PV Cell Parameters Modeling and Temperature Effect Analysis

With the wide acceptance of modeling a PV cell by a single diode, a series and parallel resistors; many researchers have discussed different mathematical forms and iterative techniques to extract the values of these model elements depending on the key parameters provided by the manufacturer datasheet. This paper avoids iterative techniques and obtains the values of the five parameters of the one diode model by developing closed form expressions. The maximum error produced by this technique is 10% when compared to the exact values of the one diode model circuit built by Spice. The 10% maximum error has occurred during the estimation of the reverse saturation current (Io) of the diode, nevertheless, it should be mentioned that even for this same parameter the model outperforms many iterative dependent works. Furthermore, this paper discusses the effect of temperature on the operation performance of PV cells. In particular, the temperature effect on the open circuit voltage, the short circuit current, the fill factor, the reverse saturation current, and the conversion efficiency was modeled and evaluated for different brand technologies.


Introduction
Solar energy is becoming one of the most important sources of energy and it is expected that its share of globally produced energy will grow fast. Thus, there is a need for continuous development of PV cells at the implantation level which ranges from different materials to different fabrication techniques. All aim at improving the efficiency of light conversion to electrical current. On the utilization level, the performance of a PV cell may be improved by operating it at optimal conditions which requires a full understanding of its behavior. In addition, it helps in the prediction of the performance under different operating conditions.
To have a good insight on the behavior of a PV cell, the cell should be modeled accurately and the parameters of the equivalent circuit have to be predicted precisely. The problem of modeling PV cells has been tackled by many researchers for many years. In fact, researchers have introduced different circuit models for PV cells; these models have various degrees of complexity and accuracy. The simplest PV cell model is the ideal model which consists of a current source and a diode (Cuce et al. 2013;Piazza et al. 2009;Rodrigues et al. 2011;Luque et al. 2003) as illustrated in Fig. 1. The current source accounts for current generated by the incidence of light (IPh) which depends mainly on the solar irradiance. The diode represents the recombination current in the quasi-neutral regions.
To improve the model accuracy, the cell's internal resistance is accounted for by adding a series resistance Rs which represents the effect of the semiconductor layers (p and n layers), the contacts and the metal connections (Wolf et al. 1963). Additionally, the leakage current is modeled by adding a parallel resistance Rp. Fig. 2 shows the schematic of this model which has been adopted by many researchers (Fesharaki et al. 2011;Petreus et al. 2011;and Sabadus et al. 2017). The value of Rp is high for good cells, but for defective ones the value of Rp is usually low. The impact of low value Rp is more significant when Iph is small i.e., when the irradiance levels are low since I will be low and consequently the produced power will also be low.   A more accurate and more complicated model was introduced in (Sah et al;1957). In this model a second diode is added on parallel as presented in Fig. 3 to reflect the effect of the recombination current in the depletion region.
A comparison study has been conducted to compare the performance between the one-diode and two-diode models (Bouraiou et al. 2015). Experimental and simulated results showed that there is a very limited accuracy difference between the two models. And with the consideration of the reduced complexity of one-diode model compared to the two-diode model, this work finds closed form expressions for the five parameters (namely Rs, Rp, Iph, Io, and n) of the one-diode model depending on the data provided by the datasheet.
Considering the one-diode model shown in Fig. 2, the following well-known equation 1 can be introduced Where I is the output current of the PV cell, V is the output voltage of the PV cell, Io is the reverse saturation current of the diode, n is the diode ideality factor, Vt = kT/q is the thermal voltage where k is the Boltzmann constant and is equal to 1.38x10 -23 Joules/K, T is the absolute temperature of the p-n junction in Kelvins and is equal to 273 + temperature in o C and q is the magnitude of charge of an electron and is equal to 1.6x10 -19 Coulombs. Usually, a datasheet of the PV cells lists three notable parameters on the current-voltage (I-V) characteristic: Voc, Isc (the output open circuit voltage and the short circuit output current respectively) and the maximum power Pm point which is determined by Vm and Im as shown in Fig.  4. Depending on these parameters specified in the datasheets, the five parameters of the single diode model are obtained.
Extracting the five parameters from the single diode model has been tackled by many researchers. To extract the five parameters, the authors of (Blas et al. 2002;Brano et al. 2013;and Chan et al. 1986) have introduced two additional parameters, Rso and Rsho, which are the reciprocal of the gradient at the open-circuit and short circuit points respectively. In (Blas et al. 2002;and Brano et al. 2013) a set of equations has been developed to be solved simultaneously to extract the five parameters while in (Chan et al. 1986), a linear approximation around Voc and Isc was used to find the Rso and Rosh.
The five parameters were extracted in (Appelbaum et al. 2014;and Sabadus et al. 2018) by utilizing Newton-Raphson Method to solve the produced equations. A systematic method for extracting the five parameters was proposed in (Zhu et al. 2011). This method depends on analyzing the sensitivity of the parameters which are bounded by a specific range for each. The five parameters were predicted using Matlab/Simulink in (Bellia et al. 2014) where a detailed Simulink model was built for each parameter e.g., Iph, Io, etc.
All of the above-mentioned methods depend on some kind of iteration technique to extract the values of the five parameters. Nevertheless, the work in this paper develops closed form analytical expressions for each of the parameters which simplifies the calculations and gives insight on the affecting factors.

The value of Rs
For the short circuit condition, the output voltage V=0 and thus the voltage across the diode is equal to Isc × Rs which is a very small voltage to turn the diode fully on. The expression developed in (Sabadus et al. 2017) to calculate Rs depending on Voc, Im, Vm and Isc is adopted in this paper since it depends only on the datasheet information and it shows a good accuracy when compared with Spice simulation results. (2)

The value of the ideality factor n
For the open circuit condition, where the output voltage V=Voc, the current through Rp is a multi-magnitude less than that of the diode current where the diode will be fully on in this case. Therefore, using the expression in (1) at the open circuit condition, the diode current Idoc can be approximated as stated in Eq (3).
Again, the current through Rp at the maximum power condition ((Vm+ImRs)/Rp)) is very small in comparison to Im and Isc, so the diode current at the maximum power condition Imd can be approximated as By taking the ratio of (3) over (4) the following expression is obtained The expression in (5) has only n as an unknown variable, solving (5) for n yields = 6 .< 7 . 54 ; =4 ?< JK) , .< @, ; , .< 14 0 The expression in (6) estimates the value of the diode quality as a function of Im, Vm, Isc and Voc, which are given by the datasheet and Rs which has been estimated earlier in (2).

The value of the diode reverse saturation current Io
Since the value of n is known, the value of Io can be simply found by substituting the value of n as obtained from (6) in (3) and solving for Io.

The value of the parallel resistance Rp
The value of Rp can be estimated by applying (1) In (8), all of the values except Rp are either provided by the data sheets or have been found through the previous equations and as such solving (8) According to (11), the margin for error introduced in (10) will be very small since the ratio of Rs/Rp is very small.

Simulation and result discussion
Since it has been proven that the one diode model represents the behavior of PV cells, thus in order to verify the accuracy of the developed expressions we compare them to the simulation results of the one-diode model. For the simulation to be realistic, the values of Io, Rs, n and Rp, are chosen to be close to those measured (the value of n has been estimated not measured) in (Sabadus et al. 2017) as such we built the circuit in Fig. 5.
The circuit has been simulated to obtain the performance values that are usually provided by the datasheet i.e. Isc, Voc, Im and Vm. The datasheet parameters, as obtained from Spice simulation of the PV cell model-circuit in The results presented in Table 1 show that the proposed models provide a good accuracy with the maximum error of 10% occurring for the calculation of Io which outperforms the accuracy presented in (Sabadus et al. 2017). Although the work in (Sabadus et al. 2017) provided better accuracy for Rs, it doesn't estimate any value for Rp. Moreover, parameter evaluations in (Sabadus et al. 2017) depend on iterative Newton-Raphson method while the proposed models don't need any iteration. The error of the estimated value of the light generated current is zero as shown in Table 1. In fact, there is a very small error since the estimated value is 0.1500019972A, and not 0.1500A exactly. According to the obtained values of Rs and Rp the value of EIph as given by (11) is 1.1x10 -3 % for the actual values and 1.3x10 -3 % for the estimated values.  The developed model has been verified further by targeting an old technology (Easwarakhanthan et al. 1986) which is considered by many researchers as listed in (Li et al. 2016). We have chosen this technology due to the fact that its parameters are significantly different from the one considered in (Sabadus et al. 2017). Of particular importance is the value of Rp which has been estimated by previous works, as presented in Table 2, to be in the range between 40-60 Ω.
The results illustrated in Table 2 show that the proposed model provides a very good agreement with the simulated results and it outperforms many of the previously published works. Nevertheless, the value of Rp seems to be very much overestimated by the model, however when compared with the simulated value it shows a good agreement. To further verify the effect of Rp, we changed its value over a wide range from 50 Ω to 150Ω, during which the variations of the values of Im, Vm, Voc and Isc were very minor. The largest variation is in Voc value and it is < 0.4%.

Modelling and discussion of temperature effect on the PV cell performance
The effect of temperature on the PV cell operation has been studied extensively by many researchers (Fesharaki et al. 2011;Petreus et al. 2008;Easwarakhanthan et al. 1986;Dzimano 2008;Ishaque et al. 2011;and Femia et al. 2013). In the following subsections we study the influence of the cell's temperature on its open circuit voltage, short circuit current, fill factor, reverse saturation current and conversion efficiency.

The open circuit voltage
In (Fesharaki et al. 2011;Petreus et al. 2008;Dzimano 2008;Ishaque et al. 2011;and Femia et al. 2013) the PV cells I-V characteristics are plotted for different cell temperatures. The curves show that the effect of the temperature on the short circuit current Isc is negligible while its impact on the open circuit voltage is significant. In Fig. 6 we reproduce the I-V curves introduced in (Fesharaki et al. 2011;Petreus et al. 2008;Dzimano 2008;Ishaque et al. 2011;and Femia et al. 2013) where the open circuit voltage has been plotted as a function to the cell temperature. In Fig. 6, we have normalized the cell's temperature to its maximum temperature (Tmax) as provided by the considered reference; and the open circuit to its value at the standard temperature (Vocn). These curves show that there is a nearly linear relationship between the temperature T/Tmax and Voc/Vocn, as such we may express Voc as Where R = V4 ?< VW represents the slope of the considered curve in Fig. 6 and it is usually provided by the manufacturer as the open circuit voltage coefficient. In fact, Kv represents the sensitivity of the open circuit voltage of a cell to its temperature variation. ∆Voc is the open circuit voltage change when the cell temperature changes by ∆T.
Following the same arguments, the effect of the temperature on the short circuit current can be modeled as Where X = V6 .< VW , and Iscn is value of the short circuit current at the standard temperature. To investigate the results presented in (Fesharaki et al. 2011;Petreus et al. 2008;Dzimano 2008;Ishaque et al. 2011;and Femia et al. 2013), we used the expression in (13) to plot Isc as a function of temperature for different PV cell brand technologies as illustrated in Fig. 7.
Clearly the obtained current plots in the previous figure demonstrate the minor effect of the temperature on the short circuit current.

The fill factor
The Fill Factor (FF) is an important index of the quality of a PV cell since it shows the cell's ability to produce energy (Dzimano 2008;and Femia et al. 2013). FF is defined as the ratio of the maximum power (Pm = Im . Vm) of a PV cell to its Voc and Isc product (Dzimano 2008;and Femia et al. 2013).   As it is apparent from (14), the variation of FF with the temperature depends on the variation of the maximum power, the short circuit current and the open circuit voltage. The variation coefficients of these parameters are provided by the manufacturers.  Table 3 where the values of Ki and Kv are presented for different manufacturers.
Equation (15) is valid if the coefficients are given as voltage/current per o C (e.g., the SunPower case in Table  3), but if these coefficients are given as a percentage, then (15) should be rewritten to be = Z ; )>5[ 8 (W=W / )1 (16) Using Eq (15) and Eq (16) FF is plotted against the temperature for different PV cell brand technologies as presented in Fig. 8 Fig . 8 shows that the FFs of the considered brand technologies change with temperature at different paces. For example, SunPower outperforms the other three technologies for cell temperatures below 40 o C, but its FF drops faster than others as the temperature increases. On the other hand, Canadian Solar seems to have more stable values of FF as the cell temperature changes.

.< a>5[ ] (W=W / )bc `4 ?< a>5[ _ (W=W / )bc
As illustrated in Fig.6, Voc varies significantly with temperature, however Fig. 7 shows that Isc current is practically stable against the change of temperature. Therefore, the change of FF as a function of temperature is mainly due to the variability of Voc.  (15) and (16).  Although it is clear from (15) and (16) that the relationship between FF and the cell temperature is not linear, Fig. 8 suggests that this relationship is very close to being a linear one as presented in (Femia et al. 2013).
Equation (16) may be linearized by using Taylor Series around T=Tn and limiting the series to two terms which yields Where FFn is the value of the Fill Factor at the standard temperature. The accuracy is evaluated by comparing the results obtained by Eq (15) or Eq (16) and the linear equation (17) as presented in Table 4. The error in Table 4 (16) and (17) respectively. Table 4 shows that the error produced by expressing FF as a linear function of temperature is very small (the maximum error is 1.09%), thus Eq (17) can be used to estimate FF for different values of temperature with a very good accuracy. Actually, Eq (17) can be rewritten as Where Kf is the temperature coefficient of the Fill Factor and it is expressed as Table 5 The values of the temperature coefficient of the Fill Factor for different technologies as estimated by (20).
In Table 4 the values of the previously considered technologies are calculated using Eq (20). Table 5 reflects the fact that SunPower has the highest value of Kf which is compatible with the conclusion drawn from Fig. 8. Furthermore,Eq (19) can be used to check the stability of the Fill Factor of PV cells against temperature. We can notice easily that this stability increases as Kf approaches zero which yields that Kp converges to the algebraic sum of Kv and Ki. It should be noted that Kp, Kv and Ki must be expressed as percent values i.e., %/ o C.

The reverse saturation current
Considering the model presented in Fig. 1, we can see clearly that the temperature affects the current generated by the incidence of light Iph and the diode current Id.
Utilizing the approximation of Iph=Isc, we can express the value of the short circuit current Iscn at the temperature of the standard condition (Tn) as Where Ion is the reverse saturation current of the diode at Tn, Vtn is the thermal voltage and is given as Vtn= kTn/q. Let Isc be the short circuit current at the actual temperature T as defined before.
The expression shows that the reverse saturation current increases with the temperature. Nevertheless, as Tn/T→0, this current saturates and reaches its plateau value estimated by Eq (26 In Fig. 9 we plot the normalized value of Io/Ion as a function of the temperature for different PV cell brand technologies. The figure shows that for all these technologies the reverse saturation current increases approximately by 200 times as the temperature increases from 20 o C to 100 o C.

The conversion efficiency
The efficiency of a PV cell is an important criterion that shows the ability of a PV cell to convert the incident light to electricity and is expressed as (Tiwari et al. 2010 The illumination intensity (Ii) and the PV cell/panel area (Ac) are independent of the PV cell temperature, which leaves the dependence of η on the temperature to be because of the temperature effect on Isc, Voc and FF. As the change of Isc with the temperature is negligible, the change of the value of η due to the temperature is therefore mainly because of the variations of Voc and FF. Since Isc ≈ Iph, then α can be seen as the solar panel's conversion efficiency between the illumination intensity and the generated current per unit area.
As a result, the relationship between η and the temperature can be developed by replacing the values of Voc and FF from Eq (12) and Eq (19) respectively in Eq (28).
The expression in Eq (30) shows that the relationship between the efficiency and the temperature is nonlinear due to the existence of Kη2(T-Tn) 2 . Nevertheless, (30) has two other terms: α Vocn FFn and Kη1(T-Tn). The first term is a constant that represents the efficiency at the standard temperature; the second term represents a linear relationship between η and the temperature.
To further understand the behavior of the efficiency variation with the temperature change, we estimated the impact of Kη1(T-Tn) and Kη2(T-Tn) 2 on η for different PV cell brand technologies and temperatures as presented in Table 6. Note the values of Kη1 and Kη2 are normalized to α.  Table 6 shows that the impact of the second order term is insignificant compared to the first order term (about 0.004 at its maximum when the operating temperature is 70 o C) and basically, the effect of the temperature on the efficiency is dominated by the first order term i.e., Kη1 (T -Tn). Hence, the effect of the temperature on a PV cell performance can be approximated by Where ηn is the performance at the standard conditions and Kη1 is as defined before. It worth mentioning that Kη2 is positive, i.e., it works toward improving the performance, however, this effect is insignificant as demonstrated in Table 5.

Conclusion
Closed form expressions have been developed in this paper for estimating the five parameters that form the one-diode model of a PV cell. As a test vehicle, a one-diode model circuit that represents a real-life PV cell has been built and simulated using Spice to extract the information that is usually provided by the datasheet.
The proposed expressions provide good accuracy with a maximum error of 10% when compared to the model circuit. Although the developed expressions are closed form ones; they are still outperforming many of the published results. The outcome of this work can be used as is to provide a fast calculation; or it can be used to offer excellent estimation of the initial value for iterative methods which helps iterative algorithms to reach a fast convergence when more accuracy is required for some of the parameters.
The effect of the temperature on the cell's performance is tackled by analyzing the behavior of the open circuit voltage, the short circuit current, the reverse saturation current, the fill factor and the efficiency as functions of the temperature. As a result, closed form expressions are developed, each one describing the relationship between the above-mentioned parameters respectively and the cell temperature. All these expressions depend on the values usually provided in the datasheet. All the parameters except for the reverse saturation current, have been approximated to simple linear equations which provide good insight and accuracy. The reverse saturation current shows an exponential dependence on the temperature. The models presented in this paper are indeed simple and suitable for hand calculations, nonetheless they provide a very good accuracy.