Systems Engineering Methodology for Verification of PV Module Parameter Solutions

Numerous sources provide methods to extract photovoltaic (PV) parameters from PV module datasheet values. The inputs are the number of series cells <inline-formula> <tex-math notation="LaTeX">$\text{N}_{\mathrm {s}}$ </tex-math></inline-formula>, open circuit voltage <inline-formula> <tex-math notation="LaTeX">$\text{V}_{\mathrm {oc}}$ </tex-math></inline-formula>, maximum power voltage <inline-formula> <tex-math notation="LaTeX">$\text{V}_{\mathrm {mp}}$ </tex-math></inline-formula>, maximum power current <inline-formula> <tex-math notation="LaTeX">$\text{I}_{\mathrm {mp}}$ </tex-math></inline-formula>, and short circuit current <inline-formula> <tex-math notation="LaTeX">$\text{I}_{\mathrm {sc}}$ </tex-math></inline-formula>. The 5 Parameter Model solutions outputs are diode ideality factor <inline-formula> <tex-math notation="LaTeX">$\eta $ </tex-math></inline-formula>, series resistance <inline-formula> <tex-math notation="LaTeX">$\text{R}_{\mathrm {s}}$ </tex-math></inline-formula>, parallel resistance <inline-formula> <tex-math notation="LaTeX">$\text{R}_{\mathrm {p}}$ </tex-math></inline-formula>, photon light current <inline-formula> <tex-math notation="LaTeX">$\text{I}_{\mathrm {L}}$ </tex-math></inline-formula>, and diode reverse saturation current <inline-formula> <tex-math notation="LaTeX">$\text{I}_{\mathrm {o}}$ </tex-math></inline-formula>. The parameter solution requires solving three simultaneous transcendental equations for <inline-formula> <tex-math notation="LaTeX">$\eta $ </tex-math></inline-formula>, <inline-formula> <tex-math notation="LaTeX">$\text{R}_{\mathrm {s}}$ </tex-math></inline-formula>, and <inline-formula> <tex-math notation="LaTeX">$\text{R}_{\mathrm {p}}$ </tex-math></inline-formula> and additional calculations for <inline-formula> <tex-math notation="LaTeX">$\text{I}_{\mathrm {L}}$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$\text{I}_{\mathrm {o}}$ </tex-math></inline-formula>. One of the primary tenants of Systems Engineering, verification, was applied to parameter solution results to check for physical and model fitness. This manuscript provides novel methods to verify parameter results and applies them to available solutions.


I. INTRODUCTION
Typically, the analysis of a PV system uses the 5 Parameter Model [1]. A notional diagram is shown in Fig 1. Photon light input generates the light current I L . The generated light current conducts through the diode as I d , the parallel resistance as I p , and the external load as I e . Using Kirchhoff's Current Law, Equation (1) was derived.
Equation (1) is modified to a format with the external current on the left, the diode current I d , replaced with the Shockley diode equation, and the parallel current I p shown in terms of the circuit parameters. This yields a PV module primary 5 Parameter Equation (2). V e + I e R s R p (2) • I e is the external current.
• I L is the photon light current.
• I o is the diode reverse saturation current.
• V e is the external voltage.
The associate editor coordinating the review of this manuscript and approving it for publication was Giambattista Gruosso . • R s is the series resistance.
• ϒ is the PV module's thermal voltage (ϒ = N s η v t ) (The number of series PV cells N s , the diode ideality factor η, and thermal voltage v t .) • R p is the parallel resistance.

II. MODULE CONDITIONS
PV module datasheets provide five characteristics, N s, the number of series PV cells in the module, the open circuit voltage V oc , the voltage at maximum power V mp , the current at maximum power I mp , and the current at short circuit I sc .  with an irradiance of 1,000 W/m 2 [2]. At STC, the characteristic values are fixed and static. Fig 2, which plots Equation (2), shows the locations of the open circuit, short circuit, and maximum power point conditions on the PV I-V (current-voltage) curve. In addition, the power curve, P=IV, was plotted.
Manipulation of the PV equation under open circuit, short circuit, and maximum power conditions result in the following equations in terms of the photon current I L . Following the methods established in [3], the simplified Equations for short circuit, Equation (3), open circuit Equation (4), and maximum power Equation (5) (3) and (4), derives a solution for I o in terms of datasheet characteristics and unknown parameters R s , R p , and Υ .
With Equation (3) to find I L and Equation (6) solving for I o , the three remaining unknowns, Rs, Rp, and ϒ, require three additional equations. Following the method described in [4] and combining Equations (3), (4), (5), and (6), a solution for I mp shown in Equation (7) was derived. The transcendental equation is in terms of unknowns R s , R p , and ϒ and known datasheet characteristic values of N s , V oc V mp , I mp , and I sc .
Two more equations were derived by using the characteristic of the PV I-V and Power curves. The two locations chosen are at the short circuit and the maximum power points, identified in Fig 2. Taking the derivative of the PV equation at short circuit, manipulating, and simplifying, Equation (8), another transcendental was derived. This is the second equation.
The third equation uses the power curve, where the derivative is zero at maximum power. Equation (9), also transcendental, provides the required third for the solutions.
By solving the three simultaneous transcendental equations (7), (8), and (9), parameter solutions for the values of R s , R p , and ϒ were found. The solutions for η, I L , and I o , were then obtained by simple calculations. The diode ideality factor η was found using ϒ and solving equation (10). The factors are k Boltzmann's constant, the temperature T in Kelvin, the charge of an electron q, and the number of series PV cells N s . Equation (3) calculates I L, and Equation (6)

IV. SOLVING FOR 5 PV PARAMETERS A. DATASHEET VALUES
Various sources utilized different methods to solve these three simultaneous transcendental equations. The Kyocera KC200GT PV module data was utilized to demonstrate this error analysis. Table 1 provides datasheet characteristics for the KC200GT PV module [5]. The fixed characteristics are N s , the count of the number of series connection PV cells, and the others obtained from testing at STC. Table 2 summarizes the solution set, the methods used to solve for the 5 parameters, and the source of the solutions. The   authors analyzed the first two, and the following were from a collection of 13 different sources. A total of 20 parameter solution sets were analyzed.

C. SOLUTION PARAMETER SETS
With PV data from the KC200GT PV module listed in Table 1 and the solutions provided by the sources listed in Table 2, sets of the 5 parameter solution values were copied into Table 3. The table lists the solutions of the 5 parameters η, R s , R p , I L , and I o . The parameter digits listed were those provided from each source solution method and subsequently used in the error calculations.

D. REPORTED SOLUTION ERRORS
Many of the sources reported solution error results. Table 4 summarizes the source set #, error calculation methods, and the reported errors. For consistency, all errors were, when required, converted and presented in percent. RMS is Root Mean Square, and RMSE is Root Mean Square Error. When ''Graph'' was listed, the error referenced a graph in the source, which reported different error values depending on conditions.  Table 2, except for the value of diode ideality factor η solved by method #2, passed these physical parameter checks.

1) DIODE IDEALITY FACTOR
An ideal diode has a value of 1.0, and for silicon devices, a reasonable number is 1.2 to 1.3 [8]. To enable support for a wider range of solutions, the diode factor limit was set from 1.0 to 1.6. For example, in set #2, the MATLAB ''PV_Array.slx'' provided a relatively low error solution, but the solved diode ideality factor of 0.60957 is not physically reasonable.

2) SERIES RESISTANCE
The series resistance R s is the contributions of the PV cell material, contact between the PV cell to metal, and the metal 44256 VOLUME 11, 2023 Authorized licensed use limited to the terms of the applicable license agreement with IEEE. Restrictions apply.
contact path resistances. Values for R s are bound by the ideal, but not physically possible 0 on the low end, and the slope of the I-V curve from V oc to the maximum power point on the high end. For the KC200GT, the slope from V oc to the maximum power point (V oc -V mp )/I mp establishes the maximum series resistance of 0.867 for R s . Typical PV modules have external 2.6 mm (#10 AWG) copper cables with a resistance of about.006 per meter. One meter of the connecting cable has a resistance of 0.006 . Other resistance, such as the PV cell material, interfaces, and conductive traces, increase this resistance. The calculated lower limit was set to 1/50 of the R s maximum or an R s minimum of 0.017 .

3) PARALLEL RESISTANCE
The parallel resistance R p sources include manufacturing defects and leakage paths around the PV cells. From Fig 2, the minimum resistance R p is the inverse slope of the IV curve at short circuit. For the KC200GT, the slope from the I sc to the maximum power point V mp , I mp is calculated as V mp /(I sc -I mp ) or 43.8 . This calculation set the lower bound for R p . The upper limit is controlled by parallel resistance paths. Using the ratio determined for R s , the R p upper limit was set at 50 times for an R p maximum of 2,190 .

B. RMS 5 ERROR EQUATION VERIFICATIONS
The verification checks model fit by calculating errors on the three primary points of the I-V curve defined by Equation (2) The overall merit of the solution set was judged by the RMS error of the results of the 5 error equations. The 'RMS 5 Error' calculation treated all five error results with equal weights, as shown in RMS 5error Equation (16), as shown at the bottom of the page. Table 5 shows the method # from Table 2, the 5 absolute error calculations in percent for Equations (11), (12), (13), (14), and (15), and the overall absolute RMS 5 Error in percent from Equation (16).

VI. DISCUSSION OF ERRORS A. CURRENT CHECK AT SHORT CIRCUIT
The first check compares the solved I L parameter versus the I L calculated with Equation (11). This is an easy check, and generally, the errors were low since the controlling factor is the ratio of R s to R p . Any reasonable solution should have a very low error for this check. Except for parameter set #20, with an error of 3.05%, all errors were under 1%, with an average of 0.11% and several 0.00%.

B. CURRENT CHECK AT OPEN CIRCUIT
The check for current at open circuit uses Equation (12) which includes an exponential function. Because of the exponential, any error in the solution for η will significantly impact the results. The exponential result is then multiplied by the diode reverse saturation current I o to determine the diode current I d . This means a low error solution must include accurate values of both η and I o . These calculations showed that several solution sets contained significant errors for this verification test. For set #17, the diode factor η of 1.277 is a reasonable value but, combined with an I o of 6.9127 nA, produces an unreasonable diode current I d of only 802 mA.

C. CURRENT CHECK AT MAXIMUM POWER
The third check utilized Equation (13), which also includes an exponential factor. In the case of maximum power, the diode current I d , calculated with the exponential function and I o , was a smaller part of the overall current compared to the open circuit case. This was reflected, for example, in set #8, where the current error at maximum power was 53.3% versus -806% for the open circuit current error.

D. VOLTAGE CHECK AT OPEN CIRCUIT
The influence of the exponent, diode ideality factor η, and reverse saturation current is also important in this error verification. In the worst cases, sets #4, #5, #6, and #7 contain errors of almost −1,300%.

E. VOLTAGE CHECK AT MAXIMUM POWER
The fifth check includes the influence of the exponent but with a lower effect than in the open circuit check. The worst cases are again in sets #4, #5, #6, and #7, with errors of almost -1,100%.

F. OVERALL RMS 5 ERROR
Set #1 shows the lowest error, which to 3 decimals was 0.00%. The worst cases were sets #4 and #7, with an error of 830%. Table 5 shows that solutions with a high error have solved for an inferior value for the diode current I d , found from the diode ideality factor η and the diode reverse saturation current I o .

VII. ANALYSIS OF ERROR SENSITIVITY
Solution sets #1 and #15 parameters are similar, with the only difference in the number of digits provided by each. In set #1, all the solved values were calculated and presented with 10 digits. In set #15, the values of η, R s , and I o were stated to 3 digits, R p to 6 digits, and I L to 4 digits. This difference led to the question of how the number of digits used in a solution affects the error calculation results. For solution set #1, Table 6 shows the number of solution digits starting with 10. Subsequent rows rounded the solution to 7, 6, 5, 4, 3, and 2 digits for each parameter. Table 7 shows the resulting errors found by equations (11) to (16) using the input parameter values from the original 10 to 2 digits. Using all 10 digits resulted in calculated errors   of 0.000% for all equations. Rounding the solution to 4 digits calculated, for example, an Error oc of 0.817% and an overall RMS 5 error of 0.731%.
For set #1, an analysis of Table 7 data shows little error difference between 10 and 6 digits. Five digits provide a high precision result of under 0.2% for the RMS 5error calculation. The IEC test standard for PV Modules specifies 0.2% measurement accuracy [22]. Four digits provide a reasonable solution with an error of under 1%.

VIII. RECOMMENDATIONS
It is recommended that any solution in calculating values for the 5 Parameter model (η, R s , R p , I L , and I o ) should include verification checks. A solution that results in unreasonable physical values or high RMS 5 Error should be re-evaluated. Any solution with less than 4 digits can provide unreliable results and should not be used.
Since the solution to the RMS 5 Error method requires solving the transcendental equations (14s) and (15s) to find the voltage errors at open circuit and maximum power point, an alternative RMS 3 Error method can be used. This method only checks the RMS errors from Equations (11), (12), and (13). The error calculation is Equation (17). Table 8