A Novel High Accuracy PV Cell Model Including Self Heating and Parameter Variation

This paper proposes a novel model for a PV cell with parameters variance dependency on temperature and irradiance included. The model relies on commercial available data, calculates the cell parameters for standard conditions and then extrapolates them for the whole operating range. An up-to-date review of the PV modeling is also included with series and parallel parasitic resistance values and dependencies discussed. The parameters variance is analyzed and included in the proposed PV model, where the self-heating phenomenon is also considered. Each parameter variance is compared to the results from different authors. The model includes only standard components and can be run on any SPICE-based simulator. Unlike other approaches that consider the internal temperature as a parameter, our proposal relies on air temperature as an input and computes the actual internal temperature accordingly. Finally, the model is validated via experiments and comparisons to similar approaches are provided.


Introduction
PV cells have been extensively studied in the last decades as solar energy is more and more accepted as a viable alternative to traditional energy sources. Rauschenbach [1] is a reference work, addressing the principles of PV energy conversion. Patel [2] covers a wider area, dealing both with wind and solar energy. The second edition of the "Handbook of Photovoltaic Science and Engineering" [3] by Luque and Hegedus is another reference book, providing rich details of all aspects regarding solar energy. Chapter 18 deals with solar cells and modules measurements and models. Aparicio et al. [4] use free software for modeling PV cells, while Sumathi et al. [5] and Khatib and Elmenreich [6] explain how to model a PV cell or array using MATLAB. Honsberg and Bowden [7] offer an online book with examples, intended for researchers and students, while in another online resource, Van Zeghbroeck [8] inspects in detail the main information of the semiconductors theory and devices. Modeling the PV behavior is useful for system design, planning, research and training. The goal of this work is to develop an accurate model for a PV cell, expandable to a whole module, using affordable tools and taking into account parameters variations. LTSpice [9] was chosen as the simulation tool due to its free cost and wide acceptance, Visual Studio Express [10], also a free tool, was used for parameters estimation and solution validation. Finally, S-Math Studio [11] was selected for the trial and error different evaluations. The solution implies a reasonable computing power and provides fast convergence. The model itself is portable, as it uses only standard components and is also vendor independent. The input data is usually provided from the manufacturer's datasheet or can be obtained via experiments.
Unlike other approaches, the model uses the ambient/air temperature and based on the irradiance it calculates the internal (silicon) temperature and provides the actual values of the parameters. Referring to Figure 1, according to Kirchhoff's current law (KCL), one can write: where is the diode reverse saturation current, is the electron charge, is the Boltzmann constant, is the actual silicon temperature and is the ideality factor of the diode. Current linearly depends on irradiation and temperature [15,18]: At the maximum power point, using (1), the maximum power can be derived [22]: Even in (1) and (3) is considered equal to , a more accurate formula for is [22]: A good overview of the PV cell performance can be found in [12], where an empirical formula for the fill factor is introduced, considering the single diode model:

Materials, Methods and Equipment
For our experiments we have chosen a high efficiency low cost monocrystalline Silicon PV solar cell, unmounted in panels [39]. The datasheet of the PV cell offers a limited amount of data, summarized in Table 1.  Referring to Figure 1, according to Kirchhoff's current law (KCL), one can write: where I o1 is the diode reverse saturation current, q is the electron charge, k is the Boltzmann constant, T is the actual silicon temperature and a 1 is the ideality factor of the diode. Current I ph linearly depends on irradiation and temperature [15,18]: At the maximum power point, using (1), the maximum power P mp can be derived [22]: Even in (1) and (3) I ph is considered equal to I sc , a more accurate formula for I ph is [22]: A good overview of the PV cell performance can be found in [12], where an empirical formula for the fill factor FF is introduced, considering the single diode model: FF = qV oc a 1 n s kT − ln 0.72 + qV oc a 1 n s kT qV oc a 1 n s kT + 1 (5)

Materials, Methods and Equipment
For our experiments we have chosen a high efficiency low cost monocrystalline Silicon PV solar cell, unmounted in panels [39]. The datasheet of the PV cell offers a limited amount of data, summarized in Table 1. It is important to note that V mp = (0.75, . . . , 0.9)V oc for any solar cell. This is a good starting point for any simulation or MPPT algorithm implementation.
The data from the datasheet is confusing, as: 1.
The claimed maximum power (5.21 W) differs from V mp I mp = 5.01 W. This latter value will be considered subsequently.

2.
The claimed fill factor (81.90%) differs from the standard definition FF = V mp I mp V oc,re f I oc,re f = 77.82%.
The irradiance was measured with a Klipp and Zonen SHP1 pyrheliometer with integrated temperature sensor for temperature compensation. The internal silicon temperature was determined with a FLIR E8 infrared camera and a PT1000 temperature sensor on the rear of the PV cell. In order to obtain reliable data, the PT1000 temperature sensor was glued with high thermally conductive adhesive to the backside metal coating of the PV cell. The ambient temperature was measured using the National Instruments NI USB T01 interface. Due to the extremely low internal series resistance R s , several series cells were carefully wired and a Kelvin connection had to be used for voltage measurements. The measurements were performed under real life conditions, when the solar irradiance was maximum with the PV cells oriented toward the sun on 45 degree inclined support. The load was an ET Instrument ESL-Solar, configured in MPPT mode.
The accuracy of the irradiance measurement relies on the Klipp and Zonen SHP1 pyrheliometer with zero offset due to temperature change, temperature dependence of sensitivity below 0.5% from −30 to +60 • C and nonlinearity below 0.2% for up to 1000 W/m 2 irradiance. The accuracy of the temperature sensor for the ambient temperature and PV cell backside coating is 0.6 • C in the measurement range of 0-60 • C. The FLIR E8 infrared camera has an accuracy of 2 • C with thermal sensitivity below 0.06 • C. The accuracy of the FLIR infrared camera is lower than the measurements taken with the temperature sensors but offers the advantage of contactless measurement which is important for PV cell front side measurements. However, the existence of hot spots can be easily observed with this method. The electrical noise was minimized using the following digital busses for communication: Modbus for irradiance and PV cell back side temperature measurement, USB for ambient and PV cell front side temperature measurements.

Classical Model Solving
Several ways for solving the equations have been proposed. In [17] de Blas et al. assume the operation of a solar module at high irradiance levels and deduce the parameters accordingly. De Soto et al. [20] show how the values of parameters for the five-parameter model can be determined for four different cell technologies. The methods introduced in [22,26] are somehow similar, the parameters being precisely extracted by letting the resistance values converge to the actual value in repeated numerical loop computations. Ishaque and Salam [27] use the differential evolution for parameters extraction. Saloux et al. [28] introduce a model able to predict the short-circuit current, the open-circuit voltage and the maximum cell power. Tian et al. [30] concentrate on an extension of the PV cell model for modules and arrays, for monocrystalline and polycrystalline silicon, used for shading effect investigation under different operating conditions.
One of the difficulties is the implicit nature of Equation (1) regarding I. This has been addressed by various techniques, ranging from pure mathematical approaches (including Lambertian W-Function, [31]) to pure numerical solutions, mainly in MATLAB [5,6,16,35].
Later models take into account the parameters variation with temperature and irradiance. Radziemska and Klugmann [40] analyze the temperature influence on the I-V curves of the PV cell, while Tsuno et al. [41] include also the irradiance in the analysis. Singh et al. [42] also present the temperature influence, while Cuce and Bali [43] focus on parameters variation in humid climates. The Lambert W-function is again used by Ghani and Duke [44] to estimate the parasitic resistances of the PV cell. The correlation between the temperature and PV cell efficiency is addressed in [45,46]. Temperature and irradiance influence on the PV cell characteristics via experiments is presented in [47,48]. Araneo et al. [49] propose a model to predict the PV cell temperature based on date, time and geographical position. Temperature and irradiance influence are also investigated in two recent papers by Chander et al. [50] and Aller et al. [51]. However the most common approach considers the internal PV temperature as an independent parameter and plots the I-V family curves for different temperatures. This aspect will be covered in the subsequent sections. It has to be stressed out that the exponential nature of Equation (1) determines that a small variation in any of the terms involved in the exponential term to substantially modify the final result. This aspect will be addressed in Sections 5 and 6.
A VB.net application has been developed by the authors in order to numerically solve and compute the model parameters. The application can be downloaded from http://tess.upt.ro. Figure 2 depicts a print screen for the initial parameter passing (a) and the results (b).
These limits are important to set reliable ranges for the algorithm. The final results are listed in Table 2. In our case, using the values taken from the datasheet for n s = n p = 1, it results that: These limits are important to set reliable ranges for the algorithm. The final results are listed in Table 2.  [31] offer the R sh formula (with R s as an argument, considering n s = 1): For the above data, the R sh formula (8) yields a result of 59.43 Ω, compared to the actual value of 73.19 Ω.

Parameters Variation for Different Conditions
The parameters in Equations (1)-(4) are not constant over the environmental conditions, as I o , R s , R sh , a i , E g depend on temperature and irradiance. A brief review of these dependencies is provided bellow.

Diode Saturation Current-I o1
Phang et al. [13] show that if I pv is below 10 A, I o1 can be derived as in (10): I o1 in (10) yields a very good result of 1.3969 nA vs. 1.39427 nA obtained in Table 2. Gow and Manning [15] were among the first to claim that: The temperature dependence of this current is more detailed expressed by [16,20]: where V g is the bandgap voltage of the semiconductor (V g = 1.1, . . . , 1.3 V for Si at 25 • C).
I o1,re f can be derived from (1) at the reference temperature as: I o1,re f = I sc,re f exp qV oc,re f a 1 n s kT re f − 1 (13) According to Vilalva et al. [22], I o1,re f can be further improved: In subchapter 4.6.3 of [8], Van Zeghbroeck states an equation in which I o1,re f can be derived from, that can offer an alternate way to estimate I o1,re f : This proves to be not very accurate in our case, as with the values from Table 1, I o,re f from (15) results 0.085 nA, quite far from the actual value (1.39427 nA).

Band Gap Energy and Bandgap Voltage-E g , V g
Van Zeghbroeck [8] shows that the bandgap energy, E g , exhibits a small temperature dependence as in (16).
From (16), E g,re f = 1.121 eV for silicon cells at 25 • C. This is the value considered in Table 1.
In contrast, Kim et al. [23] define the variance for E g for silicon to be: Both (16) and (17) fit in the [1.1, . . . , 1.3] V interval specified when Equation (12) was introduced. In our approach shown in Figure 3, we adopted the Van Zeghbroeck proposal because it will finally lead to a more realistic value for a 1 and close to the linear approximation of E g against temperature suggested by Radziemska and Klugmann [40], which indicate a temperature coefficient According to Vilalva et al. [22], , can be further improved:

Band Gap Energy and Bandgap Voltage-,
Van Zeghbroeck [8] shows that the bandgap energy, , exhibits a small temperature dependence as in (16).
From (16), , = 1.121 eV for silicon cells at 25 °C. This is the value considered in Table 1.
In contrast, Kim et al. [23] define the variance for for silicon to be: Both (16) and (17) fit in the [1.1, …, 1.3] V interval specified when Equation (12) was introduced. In our approach shown in Figure 3, we adopted the Van Zeghbroeck proposal because it will finally lead to a more realistic value for and close to the linear approximation of against temperature suggested by Radziemska and Klugmann [40], which indicate a temperature coefficient d d ⁄ = −2.3 × 10 eV K ⁄ .

Series Resistance-R s
Honsberg and Bowden [7] show that R s does not influence V oc , but close to the open-circuit voltage, the I-V curve is affected by R s . An initial estimation for R s is to find the slope of the I-V curve at the open-circuit voltage point (18): In our case, R so = 11 mΩ (while R s = 3.8 mΩ, as it will later be shown). Cuce and Bali [43], Cuce et al. [47] and Singh et al. [42] claim that R s linearly decreases with the temperature. Obviously, reducing R s yields an increase in the output current.
A PV Cell model is also available in MATLAB Simscape [52]. It consists of the same circuit as in Figure 1, where the user can choose between: • An 8-parameter model, where Equation (1) describes the output current • A 5-parameter model that neglects D 2 in Figure 1 and the value of the shunt resistor is infinite.
Both models adjust the resistance values and current parameters as a function of temperature. Resistance R s is assumed to be given by (19): where k Rs is the temperature exponent for R s . k Rs is 0 by default and when modified has to be positive. Figure 4 summarizes all these above dependencies. In order to have the results in the same range, Cubas et al. [31] and Cuce et al. [47] results were scaled, and Equation (19) was re-written as in (20), interchanging T with T re f and k Rs was estimated as 4.9 for the best fit. A linear dependency is easy to implement, but might also lead to results not physically true (for example Cuce et al. [47] data lead to negative R s resistances for temperatures over 75 • C and so does Cubas et al. [31] over 97 • C).
Cuce and Bali [43], Cuce et al. [47] and Singh et al. [42] claim that linearly decreases with the temperature. Obviously, reducing yields an increase in the output current. A PV Cell model is also available in MATLAB Simscape [52]. It consists of the same circuit as in Figure 1, where the user can choose between: • An 8-parameter model, where Equation (1) describes the output current • A 5-parameter model that neglects in Figure 1 and the value of the shunt resistor is infinite.
Both models adjust the resistance values and current parameters as a function of temperature. Resistance is assumed to be given by (19): where is the temperature exponent for . is 0 by default and when modified has to be positive. Figure 4 summarizes all these above dependencies. In order to have the results in the same range, Cubas et al. [31] and Cuce et al. [47] results were scaled, and Equation (19) was re-written as in (20), interchanging with and was estimated as 4.9 for the best fit. A linear dependency is easy to implement, but might also lead to results not physically true (for example Cuce et al. [47] data lead to negative resistances for temperatures over 75 °C and so does Cubas et al. [31] over 97 °C).
where | | = | |. The linear law (21) was adopted for and we chose = −0.01 K , again for the best fit. The linear law (21) was adopted for R s and we chose α Rs = −0.01 K −1 , again for the best fit.
Honsberg and Bowden [7] and Jung and Ahmed [25] showed that the shunt resistance of a solar cell can be determined from the slope of the I-V curve close to the short-circuit point, yielding a fair approximation for R sh : From our experimental data, R sho = 73.18 Ω, very close to the accurate solution R sh = 73.19 Ω, as it will later be illustrated.
Cuce and Bali [43] and Cuce et al. [47] claim that the shunt resistance linearly decreases with temperature. They explain this decrease in terms of a combination of tunneling and trapping-detrapping of the carriers through the defect states in the space-charge region of the device. These defect states act either as recombination centers or as traps depending upon the relative capture cross sections of the electrons and holes for the defect. Temperature dependency for R sh is however more complicate.
R sh is again modeled in MATLAB Simscape like (23): where k Rsh is the temperature exponent for R sh . k Rsh is 0 by default and when modified has to be positive. Figure 5 summarizes all these R sh equations. In order to bring the results in the same range, Cubas et al. [31] and Cuce et al. [47] dependencies were scaled, and Equation (23) was re-written as in (24) interchanging T with T re f and k Rsh was estimated as 8 for best fit.
where k Rsh = |k Rsh |.  [7] and Jung and Ahmed [25] showed that the shunt resistance of a solar cell can be determined from the slope of the I-V curve close to the short-circuit point, yielding a fair approximation for : From our experimental data, = 73.18 Ω, very close to the accurate solution = 73.19 Ω, as it will later be illustrated. Cuce and Bali [43] and Cuce et al. [47] claim that the shunt resistance linearly decreases with temperature. They explain this decrease in terms of a combination of tunneling and trappingdetrapping of the carriers through the defect states in the space-charge region of the device. These defect states act either as recombination centers or as traps depending upon the relative capture cross sections of the electrons and holes for the defect. Temperature dependency for is however more complicate.
is again modeled in MATLAB Simscape like (23): where is the temperature exponent for . is 0 by default and when modified has to be positive. Figure 5 summarizes all these equations. In order to bring the results in the same range, Cubas et al. [31] and Cuce et al. [47] dependencies were scaled, and Equation (23) was re-written as in (24) interchanging with and was estimated as 8 for best fit.
where | | = | |. Although influence is small in the overall model, for an accurate modeling and especially for larger temperature ranges linear variation is not realistic. Therefore in our model described by (24) we chose = 8. Although R sh influence is small in the overall model, for an accurate modeling and especially for larger temperature ranges R sh linear variation is not realistic. Therefore in our model described by (24) we chose k Rsh = 8.

Ideality Diode Factor-a 1
Some authors consider the ideality factor as being constant over the operating temperature range and with a generic value for a 1 in the interval [1, 1.5] for every kind of cell [22,31]. Cuce et al. [29] propose a 1 = 1.2 for monocrystalline silicon cells, and a 1 = 1.3 for polycrystalline ones. Some studies indicate a linear decreasing with temperature [18]. Cubas et al. [31] say that "the lack of accuracy produced when considering the ideality factor as constant is generally reduced, given that variations of this parameter only affects the curvature of the I-V curve." This is arguable, as a 1 interferes in an exponential dependency and small variations of a 1 lead to significant changes in I o1, and finally in I. One might say that picking a random a 1 in the above specified range will be balanced by a different I o , so only the pair (I o , a 1 ) matters. However this approach is misleading, as it may induce impossible physical solutions.
Phang et al. [13] have the following proposal: De Blas et al. [17] suggest that: E. Saloux et al. [28] somehow simplify (26) as below: In the algorithm of Villalva [53], a different formula is introduced. Considering that for crystalline silicon E g = 1.8 J, V g becomes 1.1235 V and the following formula provides a good result (Formula (28) was adapted from [53], as the additional presence of the n s in the initial formula provided correct results only for n s = 1), thus eliminating a trial and error time consuming for the initial guess of a 1 : The results for a 1 are summarized in Table 3, with a very good correlation between (25), (26) and (28). This is the reason we have adopted the Villalva value of 1.2034. Xiao et al. [18] specify a linear decreases of the ideality factor with the temperature for the Shell ST40 module, ranging from 1.85 to 1.15, corresponding to 5 to 45 Celsius degree variance respectively. From the data plotted in their work, the following law can be adopted: Such approach must be taken with extreme care, as it is a common practice to operate often at temperatures higher than 48 • C, where (29) yields a 1 = 1 (or 0 at 100 • C) De Soto et al. [20] come with a different proposal: which has a wrong slope. For a proper variation T and T re f should be reversed as follows: Our experiments presented in Figure 6 yielded a different result, closer to reversed Soto (31), according to the following linear dependency: Energies 2018, 11, x FOR PEER REVIEW 11 of 21 Such approach must be taken with extreme care, as it is a common practice to operate often at temperatures higher than 48 °C, where (29) yields = 1 (or 0 at 100 °C) De Soto et al. [20] come with a different proposal: which has a wrong slope. For a proper variation and should be reversed as follows: Our experiments presented in Figure 6 yielded a different result, closer to reversed Soto (31), according to the following linear dependency:

Self Heating Phenomenon
It is a common practice to express the internal cell temperature, based on Normal Operating Cell Temperature (NOCT) data, when the module is mounted 45° from horizontal.
The internal temperature of the PV was of permanent concern for the researchers [40][41][42]46], but in most situations just an uncorrelated dependency is studied. Simply the temperature dependency of the I-V characteristic without acknowledging neither the real, actual temperature of the PV nor parameter variation is considered. Advanced simulators software packages include such features, MATLAB Simscape [52] being one of them.
In a recent work, Krac and Górecki [54] introduced a thermal model for the PV cell, where the self heating is modeled. The thermal behavior is modeled by a thermal resistor and a thermal capacitor, a voltage source related to the ambient temperature and a current source that represents the total dissipated power within the PV. They claim that "for the maximum allowable value of the panel forward current (equal to 8 A), a self heating phenomenon causes an increase in the panel temperature value equal only by 12 °C." In our experiments, we acquired a rather extended influence, ranging from 20 to 30 °C.
Opposite to [55,56], the power dissipated by the PV cell is taken into account from the dissipative

Self Heating Phenomenon
It is a common practice to express the internal cell temperature, T cell based on Normal Operating Cell Temperature (NOCT) data, when the module is mounted 45 • from horizontal.
The internal temperature of the PV was of permanent concern for the researchers [40][41][42]46], but in most situations just an uncorrelated dependency is studied. Simply the temperature dependency of the I-V characteristic without acknowledging neither the real, actual temperature of the PV nor parameter variation is considered. Advanced simulators software packages include such features, MATLAB Simscape [52] being one of them.
In a recent work, Krac and Górecki [54] introduced a thermal model for the PV cell, where the self heating is modeled. The thermal behavior is modeled by a thermal resistor and a thermal capacitor, a voltage source related to the ambient temperature and a current source that represents the total dissipated power within the PV. They claim that "for the maximum allowable value of the panel forward current (equal to 8 A), a self heating phenomenon causes an increase in the panel temperature value equal only by 12 • C." In our experiments, we acquired a rather extended influence, ranging from 20 to 30 • C.
Opposite to [55,56], the power dissipated by the PV cell is taken into account from the dissipative elements, which are resistive in our model. The energy flows from the two current sources to the resistors and the external circuit. Two or three current sources (or even diodes) are used in order to model different phenomena that take place inside the PV cell, the photoelectric effect and the behavior of p-n junction [8].

Open Circuit Voltage-V oc
Ishaque and Salam [27] propose for the V oc,cell the following variation (34), which proves good correlation with the datasheet info and experimental data-see also Figure 7: Energies 2018, 11, x FOR PEER REVIEW 12 of 21 resistors and the external circuit. Two or three current sources (or even diodes) are used in order to model different phenomena that take place inside the PV cell, the photoelectric effect and the behavior of p-n junction [8].

Open Circuit Voltage-
Ishaque and Salam [27] propose for the , the following variation (34), which proves good correlation with the datasheet info and experimental data-see also  Even (34) is not necessary for the model, it is another starting point for computing .

The New Proposed PV Cell Model
The proposed model is presented in Figure 8. The upper section consists of standard elements, while the thermal modeling is ensured by the lower section. Here the current source labeled simulates the power dissipated in the cell, the voltage labeled is the cell temperature and the air temperature is modeled by the voltage source . The thermal resistance models the thermal flow through the system structure, in our case the PV cell. The thermal capacitance models the thermal inertia of the PV cell. Both and emulate all thermal transmission phenomena (conduction, convection and radiation) and depend of the materials, the finishing of the surfaces and on the mechanical dimensions of the system. The practical LTSpice model implementation is depicted in Figure 9 and can be found at http://tess.upt.ro. The upper circuit addresses the standard conditions (for reference and validation), while the middle section deals with the thermal model of the PV solar cell. The power associated with the circuit also includes the power due to the irradiance scaled with the cell area and the electrical power dissipated in and . Even (34) is not necessary for the model, it is another starting point for computing a 1 .

The New Proposed PV Cell Model
The proposed model is presented in Figure 8. The upper section consists of standard elements, while the thermal modeling is ensured by the lower section. Here the current source labeled P d simulates the power dissipated in the cell, the voltage labeled T j is the cell temperature and the air temperature is modeled by the voltage source T amb . The thermal resistance R th models the thermal flow through the system structure, in our case the PV cell. The thermal capacitance C th models the thermal inertia of the PV cell. Both R th and C th emulate all thermal transmission phenomena (conduction, convection and radiation) and depend of the materials, the finishing of the surfaces and on the mechanical dimensions of the system. The practical LTSpice model implementation is depicted in Figure 9 and can be found at http://tess.upt.ro. The upper circuit addresses the standard conditions (for reference and validation), while the middle section deals with the thermal model of the PV solar cell. The power associated with the circuit also includes the power due to the irradiance scaled with the cell area and the electrical power dissipated in R s and R sh .  The thermal parameters and were extracted from experimental data, similar to [54]. After a set of data was acquired, the temperature against time curve variation was fit and the time constant and the steady state value were determined. Unlike Górecki and Krac [55,56], we considered   The thermal parameters and were extracted from experimental data, similar to [54]. After a set of data was acquired, the temperature against time curve variation was fit and the time constant and the steady state value were determined. Unlike Górecki and Krac [55,56], we considered The thermal parameters R th and C th were extracted from experimental data, similar to [54]. After a set of data was acquired, the temperature against time curve variation was fit and the time constant and the steady state value were determined. Unlike Górecki and Krac [55,56], we considered no dissipated power occurs in the BD2 current source of the model in Figure 9, as it makes no physical sense. Figure 10 exhibits the simulated and the measured internal temperature of the PV cell and the dissipated power variation. It is worth mentioning that the corresponding NOCT for the temperature in Figure 9 is 47.2 • C.

Experimental Results
Energies 2018, 11, x FOR PEER REVIEW 14 of 21 no dissipated power occurs in the BD2 current source of the model in Figure 9, as it makes no physical sense. Figure 10 exhibits the simulated and the measured internal temperature of the PV cell and the dissipated power variation. It is worth mentioning that the corresponding NOCT for the temperature in Figure 9 is 47.2 °C.  Table 4 summarizes the main results for both the proposed model and the experiments performed for the PV cell. It can be observed that perfect agreement between the simulated and measured results is achieved. The final validation of the model is presented in Figures 11 and 12. Here the I-V and P-V characteristics of the PV cell are plotted at the reference temperature and at the operating temperature.  Table 4 summarizes the main results for both the proposed model and the experiments performed for the PV cell. It can be observed that perfect agreement between the simulated and measured results is achieved. The final validation of the model is presented in Figures 11 and 12. Here the I-V and P-V characteristics of the PV cell are plotted at the reference temperature and at the operating temperature.

Experimental Results
Experimental data is represented with markers while the lines correspond to simulated results with the model proposed. A good correlation between the model and the experiments can be noticed. Figure 12a displays the serial resistance R s influence on the output current and power. The solid lines graphs correspond to a fixed R s while the dashed lines correspond to variable R s with all the parameters included. At MPP a 98 mW power increase was observed. As estimated before, R sh has a minor influence on the PV output-only 4.6 mW power decrease at MPP could be noticed, as displayed in Figure 12b. It is worth mentioning that in all cases the model self-computes the appropriate values for R s and R sh based on the predicted internal temperature.
Energies 2018, 11, x FOR PEER REVIEW 15 of 21 Experimental data is represented with markers while the lines correspond to simulated results with the model proposed. A good correlation between the model and the experiments can be noticed. Figure 12a displays the serial resistance influence on the output current and power. The solid lines graphs correspond to a fixed while the dashed lines correspond to variable with all the parameters included. At MPP a 98 mW power increase was observed. As estimated before, has a minor influence on the PV output-only 4.6 mW power decrease at MPP could be noticed, as displayed in Figure 12b. It is worth mentioning that in all cases the model self-computes the appropriate values for and based on the predicted internal temperature.
(a) (b) has no significant influence on the performance.
PV arrays compared (Table 5)  All the data from Table 5 was processed with the above proposed algorithm and the results are listed in Table 6, along with similar results from other researchers. Experimental data is represented with markers while the lines correspond to simulated results with the model proposed. A good correlation between the model and the experiments can be noticed. Figure 12a displays the serial resistance influence on the output current and power. The solid lines graphs correspond to a fixed while the dashed lines correspond to variable with all the parameters included. At MPP a 98 mW power increase was observed. As estimated before, has a minor influence on the PV output-only 4.6 mW power decrease at MPP could be noticed, as displayed in Figure 12b. It is worth mentioning that in all cases the model self-computes the appropriate values for and based on the predicted internal temperature.
(a) (b) has no significant influence on the performance. PV arrays compared (Table 5)  All the data from Table 5 was processed with the above proposed algorithm and the results are listed in Table 6, along with similar results from other researchers. All the data from Table 5 was processed with the above proposed algorithm and the results are listed in Table 6, along with similar results from other researchers. The final validation of the model was by applying the introduced model and computation method for the MSMD290AS-36.EU monocrystalline PV cell array and compare the results to the ones provided by Cubas et al. [31], as shown in Figure 13. As it can be seen, a good correlation exists between the two approaches.  The final validation of the model was by applying the introduced model and computation method for the MSMD290AS-36.EU monocrystalline PV cell array and compare the results to the ones provided by Cubas et al. [31], as shown in Figure 13. As it can be seen, a good correlation exists between the two approaches.

Conclusions
A new thermo-electrical model for the PV cell was introduced. Only free available tools were used during modeling. The literature analysis proved discrepancies between authors when studying parameters variation and a more precise model is proposed in this paper.
The model proved to be accurate, while considering parameter variation and selfheating phenomenon. To the best of our knowledge this is the first time when all these parameters are included in a PV model. The internal silicon operating temperature at 1 Sun (with the ambient temperature being 20 °C) is 54 °C predicted by our model and validated by measurements performed with the FLIR and the PT1000 sensors.
As other authors have mentioned, influence is relatively reduced in the model. However proved to be a major factor. displayed a small variance with temperature. Resistance influence is important but sometimes shadowed by the wiring. The proposed model was accurately confirmed and validated by the experiments. Greek Symbols α Rs Series resistance temperature coefficient (linear law)