The input dataset consists of 100 voltage-current pairs measured from the short-circuit region to the open-circuit region of a commercial crystalline-silicon PV module. The dataset represents one operating condition at STC. Accordingly, the present validation should be interpreted as a single-condition demonstration rather than as evidence of robustness across different PV modules or operating conditions. In PV testing, STC conventionally corresponds to 1000 W/m² irradiance, 25 cell temperature, and the AM1.5 reference spectrum. Therefore, the extracted parameters should be interpreted as reference-condition parameters rather than as complete outdoor parameters. Because only one I-V curve is used, the extraction problem is deliberately constrained to the reproduction of the measured STC curve. No effort is made to derive complete parameter functions dependent on irradiance or temperature from only this one dataset. This distinction is essential since outdoor forecasting is based on the need for further temperature and irradiance translation equations and multi-case validation. The present study does not involve an independent experimental measurement campaign; rather, it focuses on parameter extraction from an available STC I–V dataset. Detailed information on the original measurement instrumentation, calibration procedure, measurement uncertainty, and repeatability was not available for the present analysis.

The main electrical characteristics obtained directly from the measured I–V and P–V curves are summarised in Table 1.

The electrical power at each measured point is calculated as:

The fill factor is calculated as:

Thus, the supplied curve corresponds to an approximately 286.6 W PV module at STC.

The choice of the five-parameter SDM model can be justified since it is a compact model with a meaningful physical interpretation of the model curve. Unlike the ideal current-source model, the SDM incorporates diode behaviour together with series-resistance losses and shunt-leakage effects. And unlike the double-diodes or triple-diodes cascade models, this one does not have any additional variables which cannot be classified using only one curve without additional assumptions.

The model divides the electrical current into three theoretical elements: current produced by light, diode current, and current through leakage. This structure allows the fitted parameter set to be interpreted in relation to the shape of the I-V curve: fixes the current plateau, and a control the exponential knee, influences the high-voltage slope and fill factor, and influences the low-voltage slope and leakage behaviour.

The five-parameter single-diode model is expressed as:

where is the module current, is the module voltage, is the photocurrent, is the diode reverse saturation current, is the series resistance, is the shunt resistance, and is the modified diode ideality factor. The modified ideality factor is:

Here, is a module-level modified diode ideality parameter expressed in volts, incorporating the effective diode ideality factor , the equivalent number of series-connected cells , and the cell thermal voltage .

The thermal voltage is:

At STC, K and the thermal voltage of one cell is VT=0.0256926 V

Using J/K and C;

For a 60-cell equivalent crystalline-silicon module:

After extraction, the effective diode ideality factor is recovered from:

Each parameter of the five-parameter single-diode model has a distinct physical meaning and affects a specific region of the I-V curve. The photocurrent mainly determines the current level near the short-circuit condition and is therefore closely associated with the incident irradiance. The diode reverse saturation current governs the exponential current loss and has a strong influence on the open-circuit region. The modified diode ideality factor a, or equivalently the ideality factor n, controls the curvature and sharpness of the knee region. The series resistance is an indicator of losses in the cells, contacts, interconnects, and wires. Increasing lowers the fill factor. The resistance compared to shunt currents describes the leakages in the junction or the whole module. A low value of allows more shunt current to move through the solar cell, resulting in worse stability of the low-voltage plateau. The physical interpretation, main curve influence, and practical implication of each SDM parameter are summarised in Table 2.

PV parameter extraction methods can be divided into five distinct classifications based on the type of input data, solving strategy, and curve information level included in the extraction process. Characteristic-point analytical methods use datasheet or measured values such as , , MPP, and temperature coefficients. These methods are fast and easy to implement, but they usually require simplifying assumptions and may not reproduce the complete I-V curve accurately. Analytical slope techniques utilise local derivatives in the near vicinity of open circuit and short circuit situations in order to calculate resistance parameters. Despite the intuitive nature of these techniques, the validity of their results may vary greatly depending on environmental noise, smoothing of the curve, and digitisation errors. Numerical deterministic methods can be used to solve nonlinear equations or to directly minimise the residual of the function. They may yield correct results provided that the initial guesses, limits, and convergence criteria are appropriately selected. Metaheuristic methods search the parameter space using population-based or nature-inspired optimisation algorithms and can handle non-convex objective functions, but they often require many function evaluations and careful adjustment of algorithmic control parameters. Hybrid methods combine analytical initialisation with numerical or metaheuristic refinement in order to improve convergence and robustness. The main features, advantages, limitations, and suitability of these approaches for the present work are summarised in Table 3.

The computational effort reported in Table 3 is expressed qualitatively because absolute execution time depends strongly on the implementation, hardware, dataset size, stopping criteria, and initialisation strategy. Analytical approaches generally require the least computational effort, whereas deterministic full-curve least-squares fitting requires repeated residual evaluation over all measured points. Metaheuristic and hybrid approaches typically involve a larger number of objective-function evaluations and therefore require greater computational effort.

The method used in the present study belongs to the full-curve deterministic least-squares family. It is selected because the available dataset contains the complete measured I-V curve and because the objective of this study is to obtain a transparent, reproducible, and physically interpretable parameter set rather than to benchmark stochastic optimisation algorithms.

The implicit single-diode model was rearranged into residual form. The diode and shunt current terms are defined as:

The residual at each measured point is:

For an exact model, for all measured points. The parameter vector is:

The optimisation problem is:

This formulation fits the implicit model directly to the measured voltage-current pairs. Because all 100 data points are used, the extracted parameter set is constrained by the low-voltage plateau, knee region, and open-circuit side simultaneously.

The nonlinear least-squares problem is solved using physically constrained parameters. The initial values used in the optimisation are summarised in Table 4.

The initial diode saturation current is estimated from the open-circuit condition:

Using the values given in Table 4, the initial diode saturation current is calculated as

To improve numerical conditioning, , , and are handled in logarithmic form during optimisation. This prevents negative resistance or saturation-current values and improves convergence because these parameters can span several orders of magnitude.

The overall parameter-extraction procedure and the sequence in which the governing equations are applied are summarised in Figure 1.

After the parameters are extracted, the model current at any imposed voltage is obtained by solving the implicit equation:

where and follow Equations (8) and (9) for the selected voltage value.

The power curve is then reconstructed from:

The model maximum-power point is identified by maximising over the interval .

The efficacy of the extracted model is assessed using both characteristic-point and full-curve indicators. The pointwise current error is defined as:

The root-mean-square error is:

The mean absolute error is:

The coefficient of determination is:

The maximum-power error is:

The specified metrics were chosen because of the fact that RMSE punishes more for bigger deviations, MAE allows calculating average absolute current error in an easy manner, represents global similarity of the curve, while maximum power error shows how relevant the model is for energy and MPPT issues.

The received parameter set is summarised in Table 5. It is shown that the obtained solution is able to mimic the experimental curve with a very small number of residuals for the entire range of the voltage.

The diode ideality factor was calculated as:

The equation of the fitted model is given by Equation (3) based on the values in Table 5. To avoid a lengthy equation, the fitted current balance is divided into three components:

The newly extracted model shows the characteristics of the measured I-V curve quite effectively in terms of important points, as demonstrated in Table 6.

The percentage error in maximum power is:

The model also satisfies the maximum-power-point slope condition. At the fitted MPP:

The maximum-power error is quite insignificant at 0.009%, which indicates that the model has managed to maintain the actual power characteristics of the measured curve. Moreover, the small difference in voltage and current at MPP is negligible, since some variations of the and have hardly any influence on the peak maximum power on the P-V curve.

The goodness of fit of the extracted single-diode model is studied by comparing the calculated current values to the experimental ones for the entire region of analysed voltage values from short circuit to open circuit. This is important because the model may show satisfactory correlation with such points like , , or maximum power point, but provide noticeable discrepancies at other points.

The statistical error indicators obtained from the full-curve comparison are presented in Table 7. The root-mean-square error is only A, while the mean absolute error is A. These values are extremely small compared with the module current level of approximately 9.3 A, indicating that the reconstructed curve follows the measured curve with very high precision. The maximum absolute current error is A, which confirms that even the largest pointwise deviation remains below one milliampere. In addition, the coefficient of determination was R²=0.99999996, showing an almost perfect agreement between the measured and modelled current values.

To clarify the relationship between the curves both measured and calculated, corresponding values of voltage and current have been checked and presented in Table 8. Chosen points were distributed in such a way as to cover the low voltage current plateau, the knee region, the high power region, and the reverse current area of the characteristic. For V=0 V, the model shows 9.29989 A as compared to 9.300 A short circuit current. In the high power range case, for instance, at V=33.116 V, the measured current equals 8.653 A while the one predicted by the model is equal to 8.65350 A. Near the open circuit at V=39.500 V, the model current is only 0.00006 A, which is practically equal to zero and, hence, corresponds to the measured open circuit condition.

The representative errors in Table 8 show that the deviations are small and randomly distributed around zero, rather than being concentrated in a particular voltage region. This behaviour indicates that the extracted parameter set does not only force agreement at a few selected points but also preserves the overall shape of the measured I-V curve. The current plateau at low voltage, the transition through the knee region, and the rapid current decrease near open circuit are all accurately reproduced.

The graphical comparison in Figure 2 further confirms this agreement. The fitted I-V curve almost overlaps the measured curve across the entire operating range, demonstrating that the full-curve least-squares formulation successfully captures the electrical behaviour of the commercial PV module at STC.

Similarly, Figure 3 shows that the reconstructed P-V curve follows the measured power curve closely and reproduces the maximum-power region with negligible deviation. This is particularly important because the accuracy of the P-V curve directly affects maximum-power prediction, MPPT studies, and energy-yield simulations.

Overall, the full-curve accuracy indicators, the representative pointwise comparison, and the graphical agreement demonstrate that the extracted five-parameter single-diode model provides a highly accurate reconstruction of the measured PV module behaviour. The very low RMSE and MAE values confirm the numerical efficacy of the fitting procedure, while the near-unity R² value shows that almost all variation in the measured current data is explained by the reconstructed model. These findings support the suitability of the proposed full-curve residual-based extraction method for accurate PV module modelling under STC.

The fitting results presented in Section 3.3 establish the numerical accuracy of the extracted model and provide the basis for the following methodological comparison. Compared with datasheet-based analytical methods, the present approach uses richer experimental information because it includes all measured voltage-current pairs in the objective function. Datasheet and characteristic-point methods are useful when raw I-V data are unavailable, but they usually rely on simplifying assumptions and may not reproduce the intermediate regions of the curve accurately. In particular, a method that matches only , , and the maximum-power point can still produce deviations in the plateau, knee, or open-circuit regions. The present full-curve approach reduces this weakness by distributing the fitting requirement over the entire measured curve. Slope-based methods provide a physically intuitive approach to estimating resistance terms by using local derivatives near the short-circuit and open-circuit regions. However, these methods are highly sensitive to curve smoothing, measurement noise, data resolution, and rounding. This is especially important for the present dataset because the low-voltage current plateau is nearly horizontal and the measured current values are rounded. Under such conditions, local slopes may be weakly observable, and direct slope-based resistance extraction can yield unstable estimates. The residual-based fitting method is therefore more appropriate because it does not depend only on a small number of local derivative values. Compared with Lambert W-based analytical formulations and explicit rearrangements, the present formulation keeps the implicit SDM structure and directly minimises the current-balance residuals. Lambert W approaches are mathematically useful and can improve analytical tractability, but the full-curve residual formulation is easier to connect with measured experimental data because each measured voltage-current pair contributes directly to the optimisation problem. This makes the procedure transparent and easy to audit. Metaheuristic and hybrid optimisation algorithms have been widely used for PV parameter extraction because they can search complex and non-convex parameter spaces with fewer derivative requirements. These methods are particularly valuable when multiple I-V curves, degraded modules, partial shading conditions, or highly nonlinear objective functions are involved. However, they may require many function evaluations and exhibit stochastic variability across runs. In the present study, the measured curve is smooth, single-condition, and well-behaved; therefore, a physically constrained deterministic least-squares method provides excellent accuracy without the additional complexity of population-based optimisation. The aim of the present work is therefore not to claim that deterministic least-squares fitting is universally superior to all metaheuristic or hybrid techniques. Rather, the results show that when a complete, smooth I-V curve is available, a transparent full-curve residual formulation can provide an accurate, reproducible, and computationally efficient parameter-extraction route. This makes the method suitable for practical PV module modelling, MPPT studies, reference-curve generation, and simulation-based performance assessment.

Although the fitted model reproduces the measured curve with very high numerical accuracy, the extracted parameters should be interpreted with appropriate caution. In the five-parameter single-diode model, several parameters influence overlapping regions of the I-V curve. For example, the diode reverse saturation current and the modified ideality factor a jointly control the exponential behaviour near the knee and open-circuit regions, while the series resistance and shunt resistance influence curve slopes that may be weakly visible in a smooth or rounded dataset. Therefore, a very low fitting error does not automatically mean that every parameter has been independently and uniquely identified. A formal quantitative identifiability analysis, including confidence intervals, parameter-correlation assessment, and multi-start tests with different initial conditions, was not performed in the present study and is therefore not claimed. Such analyses are considered an important extension of the present work, particularly when multi-condition I–V data and measurement-uncertainty information are available. This issue is particularly relevant for the present module curve. The extracted photocurrent =9.299890 A is physically consistent with the measured short-circuit current of 9.300 A, confirming that the low-voltage current plateau is well represented. Similarly, the extracted , a, and effective ideality factor n=1.5528 reproduce the knee and open-circuit behaviour successfully. However, the very low series resistance =1.4682×10^-4 Ω and very high shunt resistance =1.3029×10⁶ Ω should be regarded as effective fitted parameters rather than direct physical measurements of contact resistance or leakage resistance. The reason is that the measured I-V curve remains almost flat over a wide voltage interval and then decreases sharply near the knee region. Such a curve shape mathematically favours a very small and a very large , especially when the current values are rounded. This does not diminish the model's usefulness for curve reconstruction, as the reconstructed I-V and P-V curves agree extremely well with the measured data. However, it means that the resistance values should not be used alone to make strong diagnostic claims about contact degradation, leakage pathways, soldering defects, or ageing mechanisms. The main practical implication is that the extracted SDM can be used confidently as a high-fidelity STC reference model for the investigated PV module. It can support module-level simulation, maximum-power prediction, MPPT algorithm testing, and comparison between measured and expected I-V curves. Because the model reproduces both the I-V and P-V curves with very small errors, it can also serve as a baseline for future studies involving degradation, fault detection, or performance comparison. Nevertheless, the model should not be interpreted as a complete outdoor performance model. The parameter set was extracted from a single I-V curve measured at STC. Therefore, the robustness of the extraction procedure across different modules, datasets, irradiance levels, and temperatures has not been established in the present study. Outdoor operation involves variations in irradiance, cell temperature, wind speed, spectrum, soiling, humidity, and ageing. Therefore, annual energy-yield prediction or field-performance simulation would require additional translation equations or multi-condition parameter extraction using I-V curves measured at different irradiance and temperature levels.

Future work should therefore extend the present full-curve extraction framework to multiple operating conditions. Fitting several I-V curves simultaneously would make it possible to evaluate how , , , and a change with irradiance and temperature. In addition, uncertainty analysis, sensitivity analysis, and confidence-interval estimation should be included to determine which parameters are strongly identifiable and which are only weakly constrained by the available data. Such extensions would improve the physical interpretation of the extracted parameters and strengthen the model's use for diagnostics and long-term PV performance prediction. Overall, the results confirm that the proposed full-curve extraction method is highly effective for reconstructing the measured STC behaviour of the commercial PV module. At the same time, they show that numerical accuracy and physical identifiability are not identical. The extracted model is therefore most reliable as an accurate STC curve-reconstruction tool, while broader physical diagnosis and outdoor prediction require additional measurements, uncertainty analysis, and multi-condition validation.