Simple PV Modeling Under Variable Operating Conditions

Photovoltaic (PV) panels are increasingly used to convert sunlight into electricity, as a source of sustainable energy. It can be used in a wide variety of applications ranging from the well-known power generation to the emerging energy harvesting in Internet of Things (IoT). Hence, an accurate model is required to evaluate and predict the performance of the PV panel. However, the non-linear characteristics of PV panels make the modeling of their electrical response a challenging task. In the literature, most of the previous PV models have been developed for large wattage PVs under high irradiance, or for small wattage PVs under lower irradiance. Those that can model both usually require more information, including the I-V curves data at different irradiances, which is not always provided by the manufacturers. Therefore, this paper presents a simple PV modeling that can be applied for different wattage panels at different operating levels of irradiance, using only the commonly provided datasheet values at standard test condition (STC). The model uses the characteristic points translation technique to translate the short circuit current, open circuit voltage and maximum power voltage points, at STC to other operating conditions. These translated values are then used by the parameter extraction technique to extract the model’s parameters. The proposed model’s techniques can model the losses across the resistors at low irradiance, which reduces the error. The accuracy of the proposed model is validated using two representative commercial PV panels. Results are generated for the proposed model and other comparative works. The results show that the proposed model can improve the accuracy over the other compared works, with a consistent percentage difference of below 5% across all levels of irradiances for both panels.


I. INTRODUCTION
Photovoltaic (PV) panels are increasingly used to convert the abundant and freely available sunlight into electrical energy. Research and development over the years has yielded vast improvements in their efficiency and lowered their costs, leading to accelerating deployments. Today, PV panels are used in many applications. They range from the high-power industry of energy generation all the way to the low-power emerging energy harvesting for powering Internet of Things (IoT) devices. The former addresses the urgent issue of global warming, caused by the expanding consumption of traditional fossil fuels [1]. While the latter is a key enabling technology The associate editor coordinating the review of this manuscript and approving it for publication was Giovanni Pau .
for the successful deployment of IoT devices, whose numbers are expected to be in the billions.
A PV panel is constructed with wafer-based silicon cells and it can be connected either in series or parallel configurations to yield the desired voltage and current. There is a wide variety of PV panels, from high to low wattages for different applications. They also operate across different operating conditions from high to low irradiances. Furthermore, their non-linear characteristics make their accurate modeling even more challenging. Nevertheless, an accurate PV model is crucial in assessing the expected electrical response of various PV panels under any operating conditions, which include variations in the irradiance.
PV models use an equivalent circuit with a set of parameters that describe the behavior of PV panel. Among the PV models that have been proposed in the literature [2]- [10], one diode model with two resistors is found to be the most popular model because the model offers great accuracy yet simple. One of the benefits of this model is that it incorporates the parallel resistor, which is used to describe the leakage current. This model contains a set of five parameters to describe the PV behavior.
Many studies have proposed methods such as optimization and analytical to find the model's parameters. In these works [11]- [19], optimization methods are used to extract the parameters. The drawback of optimization methods is that they require the I-V curve information to allow the algorithm to find the best fit. The optimization methods also require setting a range limit for different parameters, which may vary for different panels. Although this method offers high accuracy, it is traded off with the model applicability as not all panel manufacturers provide I-V curve information.
To overcome some of the shortcomings, analytical methods [20]- [28] are used to extract the parameters directly and with most of them requiring only datasheet values as input. The most common values that can be found in the datasheets are the values of the three characteristic points, which are short circuit current, open circuit voltage, and maximum power points. These values are obtained at standard test condition (STC), which refers to, average solar spectrum at an air mass of 1.5, 1000 W/m 2 irradiance, and cell temperature of 25 • C. The analytical method can be applied to almost all PV panels because it only requires the datasheet values at STC, which is always provided by the manufacturer. However, their consistent accuracy across different panel wattages and irradiances can still be improved.
The analytical method in [20] has proposed a simple yet accurate parameter extraction technique for one diode model. The work uses the PV panel's maximum power point to derive the model's parameter. As a result, the model's accuracy is improved when compared to existing techniques. This work not only allows the modeling of PV panel at STC but also at other operating conditions. To achieve that, parameter scaling technique is implemented to scale the model's parameters to different operating conditions. However, the accuracy of the parameter scaling technique to degrade at low irradiance levels.
Not only that, the work in [21] has made a comparison of existing parameter scaling techniques and evaluate their performances at different irradiance levels. The results show that most of the techniques have good accuracy at STC condition. But the model's accuracy varies significantly at low irradiance levels although most techniques use a similar scaling approach. The work suggests that the parameter scaling technique should not be the primary technique to model the PV panel's behavior at different irradiance levels. In [22], the work also uses parameter scaling technique to scale the parameters. An adjustable series resistor is used to increase the model's accuracy by fitting the maximum power point as close as possible to the datasheet value. Yet, the model's percentage difference at low irradiance levels is more than 5% compared to the datasheet values. Therefore, the technique proposed in these works are not suitable for low irradiance PV modeling due to the model's lack of accuracy.
As the parameter scaling technique does not meet the expected accuracy at low irradiance levels, another analytical work in [23] uses the values of characteristic points at specific operating conditions to extract the model's parameters. Despite that, those values are obtained through experiments as the datasheet gives values only at STC. Thus, this work is unable to model the behavior of the PV panels at different irradiance levels without conducting experiments.
Alternatively, the characteristic points can be obtained through the translation technique. This technique can translate the datasheet three characteristic points to the specific operating conditions without requiring experimental values. The work in [24] has proposed a translation technique that can achieve high accuracy at high irradiance levels, but it degrades as the irradiance levels decrease. This is due to the proposed voltage correction coefficient not able to provide an accurate translation for the open circuit voltage and maximum power voltage.
For the translation work in [25], [26], a new coefficient named irradiance factor is proposed to improve the technique's accuracy. The irradiance factor can predicts the relationship between irradiance levels and open circuit voltage accurately. But there is still lack of accuracy on the maximum power voltage at low irradiance levels. The error is more than 5% when compared to the datasheet. An improvement on the maximum power voltage translation has been made in the following work [27]. The work can translate the maximum power voltage accurately by using the initial series resistor to define the relationship between irradiance levels and maximum power voltage. However, the open circuit voltage translation proposed in this work is not accurate.
The parameters extraction and characteristic points translation techniques are very important for creating an accurate PV model. As discussed previously, the work in [20] offers a simple yet accurate parameter extraction technique compared to other works, but the accuracy degrades at low irradiance. Hence, the translation techniques in [25]- [27] are required to increase the accuracy of three characteristic points at low irradiance levels. With the combination of these techniques, the model can model the behavior of PV panel at all irradiance levels. Moreover, the proposed model can also model high and low wattage panels at different operating irradiance levels using only datasheet values at STC.
Following is the organization of the paper. One diode model with two resistors is presented in detail in Section II. Section III provides the one diode model parameter extraction technique at STC. Section IV discusses about the parameter scaling technique while Section V introduces the characteristic points translation technique. In Section VI, the results of different translation techniques are presented while Section VII concludes the paper.

II. ONE DIODE MODEL WITH TWO RESISTORS
The equivalent circuit of one diode model with two resistors is shown in Fig. 1. The circuit consists of photocurrent source, I ph , the series resistor, R s , the parallel resistor, R p , and a diode. With reference to Fig. 1, the output current, I can be expressed as below by applying Kirchhoff's current law: where I d is the diode current and V is the voltage. The diode current can be further expressed by the Shockley diode equation which is described as follow: By substituting (2) into (1), the model's output current equation is described as: where I o is the diode's saturation current, q is the electron charge, N s is the number of series-connected of PV cells, a is the diode's ideality factor, k is the Boltzmann constant and T is the temperature.

III. PARAMETER EXTRACTION TECHNIQUE AT STC
The model consists of five parameters I ph , I o , a, R s , and R p which can be obtained through the parameter extraction technique proposed in work [20]. Firstly, the model's output current using (3) At short circuit condition, V = 0 and I = I sc . Therefore, At maximum power condition, V = V m and I = I m . Therefore, where I sc is the short circuit current, I m is the maximum power current, V oc is the open circuit voltage and V m is the maximum power voltage. By rearranging (4), the I ph is given by: By substituting (7) into (5), the I o is written as: Simplifying the above equation: The term exp(qR s I sc /N s akT) can be neglected due to its insignificant value compared to exp(qV oc /N s akT). By rearranging (9), I o is written as: In order to get a more accurate expression for the parameter R s and R p , the work [20] proposed to include the number of parallel-connected PV cells, N p in (3). Hence the new equation for the model's output current is: To derive R s , the term (VN p + IR s N s )/(N p N s R p ) can be neglected as the current flow across the parallel resistor is small. Then, the equation is evaluated at short circuit condition: and open circuit condition: By rearranging (12) and (13), a new expression of I ph and I o are derived and substituted into (11).The equation is written in (14).
To simplify (14), the term exp(qI sc R s /N p akT) is assumed equal to one. This is because the value of a and T does not significantly affect the overall term. Therefore, the simplified model's output current is shown: Evaluating (15) on the maximum power condition, the expression of R s can be deduced as: On the other hand, the expression for R p can be derived from the derivative of current with respect to voltage at maximum power point: To obtain the model's output current expression, (11) is evaluated at short circuit conditions: and open circuit conditions: By rearranging (18) and (19), a new expression of I ph and I o are derived and substituted into (11). The equation is written in (20). Substituting the derivative of (20), as shown at the bottom of the page, into (17), the expression of R p is shown in (21), at the bottom of the page. For the value of parameter a, the work [20] proposed to increase its value from 0 until R p reaches its minimum positive value. After the five parameters' values have been found, the R s value can be increased to fit the maximum power point as close as possible, to the desired accuracy. Hence, the model's accuracy can be increased by using this technique

IV. PARAMETER SCALING TECHNIQUE
However, the parameters extracted at STC cannot be used to perform PV modeling at other operating conditions. This is because factors such as irradiance and temperature can affect the parameters and change the model's behavior. In the works [12] and [29]- [32], parameter scaling technique is used to scale the model parameters from STC to other operating conditions. The five equations are shown below: where G STC , I sc,STC , T STC , I o,STC , a STC , R s,STC and R p,STC are irradiance, short circuit current, temperature, diode saturation current, diode's ideality factor, series resistor and parallel resistor extracted at standard test condition. While G, T , I ph , I o , a, R s and R p are irradiance, short circuit current, temperature, diode saturation current, diode's ideality factor, series resistor and parallel resistor scaled at the specific operating condition. α Isc is the current temperature coefficient and β Voc is the voltage temperature coefficient of the PV panel. E g,STC is the material band gap energy where it is equal to 1.121eV for silicon cells.
According to [33], the photocurrent, I ph changes linearly with solar irradiance, and it is also affected by the operating temperature. However, the work in [34] has suggested improvising the irradiance, G to effective irradiance instead, as there are reflectance losses that may occur on the PV panel's surface. The diode's saturation current I o changes its values based on the diode theory, and a parameter is described as being proportional to the cell temperature based on the experiment [29]. Not only that, the series resistor R s , plays a critical role in PV modeling as it determines the slope of I-V curve near the maximum power point. Therefore, the work in [29] and [35] proposed to decrease the value of R s as the irradiance decreases. Nevertheless, the accuracy of R s given by this equation at much lower irradiances can still be improved.
On the other hand, the parallel resistor's R p value will affect the slope of the I-V curve near the short circuit current. An experiment has been conducted by the NIST (National Institute of Standard and Technology) to investigate the slope. Results show that the value of R p increases when the irradiance decreases. It is also can be expressed as inversely proportional to the irradiance level. To validate this technique, an experiment [21] has been conducted and tested with commercially available PV panels. Results show that most I-V curves can achieve good accuracy at high irradiance levels, where the three characteristic points are very close to the experiment value. However, the accuracy starts to decrease when the parameters are scaled to lower irradiance.

V. CHARACTERISTIC POINTS TRANSLATION TECHNIQUE
As discussed in the previous section, there is lack of accuracy in parameter scaling technique at low irradiance levels. Thus, there is a need of an alternative technique to model the behavior at different operating conditions. Translation technique is chosen in this work as it requires only datasheet values to obtain the characteristic points values at different operating conditions. Below are the four equations proposed: Many of the existing works [36]- [40] show that the short circuit current and maximum power current of the PV panel changes proportionally to the irradiance and temperature. Moreover, the proposed translation of open circuit voltage also can be seen in the works [25], [26]. The translation uses the irradiance factor, δ to increase the translation's accuracy and it can be derived by using datasheet values shown in (31).
On the other hand, the translation of maximum power voltage also can be found in the work [27]. Initial series resistor, R s0 is used to model the losses across the series resistor which helps to increase the translation's accuracy. The initial series resistor is shown in (32), and it can be found using datasheet values too. By using these four proposed translation equations, the three characteristic points can be translated to other operating conditions by using only datasheet values.
Next, some of the works have different translation equations for the open circuit voltage and maximum power voltage. This is because the voltage did not have the same response as the current. The works in [24] and [27] have proposed an equation for the open circuit voltage. The output current equation of the one diode model is evaluated at the open circuit condition and rearranged in (33). By substituting the parameter scaling from (22) and the I o equation in (34) into (33), the translation of the open circuit voltage is shown in (35).
where V th = (aN s kT/q), is the thermal voltage of the PV panel. As there is an unknown diode ideality factor, a parameter in the thermal voltage, the work [41] proposed to use the nominal operating cell temperature (NOCT) that is provided in the datasheet to derive the parameter value. The expression of a is given as follows: The translation requires the open circuit voltage at NOCT, which is at 800 W/m 2 irradiance and the ambient temperature of 20 • C. However, most manufacturers did not provide the values at NOCT as it is seldom used. Therefore, the work in [28] proposed to set the value of a equal to 1, for simplicity.
Another work, [26] also proposed a translation technique for the open circuit voltage. The new equation for I ph is shown in (37).
In order to obtain the translation for open circuit voltage, the parameter scaling from (22), (23) and (24) are required. By substituting them into (37), the translation is described as: To simplify (38), the second term can be approximately represented as the temperature dependency of the open circuit voltage as shown in (29). Compared to the works in [27], [28] and [42], irradiance factor, δ in (29) is proposed to improve the accuracy of the open circuit voltage. The a parameter is replaced by δ coefficient as it only requires datasheet values such as current and voltage temperature coefficient. To obtain the δ coefficient, the work uses (37) to evaluate operating conditions at 1000 W/m 2 irradiance and arbitrary temperature. By using the parameter scaling from (22), (23), (24) and (35) into (37) and defined as: To simplify (39), some modifications are applied to the equation, using (40) and (41): where (41) (40) and (41) into (39), a new equation in terms V oc is shown below: The term f ( T ) varies in a limited range under arbitrary temperature for multiple PV panels as presented in [26]. Since the term f ( T ) is not defined in (T − T stc = 0), therefore the corresponding limit is evaluated as follows: By solving (44), two equations, as shown in (45) and (46), and are substituted into (42) to form (31).
For the maximum power voltage translation, the works in [24] and [42] proposed that it has the similar response as the open circuit voltage. However, this translation may not be implementable if the manufacturer did not provide the NOCT values, on which the V th is derived from.
On the other hand, the work in [27] proposed an alternative translation for the maximum power voltage, as follows: (48) By replacing the translated open circuit voltage in (48) with (29) and the translated maximum power current in (28), the equation for maximum power voltage is (30). However, the value of initial series resistor R s0 is unknown. The method proposed in [27] uses (3) to find its value. If an I-V curve has a slope that is almost zero at short circuit, then the value of parallel resistor R p can be assumed as infinite. Hence, the last term in (3) can be ignored and by replacing I ph with I sc , the equation will become: When the PV model is in open circuit condition, (49) is solved as follows: By substituting (50) into (49), (51) is formed.
To simplify (51), a new coefficient is introduced where z = I sc /I o . Hence, the new current output equation is shown in (52).
By evaluating (52) at maximum power conditions, the coefficient z can be solved as (53) and the derivative can be solved as (54).
To obtain the value of R s0 , (53) is substituted into (54) and the expression is (32).
Multiple types of voltage translation equations proposed from different works are discussed. The open circuit voltage [26] and maximum power voltage [27] give a direct translation using only datasheet values which is very convenient. Therefore, the equations from (27), (28), (29), (30), (31) and (32) are implemented in this work to translate the characteristic points from STC to other operating conditions.
After obtaining the translated characteristic points, these values will be used to extract the model's parameters and the procedures are shown in Fig. 2. Newton Raphson method is used to generate the model's I-V curve. The model's values VOLUME 9, 2021  If the values did not meet the accuracy, the value of parameter R s will be increased to fit the maximum power point as close as possible.
Not only that, the proposed model also has a different and novel approach when compared to the other works in the literature. The latter implements the parameter extraction technique first to find the model's parameter at STC. This is followed by the parameter scaling/translation technique for other operating conditions. However, this results in decreased accuracy at low irradiance levels, due to the limitations of the latter technique. In contrast, the novelty of the proposed model comes from by implementing the best translation technique first to find the three characteristic points at different irradiance levels. Then, it is combined with the parameter extraction technique, which include the iteratively determined R s value. At each iteration, the R s value is increased during the parameter extraction technique process so that the corresponding I-V curve's maximum power point is as close as possible to the translated characteristic points. Hence, the order and combination of both the translation and parameter extraction techniques, are expected to improve the model's accuracy, especially at low irradiance levels.

VI. RESULTS AND DISCUSSION
Two commercial PV panels, KC200GT (200 Watts) and CNPV-5M (5 Watts) are used for the evaluation of the models. The KC200GT is chosen to represent the high power application in power generation [43], [44], while CNPV5M can represent the low power application such as IoT [45]- [48]. The values of three characteristic points between the model and datasheet are compared. The main reason for choosing these PV panels is that the manufacturer has provided the I-V curves for 1000, 800, 600, 400 and 200 W/m 2 in the datasheet. However, these information were only provided at 25 • C. Therefore, the results can only be compared at this temperature. To obtain the accurate values from the datasheet I-V curves, the GRABIT function from MATLAB is used. The specifications of both panels are summarized in Table 1.
Few works such as Al-Wahed and Abdullateef [24], Batzelis [26], and Ghandi et al. [27] are used for comparison with the proposed model. Both Wahed and Ghandi's works have implemented the parameter extraction technique and characteristic points translation similar to the proposed model. However, Batzelis did not implement the parameter extraction technique due to the complexity involved in the modeling process. Hence, Batzelis improvised the characteristic point translation technique to give a simple and straightforward voltage equation. In contrast, this improvisation drawback is that the technique can only provide the three characteristic points, rather than the entire I-V curve.
The results of the proposed model and discussed works are shown in Figs. 3-7. Fig. 3 shows the I-V curves generated from both panels at 1000 W/m 2 irradiance level. Most of the works fit very closely to the datasheet although Ghandi has a little overshoot at the open circuit voltage for 200 W panel. The overshoot is acceptable, as the percentage difference did not exceed 5% when compared to the datasheet values. However, as the irradiance level reduces, the accuracy of these works start to degrade. For example, at 800 W/m 2 for 5 W panel, Ghandi's technique has overshoot the open circuit voltage by more than 5%. The I-V curves of the other works also start to diverge after the maximum power point. Nevertheless, with Batzelis' improvised translation technique, the three characteristic points' values show a good fit with the datasheet.
At 600 W/m 2 , the discussed works' I-V curves for both panels did not differ significantly as they are able to provide At 200 W/m 2 , the I-V curves generated for both panels show that Wahed is not able to predict the PV behaviour accurately at low light conditions. Moreover, Batzelis also faced difficulties at low light conditions where the maximum power point is not accurate, especially for the 5 W panel. Similarly, Ghandi's I-V curve has exceeded the limit of 5%. It is important to note that the proposed model can accurately match the I-V curves at all irradiances for both panels.     The value of three characteristics points of both panels obtained from datasheets and the discussed works are shown in Table 2 and Table 3. At 1000 W/m 2 irradiance level, the discussed works can achieve a low percentage difference for both panels. Only Wahed has a slightly higher percentage difference of 2.660% for the 200 W panel. However, higher percentage errors are generated for the 5 W panel as the irradiance level decreases. At 800 W/m 2 , Table 3 shows that Wahed has generated a percentage difference of 5.603% on the maximum power current. It can also be noted that, Ghandi has generated a percentage difference of 7.444% on the open circuit voltage, which is more than 4 times when compared to the proposed model's 1.655%.
Starting from 600 W/m 2 , Batzelis has difficulties in accurately modeling the maximum power voltage for the 5 W panel. For example, Batzelis has a percentage difference of 5.788% due to the irradiance factor of maximum power voltage used in the translation technique. The irradiance factor uses the Lambert function and parameter scaling technique to translate the maximum power voltage when the irradiance level changes. However, some ideal state of condition assumptions are made to simplify the process. Therefore, the irradiance factor is unable to model the losses across the resistor accurately at low irradiance levels.
At 400 W/m 2 , Ghandi and Batzelis have generated inaccurate maximum power voltage and open circuit voltage for the 5 W panel. Ghandi used an iterative thermal voltage to translate the open circuit voltage. By applying it iteratively, it tries to fit the characteristic point as close as possible to the datasheet. Although its accuracy is acceptable for the 200 W panel at the same irradiance level, but the accuracy degrades for the 5 W panel, which is at 5.170%. This is because the iterative process is unable to provide an accurate fit for low wattage panels with a low number of PV cells. Moreover, it can be noted that Batzelis' accuracy for the maximum power voltage continues to suffer, with an even higher difference of 8.039%, due to the irradiance factor mentioned previously. In comparison, the proposed model provided a significant 3-fold improvement with 2.377%.
A  can accurately translate most of the characteristic points at high irradiance levels, but the percentage difference at low irradiance for both panels is relatively high. The results show a 9.931% and 9.828% differences on the maximum power voltages, and 5.580% difference on the open circuit voltage. On the other hand, the proposed model results showed differences of only 1.676% and 1.130%, which is a significant improvement of several folds.
The results in Table 2 and 3 show that the proposed model can consistently achieve an overall lower percentage difference when compared to other works at different irradiance levels. The highest percentage difference generated is 4.237% on the open circuit voltage for the 5 W panel, which is still acceptable as it is still below 5%. The proposed model achieves this by combining the best features from Fahad's parameter extraction technique and selective translation techniques from Batzelis and Ghandi. The improved accuracy of the proposed model results over the compared works highlights its importance and contribution.
Based on the discussion of the other works' performance, it can be summarized that Batzelis has the highest percentage difference for the 5 W panel's maximum power voltage due to limitations of the irradiance factor. On the other hand, Ghandi also struggles to provide an accurate open circuit voltage due to the iterative thermal voltage. Therefore, the improved accuracy of the proposed model is due to the use of translation techniques for open circuit voltage from Batzelis and maximum power voltage from Ghandi. Batzelis uses the datasheet's temperature coefficient to predict the behavior of open circuit voltage, which is better than Ghandi and Wahed's iterative thermal voltage. Furthermore, Ghandi also uses the initial series resistor to model the losses at different irradiance levels, which reduces the errors during the translation, especially for the maximum power voltage. All of these works use the same translation for short circuit current and maximum power current. Hence, the observed minimal differences among them.

VII. CONCLUSION
In this paper, one diode with two resistors PV model with improved characteristic points translation and parameter extraction techniques has been presented. The inputs in both characteristic point translation and parameter extraction techniques are based on datasheet values, which is analytical and simple. Then, the proposed model was compared with datasheet values and few similar works to verify its accuracy.
Results show that the proposed model can achieve a consistently higher accuracy, with a percentage difference of less than 5% at all irradiance levels for both PV panels. Although other works can achieve a lower percentage difference, but their accuracy lacks consistency for both panels at all irradiance levels. Therefore, it can be concluded that the proposed model can model the PV behavior at different irradiance levels for high and low wattage panels, at a consistently high level of accuracy. The investigation of the proposed model at different temperatures and with two diode model are interesting directions for future works.