This paper presents a succinct exploration of several analytical methods for extracting the parameters of the singlediode model (SDM) of a photovoltaic (PV) module under standard test conditions (STC). The paper investigates six methods and presents the detailed mathematical analysis leading to the development of each method. To evaluate the performance of these methods, MATLABbased software has been devised and deployed to generate the results of each method when used to extract the SDM parameters of various PV test modules of different PV technologies. Similar software has also been developed to extract the same parameters using wellestablished numerical and iterative techniques. A comparison is subsequently made between the synthesized results and those obtained using numerical and iterative methods. The comparison indicates that although analytical methods may involve a significant amount of approximations, their accuracy can be comparable to that of their numerical and iterative counterparts, with the added advantage of a significant reduction in computational complexity, and without the initialization and convergence difficulties, which are normally associated with numerical methods.
Photovoltaic (PV) systems offer the most direct conversion of the sunlight energy into electricity making them one of the most appealing systems for renewable energy generation [
The normalized current–voltage (I–V) and power–voltage (P–V) characteristics of a typical PV generator are shown in
An MPPT system uses a DC–DC power converter controlled by an MPPT algorithm that is normally implemented using a microcontroller [
As shown in
To estimate the model parameters of the SDM at STC, numerous methods with varying degrees of mathematical and computational complexities and accuracy have been reported in the literature, for example [
In addition, there are other strategies for the parameter extraction, such as those based on computational intelligence [
Referring to the SDM in
The first term is the photocurrent
At room temperature, the thermal voltage is about 26 mV. The ideality factor for a silicon PV cell is typically between one and two. For a typical PV module, the series resistance is in the order of
Equation (1) is an implicit and transcendental equation that would normally require a numerical solution to obtain the five parameters of the SDM. However, since numerical methods are susceptible to convergence and initialisation difficulties, several alternative analytical methods have been described in the literature. These methods aim at reducing the complexity of the implicit general equation of the SDM by introducing some approximations to develop explicit models as detailed in
The contributions of this article are: Explore the most literaturereported analytical methods for parameters extraction of the SDM, present detailed derivation of the mathematical expressions that lead to each method which are not normally detailed in the reporting literature, and synthesize the results of each method by deploying them to extract the parameters of PV modules of different technologies. Finally, the paper presents a comparison between these synthesized results and those obtained using numerical and iterative techniques.
After this introductory section,
When the entire I–V curve is available, the parameters of the SDM may be estimated using curve fitting or optimisation algorithm [
Substituting the SC point (
Similarly, substituting the OC point
Substituting the maximum power point
Another important quantity required for parameter extraction is the derivative of the module’s current with respect to its voltage, i.e.,
Solving for the derivative:
It is to be noted that the above expression for the derivative is valid at any point on the I–V curve. Another derivative used in parameter extraction procedures is that of the power with respect to voltage defined as:
At the MPP point, this derivative equates to zero:
Therefore, at the MPP the current derivative becomes:
The shunt resistance at the SC point
Similarly, the series resistance at the OC point
For the five parameters extraction process, the majority of the parameter extraction methods, for example [
In this section we present the mathematical analysis and derivation of the equations required for the extraction of the SDM parameters for six main analytical methods.
This method was developed to determine the parameters of a blue and grey solar cell using experimentally obtained I–V curve and the three salient points, i.e., the SC, OC, and the MPP, which are available in datasheets [
The photocurrent is estimated using the shortcircuit current point
The shortcircuit current
We can derive an expression to estimate the shunt resistance using the expression for derivative
Substituting for the derivative at the SC point from Equation (11) into Equation (14) we obtain:
The exponential term [
This means that the shunt resistance is estimated directly from the gradient of the I–V curve at the SC point.
To estimate the series resistance, we can use the gradient of the I–V curve at the OC voltage point. Substituting the OC point (
Substituting for the derivative at the OC point from Equation (12) and rearranging:
The term
This is not a closed form expression for the series resistance since it contains the saturation current and the ideality factor which need to be determined. The opencircuit voltage is obtained from the datasheet.
By substituting, the opencircuit Equation (4) into the shortcircuit Equation (3) and neglecting the (−1) in the exponential terms, we obtain:
This may be rearranged as:
However, since in practice at
The OC voltage and the parallel resistance can be estimated from the I–V curve however, the ideality factor is still unknown and needs to be determined.
Substituting the OC voltage equation for the photocurrent, i.e., Equation (4), into the MPP Equation (5), and after ignoring the (−1) in the exponential terms, we can write:
Since
An expression for the ideality factor can now be derived as explained below.
Substituting Equation (16) into (22):
Substituting Equation (25) into (24), we obtain:
This can be simplified to:
Therefore, the ideality factor becomes:
This is not a closed from expression because it contains the series resistance, which must be determined. This can be obtained by substituting the expression for the saturation current of Equation (25) into the expression for the series resistance of Equation (19) as:
Hence, the expression for the series resistance becomes:
Substituting this expression for the series resistance, i.e., Equation (30), into Equation (28):
This can be simplified to:
Solving for the ideality factor:
This is now a closed form expression for the ideality factor because all the variables can be obtained or estimated from the datasheet and the I–V curve of the cell. For a PV module with
This method uses the idealised SDM model shown in
The general current equation for this model is:
A the shortcircuit point
That is the photocurrent is the same as the SC current, and this is always provided by the PV module’s datasheet. At the opencircuit point
Alternatively, since in this model,
We can solve this for the OC voltage as:
Using Equation (37), the saturation current can be expressed as:
At the MPP, we can express the current as:
The derivative
The derivative at the maximum power point is:
Substituting Equation (42) into Equation (41), the current at the MPP becomes:
Substituting (37) and (43) into (40) and ignoring the (−1) in the exponential terms:
Equations (35), (38), (43) and (44) represent the main equations of the explicit model. To estimate the maximum power point voltage
The asymptotic behaviour of the I–V curve around the opencircuit and shortcircuit points were used to estimate the derivative needed in the above equation as [
Substituting Equation (46) into (45), we obtain:
Substituting Equation (47) into (40):
Simplifying the above, we obtain:
The maximum power is:
Therefore,
To estimate the ideality factor under STC conditions, we substitute Equation (39) into (40) and ignore the (−1) since the exponential terms in both is much larger than unity:
Rearranging and simplifying, we obtain:
Solving for the ideality factor at STC, we obtain:
In attempting to improve the accuracy of this method, Mahmoud et al. [
Yousef et al. [
This method assumes that the shunt resistance is very large so that it can be neglected hence, it is developed using the singlediode equivalent circuit model shown in
The shortcircuit equation is:
The opencircuit equation is:
The MPP equation is:
Referring to the equation of the shortcircuit condition, i.e., Equation (62), the second term represents the diode current. In practice, since the voltage
That is the module’s photocurrent is the same as the SC current, which is always given in datasheets.
Under opencircuit condition, the entire shortcircuit current is the diode current, i.e.,
Since the exponential term is much greater than one, we can ignore the (−1) hence, the saturation current can be expressed as:
This is, however, not a closed form expression for the saturation current since it includes the ideality factor which must be determined.
The series resistance may be found by substituting Equation (67) into the MPP Equation (64) as:
Neglecting the (−1) in the second exponential term and replacing the photocurrent by the shortcircuit current and rearranging:
Rearranging and solving for the series resistance, we obtain:
This is not a closed form expression since it includes the ideality factor, which needs to be determined.
The ideality factor is derived using the fact that the slope of the I–V curve at the maximum power point is equal to zero. Equating Equation (10) and Equation (7) at the MPP:
Rearranging as:
The above may be written as:
Substituting for the series resistance
This may be rearranged as:
Substituting for the saturation current from Equation (67):
This may be rearranged as:
The denominator on the righthand side of Equation (77) may be rewritten with the series resistance substituted by its expression from Equation (70) and simplified as follows:
Rearranging:
Therefore:
Substituting (80) into (77)
Simplifying and solving for the ideality factor:
This is a closed form expression for the ideality factor since all variables in its expression are available in the datasheet. Therefore,
This method uses information available in the datasheet to estimate the parameters of the SDM. It uses a piecewise I–V curve fitting scheme along with the four parameters PV model to evaluate them [
Using Equation (7) the derivative of voltage with respect to current
Simplifying:
At the SC point, the derivative becomes:
At the OC point, the derivative becomes:
At the MPP point, the derivative is:
Equating equations (88) with the reciprocal of the derivative defined in Equation (10):
Using Equations (3), (4), (87), (88) and (89) with a series of simplifications, the following equations are obtained to extract the five parameters
And
According to the actual measurement of a PV module, the I–V curve in the low and highvoltage zones is smooth and can be represented by straight lines therefore, the slopes of the straight lines can be considered as the differential values of the I–V curve in the two zones as explained in [
This method which is also based on the manufacturer’s datasheet, uses a reduced set of approximations compared to the previous analytical methods without increasing complexity by incorporating two boundary conditions [
Referring to Equation (3), the term
Therefore, we can write for the photocurrent:
Substituting (97) in Equation (4), and neglecting the (−1) in the exponential term of the latter, we have:
Solving for the saturation current:
Substituting equations (97) and (99) into Equation (5) and ignoring the (−1) in the exponential term in the latter:
Simplifying:
This may be rearranged as:
Considering the equation for the derivative, Equation (6), at the MPP and using Equation (10), we can write:
Substituting (97), (99) and (102) in (103):
Using equations (97) and (102), we can rewrite Equation (104) as:
From Equation (105), an expression for the shunt resistance may be obtained as follows:
Multiplying Equation (106) by
Let x be defined as:
Hence, (107) becomes:
Simplifying:
Rearranging:
Solving for the shunt resistance
Using the definition of x in Equation (108), we can rewrite Equation (102) as:
Substituting the value of the shunt resistance from Equation (112) into Equation (113), we obtain:
Cross multiplying and simplifying, the above may be rearranged as:
The lefthand side of Equation (115) may be simplified to:
The righthand side of Equation (115) may be reduced to:
Therefore, Equation (115) simplifies to:
Combining equations (102) and (118) an alternative expression for the shunt resistance is obtained as:
Using the above expression for the shunt resistance, i.e., Equation (119), an expression for the ideality factor can be derived as follows:
The expression in (15) may be reexpressed as:
Simplifying, we obtain:
Ignoring the exponential term, we obtain:
Substituting Equation (123) into Equation (120) we obtain the expression given in [
Substituting Equation (124) into (118) we obtain an explicit and closed form expression for the series resistance as:
Finally, the gradients of the I–V curve at the SC and OC points may be estimated as [
This method presented a unique strategy for developing the analytical expressions required for estimating the parameters of the SDM model [
Using the opencircuit voltage of Equation (4) and after neglecting the shunt resistance, a new expression that connects the modified ideality factor and the opencircuit voltage at STC is given as:
The modified ideality factor is assumed to vary linearly with temperature but is independent of insolation. For any arbitrary temperature, Equation (130) is adjusted as:
Using (3), (4), (5), (8) and (130), the parameters of the SDM can be extracted numerically. The method relies on using equations presented in [
All the mathematical derivations of (130)–(137) can be found in references [
To synthesize the results of the analytical methods described above, software has been developed using MATLAB to extract the five parameters of the SDM model for different photovoltaic modules. Similar software has also been developed to estimate the same parameters using two numerical methods: the first uses Newton–Raphson algorithm and the initialisation provided in [
However, it is a matter of fact that there are different combinations of the five parameters of the SDM whose I–V curve pass through the same salient points (SC, OC, and MPP) which does not necessarily imply that all these curves represent physical meaning [
Methods one and five exhibit similar performance, particularly in the higher current region of the I–V curve since both are based on the slope of the I–V curve in this region as shown in
Method 6 demonstrated good agreement with the iterative method as illustrated in
The paper presented a detailed mathematical analysis and comparative evaluation of the performance of the commonly reported analytical methods for parameters extraction of the singlediode model of a PV module. Six prevalent methods have been explored and deployed to extract the parameters of three PV modules of different PV technologies. The extracted parameters were compared with reference values extracted using numerical and iterative methods. It has been confirmed that while some methods may not be the most accurate in extracting the parameters, e.g., Methods two and three, they can still provide good agreement between their I–V curves and those obtained numerically. The reduced accuracy of Method two in the parameter extraction process could be attributed to the fact that it neglects the shunt resistance which can reduce the accuracy particularly at low levels of insolation. Methods one and five use the slope of the I–V curve about the shortcircuit point which led to both methods having similar results. Method four resulted in slight overestimation in the high current region of the I–V curve since it approximated the I–V curve by a straight line in this region to simplify the model equations. Method six resulted in good agreement with the iterative I–V curve. It is recommended that additional extensive investigation should study the dependence of the analytical methods on the technology and different materials of PV modules.
Conceptualization, N.A.; Data curation, H.I.; Formal analysis, N.A. and H.I.; Investigation, H.I.; Methodology, N.A. and H.I.; Software, N.A. and H.I.; Supervision, N.A.; Writing—original draft, N.A. All authors have read and agreed to the published version of the manuscript.
This research received no external funding.
The authors declare no conflict of interest.
Normalised experimental current–voltage (I–V) and power–voltage (P–V) characteristics of a generic PV generator.
The singlediode model of a photovoltaic (PV) generator.
The idealised singlediode model (SDM) model of a PV generator.
The fourparameters SDM.
Comparison between the I–V (
Comparison between the I–V (
Comparison between the I–V (
Comparison between the I–V (
Parameters for the Multicrystalline (Kyocera KC200GT).
Datasheet  KC200GT  LC5012M  180BA19 






8.21 A  3.2 A  3.65 A 

32.9 V  22.5 V  66.4 V 

7.61 A  2.9 A  3.33 A 

26.3 V  17.2 V  54 V 

−1.23 × 10^{−1}  −7.88 × 10^{−2}  −173 × 10^{−3} 

3.18 × 10^{−3}  2.88 × 10^{−3}  1.01 × 10^{−3} 

54  36  96 
Parameters for the Multicrystalline (Kyocera KC200GT).
Method  Parameter  

n 





Method 1  1.08317  0.27077  124  2.4885 × 10^{−9}  8.22793 
Method 2  1.81764  0  infinite  1.78074 × 10^{−5}  8.21 
Method 3  1.40991  0.19455  infinite  4.09919 × 10^{−7}  8.21 
Method 4  0.65008  0.39999  82.5508  1.14541 × 10^{−15}  8.24978 
Method 5  0.88423  0.38033  123.62  1.81544 × 10^{−11}  8.23526 
Method 6  1.00258  0.30567  130.466  4.43777 × 10^{−10}  8.22924 
Iterative [ 
1.3  0.2283  572.124  9.89443 × 10^{−8}  8.21329 
Numerical [ 
1.3405  0.2172  951.327  1.7097 × 10^{−7}  8.2119 
Parameters for the Monocrystalline (Lorentz LC5012M).
Method  Parameter  

n 





Method 1  2.0361  0.10045  206  2.0109 × 10^{−5}  3.20156 
Method 2  2.41979  0  ∞  1.3832 × 10^{−4}  3.2 
Method 3  1.76187  0.4969  ∞  3.24464 × 10^{−6}  3.2 
Method 4  0.79342  0.93255  105.70414  1.47618 × 10^{−13}  3.22823 
Method 5  1.24254  0.77359  205.22641  9.82922 × 10^{−9}  3.21206 
Method 6  0.99693  0.84024  125.53699  8.22168 × 10^{−11}  3.22142 
Iterative method [ 
1.2  0.784  186.40574  5.06574 × 10^{−9}  3.21352 
Numerical method [ 
No convergence 
Parameters for the Thin film (Sanyo 180BA19).
Method  Parameter  

n 





Method 1  0.55767  2.69004  2329  3.99938 × 10^{−21}  3.65422 
Method 2  2.06455  0  ∞  7.9701 × 10^{−6}  3.65 
Method 3  2.11483  −0.09068  ∞  1.08651 × 10^{−5}  3.65 
Method 4  0.95729  1.13544  313.85129  2.13594 × 10^{−12}  3.6632 
Method 5  1.95145  0.10657  2328.8934  3.71538 × 10^{−6}  3.65017 
Method 6  0.95589  1.41883  327.95525  2.1766 × 10^{−12}  3.66579 
Iterative [ 
1.8  0.27300  1181.56509  1.17348 × 10^{−12}  3.65084 
Numerical method [ 
No convergence 