 4). However, the new model was constructed by incorporating sub-models of different existing models in a new and unique way to yield a new model with improved accuracy. For this, different sub-models of the above

described models were combined and tested, and the combination with the best accuracy was chosen as the mode for this paper. The total absorbed energy is consisting of two components and given as

$$q\_{absorbcd} = q\_1 + q\_{2"\prime} \tag{8}$$

$$q\_1 = \mathfrak{a}\_{f\mathfrak{F}} \cdot A \cdot \Phi,\tag{9}$$

$$q\_2 = \tau\_{f\underline{\chi}} \cdot \mathfrak{a}\_{PV} \cdot A \cdot \Phi \cdot (1 - \eta),\tag{10}$$

where *q*<sup>1</sup> is the rate of thermal energy absorbed by the tempered glass layer, *q*<sup>2</sup> is the energy absorbed by the semiconductor layer, *τf g* is the transmittance of the glass layer, *αf g* and *αPV* is the absorptivity of the front glass and semiconductor layers, respectively.

**Figure 2.** Thermal condition of the PV module.

In this paper, we consider a static model, that is, we assume that the module output power predicted by the proposed model is required only with a time resolution that enables the the temperature to reach a steady state. Therefore, the HBE component which is related to the material thermal capacity is neglected and the converted energy is limited only to the electrical produced component, which is given as

$$
\eta\_{\text{convected}} = \pi\_{f\text{\%}} \cdot \mathfrak{a}\_{PV} \cdot A \cdot \Phi \cdot \eta\_{\text{\%}} \tag{11}
$$

The radiation heat losses trough both front and back surfaces are calculated using the following expressions, respectively [8].

$$q\_{\rm rnd-front} = \sigma \cdot F\_{\rm mfsky} \cdot \varepsilon\_{\rm ffront} \cdot A \cdot (T\_{fs}^4 - T\_{sky}^4) + \sigma \cdot F\_{\rm mfyr} \cdot \varepsilon\_{\rm ffront} \cdot A \cdot (T\_{fs}^4 - T\_{g\rm rund}^4). \tag{12}$$

$$\boldsymbol{q}\_{\rm rad-back} = \boldsymbol{\sigma} \cdot \boldsymbol{F}\_{\rm mbsky} \cdot \boldsymbol{\varepsilon}\_{\rm back} \cdot \boldsymbol{A} \cdot \left(\boldsymbol{T}\_{\rm bs}^{4} - \boldsymbol{T}\_{\rm sky}^{4}\right) + \boldsymbol{\sigma} \cdot \boldsymbol{F}\_{\rm mfgr} \cdot \boldsymbol{\varepsilon}\_{\rm back} \cdot \boldsymbol{A} \cdot \left(\boldsymbol{T}\_{\rm bs}^{4} - \boldsymbol{T}\_{\rm ground}^{4}\right). \tag{13}$$

The view factors are calculated using the expressions given in Section 4.3.3 (point number 10). The sky temperature is, *Tsky* = (*Tambien* − *δT*) for clear sky condition where *δT* = 20*K*, *Tsky* = *Tambient* for overcast condition [29]. The ground temperature is assumed to be equal to the ambient temperature.

Both free and forced convection mechanisms are considered in creating this thermal model. Their overall effect is calculated by combining their effect using Equation T 1.2 from Table 1. For both mechanisms, we treat the front and back surface individually because the properties of the film layer at the boundary of each one are different. The free convection heat loss is determined using Equations (14) to (18), in which the subscript *x* refers to the front (*f*) or back (*b*) surface; therefore, during implementation, the equation has to be rewritten for each surface.

$$Gr\_{\mathbf{x}} = \frac{\mathbf{g} \cdot \rho\_{air,\mathbf{x}}^2 \cdot \cos(\theta) \cdot \beta\_{\mathbf{x}} \cdot \Delta T \cdot L\_c^3}{\mu\_{air,\mathbf{x}}^2},\tag{14}$$

$$Ra\_x = Gr\_x \cdot Pr\_{x\prime} \tag{15}$$

$$Nu\_{free,f} = 0.27 \cdot Ra\_f^{0.25},\tag{16}$$

$$Nu\_{free,b} = 0.54 \cdot Ra\_b^{0.25},\tag{17}$$

$$h\_{free,x} = Nu\_{free,x} \frac{k\_{air\_x}}{L\_c},\tag{18}$$

For estimating the forced convection coefficients (for both surfaces), we modify the expressions used by Kayhan [13], given as

$$Re\_{\chi} = \frac{V\_w \cdot L\_c \cdot \rho\_{air\_x}}{\mu\_x},\tag{19}$$

$$h\_{forward,x} = \frac{k\_{air\_x}}{L\_c} \cdot (2 + 0.41 \cdot Re\_x) \cdot H\_x. \tag{20}$$

The introduced modification can be seen in Equation (20) where we added a novel coefficient (*H*), which is defined as the forced convection adjustment coefficient for both front and back surfaces. This coefficient modulates the relationship between the tilt angle and the wind effect on the amount of heat loss by forced convection. This coefficient is calculated as

$$H\_f = (1 + \cos(\beta\_{surface})) / m\_\prime \tag{21}$$

$$H\_b = \left(1 - \cos(\beta\_{surface})\right) / m\_\prime \tag{22}$$

where *m* is an empirical factor estimated with the help of measurement data. The following points explain the fundamental concept behind the coefficient *H* by considering PV module mounted with different tilt angles and assuming that the value of *m* is equal to 2.

	- **–** The front surface will undergo a maximum effect of the wind that will sweep the hot air away. This fact is ensured by Equation (21), which will be evaluated to 1. That is, the expression used for calculating the heat loss by forced convection will not be disturbed by the tilt angle.
	- **–** For a typical PV system, there are two facts: First, the system is consisting of many PV modules with a defined density. Second, PV modules are mounted close to the ground in case of flat and small tilt angles. Therefore, the wind will have no considerable effect on the back surface of the PV module. Equation (22) will be evaluated to zero for a flat surface; that is, the forced convection heat loss from the back surface will be neglected in this case.
	- **–** This implies that the wind will face resistance from the front surface of the PV module compared to the case of flat mounting. Therefore, reducing the ability to sweep out the hot air away from the surface. Equation (21) will be evaluated to 0.75. That is, the tilt angle will be a reason for reducing the amount of heat loss by forced convection.
	- **–** The lower surface will be facing the wind, which was not the case for a flat-mounted module. Equation (22) will be evaluated to 0.25. Thus, heat loss by forced convection is much higher compared to flat or small tilt angles. However, it is still lower compared to the front surface.

Therefore, we consider that the PV module tilt angle will control the amount of heat losses from both surfaced. For tilt angles between 0◦ and 90◦, the front surface heat loss by forced convection is higher compared to the back surface. Increasing the tilt angle (within this range) produces lower forced convection heat loss from the front surface and higher from the back surface.

We claim that *m* is a factor that affects the relationship between the tilt angle and the heat loss by forced convection by involving other installation parameters. These parameters include the PV modules installation density, the elevation from the ground and the thickness at the module edges at which the wind speed drops to zero. From experience, we found that this empirical factor has a value in the range between 1.5 and 2. Therefore, in this paper, we consider scanning this range with a specific resolution and running the model for each value. By increasing the resolution more, the value of m can be determined more accurately. Based on experience, We consider 0.1 as a resolution value considering a trade-off between the computational cost and accuracy. Therefore, we consider running the thermal model six times after which we decide what is the best value for *m* (by monitoring the error indication parameters) to be fixed for the module under investigation.

Once we have the value of the empirical factor *m*, we substitute it in Equations (21) and (22) to determine the forced convection adjustment coefficient for the front and back surface, respectively. For each surface, the overall convection coefficient and the corresponding rate of convection thermal energy losses can be calculated using the Equations T 1.2 from Tables 1 and 6.


**Table 6.** PV modules technical specifications, physical and installation parameters.

The proposed model also considers the following points.


$$
\Delta T\_f = q\_{f\text{front-total}} \cdot \mathcal{R}\_f.\tag{23}
$$

$$
\Delta T\_b = q\_{\text{back-total}} \cdot R\_b. \tag{24}
$$

where *q f ront*-*total* and *qback*-*total* are the total thermal losses from the front and back side of the PV module, respectively. *Rf* , and *Rb* are the thermal resistivity of the PV module, between the front and back surfaces and the active layer, respectively.

#### **6. Results and Discussion**

This model has been validated using a polycrystalline and an amorphous PV module. The validation data of the polycrystalline module has been taken from a reference [9]. Our measurement system has been used to collect the amorphous module validation data. This measurement system provides data such as PV module back surface temperature, full I–V curve, global solar irradiance, ambient temperature and wind speed. The global irradiance was measured by Delta Ohm LP RAD 03 piranometer that detect solar irradiance ranging from 0 to 2000 W/m2. The temperature of the PV module is recorded using a circuit board attached to the back side of the module with a thermal conductive adhesive. The circuit includes a temperature sensor IC (MAX6603ATB+T). The accuracy of the circuit is ±0.8 ◦C at +25 ◦C . The ambient temperature is measured using PT100 resistance thermometers. Wind speed data is taken from a wind turbine FD2.5-300 which is capable of measuring range of 0 to 60 m/s with an accuracy of ±0.3 m/s.

Table 6 shows the technical specifications and physical parameters of both modules, which are required for running the model.

To verify the model, we use measurement data including irradiances, ambient temperatures and wind speed that have been recorded for two different full days for each module. For the amorphous module, the two days were the 5th and the 11th of October. For the polycrystalline module, the two days were the 3rd and the 26th of July. The main difference between the two days of each module is the wind speed. The average wind speed is 6.14 m/s on the 26th and 2.16 m/s on the 3rd of July, while it is 2.05 m/s on the 5th and 0.77 m/s on the 11th of October. As mentioned in Table 6, each module has a different tilt angle. Based on our experience, we claim that the module tilt angle has a significant effect on the value of the thermal energy losses by forced convection. As described in the model introduced in Section 5, we introduced a forced convection adjustment coefficient (*H*) and its empirical factor (*m*). In this regard, we report that the value of *m* typically takes a value between 1.5 and 2. We calculate this factor by scanning its range and running the model with a step of 0.1.

For evaluating the proposed model and validating the results we use two error indication parameters: one is the root mean square error (*RMSE*) and the other is the correlation coefficient (*r*). These parameters are used, as shown in Table 7, to compare the module's back surface temperature for each day (entire day measurement) of the two modules with the estimated values using the proposed model for different values of *m*.


**Table 7.** Error quantifying parameters corresponding different *m* values.

Based on the results shown in Table 7, we chose a value of *m* = 1.6 for the polycrystalline module and *m* = 1.8 for the amorphous module to be used in this study, as these values give the best results. Figure 3 shows both the measured and the estimated PV module back surface temperature for the polycrystalline module, for the two investigated days: 3rd and 26th of July. Each curve includes 120 points as a result of recording the temperatures every 5 min between 9 am and 7 pm.

**Figure 3.** Measured and estimated polycrystalline module backside temperature. (**a**) 3rd July. (**b**) 26th July.

Figure 4 shows both the measured and the estimated PV module back surface temperature for the amorphous module, for the two investigated days, 5th of October (49 points between 9 am and 1 pm) and 11th of October (71 points between 9 am and 3 pm).

**Figure 4.** Measured and estimated amorphous module backside temperature. (**a**) 5th October. (**b**) 11th October.

Figure 5 shows the absolute value of the temperature difference between the measured and estimated values of the back surface temperature using the proposed model for both modules.

**Figure 5.** Absolute values of the temperature difference between the measured and the estimated values. (**a**) Polycrystalline 3rd October. (**b**) Amorphous 5th October.

Figure 6 shows the relationship between the junction temperature and both surfaces temperatures. It illustrates how these temperature differences are changing with the time of day. Therefore, it will provide a clear picture of the temperature profile across the PV module. The temperature difference to the front surface (Δ*Tf*) is ranging from 0.7 to 2.5, with an average value of 1.86 ◦C. The temperature difference to the back surface is between 1.67 and 0.22 with 0.96 ◦C as an average value.

**Figure 6.** Temperature difference between the electronic junction and the polycrystalline module (measurements used for 3rd July) front and back surfaces. (**a**) Difference to the front surface. (**b**) Difference to the back surface.

Figure 7 highlight the importance of the novel coefficient and the consideration of the tilt angle in the forced convection, introduced in this paper. The same figure also shows the effect of neglecting the wind in the thermal model. Figure 7a compares the measured back surface temperature (blue colour) to the estimated temperature with the coefficient *H* (red colour), without the coefficient *H* (black colour), and without wind effect (green colour), for the polycrystalline module (measurements used for 3rd July). Figure 7b shows the absolute error to compare different situations.

**Figure 7.** Evaluating the effect of including H in the PV thermal model (the model applied for the polycrystalline module, measurements used for 3rd July). (**a**) Compares back surface temperatures. (**b**) Compares the absolute error.

According to the results shown above, we summarise the discussion with the following points.


considered for this purpose, which includes an empirical factor (*m*). We calculate this factor by scanning its range as discussed above.



**Table 8.** Thermal model accuracy achieved for the two modules.

• For the same module, typically, Δ*Tf* > Δ*Tb*. Therefore, the top surface temperature is slightly lower than the backside temperature due to, relatively, more effective heat transfer mechanisms.

The rest of this section is dedicated to highlighting the scientific improvement that has been introduced in this work. We made a comparison between the proposed model and the results reported by different thermal models from the recent and most accurate literature using various error quantifying parameters. In this comparison, we will refer to the best results reported by the references and compare it to our model using the measurement recorded on the 3rd of July for the polycrystalline module.

• Several thermal models found in the literature use the root mean square error (*RMSE*) as an error quantifying parameter to validate the results. The models presented in [16,18,23,28,31,32,38,42,45,46] have reported *RMSE* values ranging between 4.9 and 1.1 ◦C. However, in our presented model we report a value of 0.927 ◦C.


#### **7. Conclusions**

In this paper, we introduced a novel thermal model to predict the PV electronic junction, front surface and back surface temperatures. The model has been verified using on-site measurement for two modules made with two different technology and mounted with different tilt angles. The measurements have been recorded for each module for two different days in which the average wind speed is the main difference. A novel concept has been introduced to consider the module tilt angle effect on the amount of heat loss by forced convection. The result presented in Table 8 shows that the model is able to estimate the PV module temperature with high accuracy represented by *RMSE* = 0.927 ◦C and *r* = 0.997 as the best results for both modules under the considered environmental conditions. From the same table, calculating the average of these parameters give *RMSE* = 1.1 ◦C and *r* = 0.987, which are comparable, but slightly better compared to the best results review from the literature. During the model validation phase, we found that obtaining a high level of accuracy is only possible by including a novel forced convection adjustment coefficient (H). Using the same proposed model without this coefficient gives *RMSE* = 3.02 ◦C when applying the model to estimate the junction temperature of the polycrystalline module (3rd July). The back surface temperature absolute differences between the measured and the estimated values have been calculated for both modules, which give an average value below 1 ◦C for both studied modules, considering the two days measurements for both. The following points summarise this work's conclusion.


It worth highlighting at this point that the novelty of the proposed paper is realized by introducing the new forced convection adjustment coefficient, and by reviewing the most often used existing expressions for calculating the different forms of the PV module heat losses and the related parameters and finding the proper combination of these expressions to be employed in the presented model.

**Funding:** The research reported in this paper was supported by the BME Nanotechnology and Materials Science TKP2020 IE grant of NKFIH Hungary (BME IE-NAT TKP2020), by the Stipendium Hungaricum Scholarship

**Author Contributions:** Individual author contributions are as follows: Conceptualization, A.K.A and G.B.; software, A.K.A. and G.B.; validation, A.K.A., G.B. and B.P.; writing—original draft preparation, A.K.A.; writing—review and editing, A.K.A., G.B. and B.P. All authors have read and agreed to the published version of the manuscript.

Programme, the grant EFOP-3.6.1-16-2016-00014 and by the Science Excellence Program at BME under the grant agreement NKFIH-849-8/2019 of the Hungarian National Research, Development and Innovation Office.

**Conflicts of Interest:** The authors declare no conflicts of interest.

#### **List of Symbols and Abbreviations**


#### **Greek letters**


#### **References**


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