4.2. Results on PV Technologies
In spite of the results that are presented above demonstrating that the empirical models present satisfactory performances in the proposed application approach, it is still necessary to quantify how good these performances are and what are their potentials and limitations of representing the behavior of the PV modules. One way of doing this is to compare the quality of the results that were obtained for each empirical model for each type of PV technology under study since the particular characteristics of each PV technology can lead to different conclusions.
Therefore,
Table 3 presents an overview of the quality of the results that were obtained by each empirical model for each PV technology in the proposed application approach.
Right away, the fact that the vast majority of empirical models have presented R² > 0.985 confirms their good potential for representing PV modules in the proposed application approach. It should also be noted that they demonstrated slightly greater sensitivity to behavioral variations of the PV modules of xSi, mSi, and CdTe technologies to the detriment of CIGS technology. Notwithstanding, these sensitivities that were presented by the empirical models were significantly higher than those that were verified by the interpolation method that was recommended by IEC 61853-3, which is observed to have R² < 0.95 for all technologies.
Analyzing the characteristics of the errors verified in the results that were obtained by the empirical models in the proposed application approach, some details draw attention. From the results presented for the MBE in
Table 3, it is verified that all of the empirical models demonstrated an average tendency to overestimate the maximum power output of the PV modules of silicon technologies. On the other hand, the vast majority of the empirical models demonstrated an average tendency to underestimate the maximum power output of PV modules of CdTe technology, being the only exception, the empirical model proposed by Kenny et al. [
24]. Concerning the PV modules of CIGS technology, a mixed situation is checked. While the empirical models that were proposed by Kenny et al. [
24], Randall and Jacot [
25], Heydenreich et al. [
30], and Silva et al. [
34] demonstrated an average tendency to overestimate, the others demonstrated an average tendency to underestimate.
Regarding the amplitude of these systematic errors, it is noted that they are within acceptable limits, which are higher for mSi technology and lower for CIGS technology. Even so, it is pertinent to note that these amplitudes were, in general, smaller than those that were verified by the interpolation method that was recommended by IEC 61853-3 for silicon technologies and slightly higher for thin-film technologies.
In turn, the results that were presented for the RMSE in
Table 3 demonstrate that the uncertainties that were associated with the results obtained by the empirical models in the proposed application approach were adequate for the xSi, mSi, and CdTe technologies and are not very good for the CIGS technology. For this latter, these uncertainties were twice as high as that found for other technologies.
Moreover, the fact of the uncertainties that are associated with the results obtained for xSi technology are less than 5% and, for mSi and CdTe technologies, are close to this value, it is something to be commended. This uncertainty level is in agreement with the uncertainties that are typically seen in the measurements and it aligns with that verified for the performance models available in the literature [
7,
57,
65,
67,
68].
Concerning the results presented for the RMSE in
Table 3, it is also possible to verify that the uncertainties associated with the results that were obtained by the empirical models in the proposed application approach for the xSi, mSi, and CdTe technologies are approximately four or five times lower than the respective uncertainties that were verified by the interpolation method recommended by IEC 61853-3. This difference drops to two and a half times for CIGS technology, but it is still quite significant. Accordingly, these results confirm that the empirical models can obtain excellent quality results in the proposed application approach. In fact, their accuracy and robustness are incredibly superior to those from the verified interpolation method recommended by IEC 61853-3.
In this sense, it is pertinent to highlight that:
(i) the empirical model that was proposed by King et al. [
27] proved to be the most effective in representing PV modules for silicon technologies;
(ii) the model proposed by Durisch et al. [
29] for CdTe technology; and,
(iii) the models proposed by Williams et al. [
26] and Reich et al. [
32] for CIGS technology. In turn, the model that was proposed by Rosell and Ibanez [
28] proved to be the least effective for all PV technologies under study. Additionally, it is important to note that the empirical model that was proposed by Driesse and Stein [
36] demonstrated a regular performance, although it was the only one that was developed when considering its application within the scope of the IEC 61853-1 matrix.
In view of these findings, some questions arise: “Do empirical models reach their full potential for representing PV modules in the proposed application approach?”; “Are the performance differences observed between PV technologies typical of the characteristics of the empirical models?”.
The answer to these questions can be found through
Table 4, which presents an overview of the quality of the results that were obtained by each empirical model for each PV technology in the conventional application approach.
Hence, when comparing the results that were presented in
Table 3 and
Table 4 for R², it is possible to notice that the empirical models, when applied in the proposed approach, have their potential to represent the behavior of the PV modules minimally affected compared with what would be found via the conventional application approach. This situation indicates that the respective measurement matrices referring to IEC 61853-1 made available for the PV modules under study were sufficient for properly fitting the coefficients of the empirical models and, consequently, enabling their practical application in the representation of the PV modules. However, using this restricted number of operational records to determine the coefficients affects somewhat the other performance characteristics of the empirical models.
The results that are presented for the MBE in
Table 3 and
Table 4 demonstrate that originally most empirical models do not present an average tendency as those verified in the proposed application approach. Indeed, the models that were proposed by Taylor [
23], Ransome et al. [
35] (both variations), and Driesse and Stein [
36] stand out for their good balance. The maximum power output values eventually overestimated by them are precisely opposed to the maximum power output values eventually underestimated so that no clear trend, or, in other words, no systematic error is verified in their results for the case of the conventional application approach.
In turn, the results that are presented for the RMSE in
Table 3 and
Table 4 demonstrate that the uncertainties associated with the results obtained by the empirical models for xSi, mSi, and CdTe technologies are higher than those that were verified for CIGS technology. Meanwhile, the latter’s uncertainties are not observed inadequate in the conventional application approach, as found in the proposed application approach, being close to 5% for the empirical models that were verified as more accurate.
It is also interesting to note that the uncertainty that is associated with the results obtained by the empirical models in the proposed application approach for the CIGS technology is about 5% higher than that verified in the conventional application approach, i.e., twice as large. This difference is less discrepant for the other technologies, reaching almost 3% for the mSi technology and only 1% for the xSi and CdTe technologies. Despite this, all of these differences can be considered to be very satisfactory. This is because, in the proposed application approach, only eighteen operational records were used for determining their coefficients and, in the conventional application approach, tens of thousands are used.
In addition, some curious situations can be observed when comparing the individual performances of the empirical models in the proposed application approach with those respectively verified in the conventional application approach.
The uncertainties that are associated with the results obtained by the empirical model that was proposed by Rosell and Ibanez [
28] in the conventional application approach were about 4% to 10% lower than those verified in the proposed application approach. This indicates that this empirical model requires a more comprehensive number of operational records for its effective application.
In contrast, the uncertainties that are associated with the results obtained by the empirical models proposed by Kenny et al. [
24], Heydenreich et al. [
30] and Driesse and Stein [
36] for PV modules of thin-film technologies in the conventional application approach were controversial regarding those that were verified in the proposed application approach, often being larger. This indicates that these empirical models are more sensitive to quality than the quantity of operational records used to efficiently determine their respective coefficients.
Furthermore, it is observed that the empirical model that was proposed by Kenny et al. [
24], King et al. [
27], and Driesse and Stein [
36] presented a greater limitation in the representation of the PV modules of CIGS technology in both application approaches. By analyzing the common features of these models, it is possible that their expressions respectively proposed for estimating the maximum power voltage of the PV modules are not as suitable for this technology as they demonstrate for the others.
From another perspective, the regularity and proximity of the quality verified in the results that were obtained by some empirical models in both application approach, such as those proposed by Taylor [
23], Randall and Jacot [
25], Williams et al. [
26], Durisch et al. [
29], Montgareuil et al. [
31], Reich et al. [
32], Huld et al. [
33], Silva et al. [
34], and Ransome et al. [
35] (both variations), is impressive.
Regarding them, it should be noted that the empirical models that were proposed by Williams et al. [
26] and Reich et al. [
32] demonstrated a comparable performance for all PV technologies, with the latter even being a reportedly improved version of the former. Nonetheless, their performance is slightly superior to those that were verified for the empirical model proposed by Randall and Jacot [
25], which can be considered as a reduced version of them. Similarly, the empirical model that was proposed by Silva et al. [
34], which is declared to be a simplified version of the empirical model proposed by Durisch et al. [
29], demonstrated a slightly lower performance than the latter. Additionally, in turn, each variation of the empirical model proposed by Ransome et al. [
35] demonstrated a performance that was practically equal.
Besides that, it is necessary to point out that the empirical model that was proposed by Taylor [
23], even though it has been developed for almost 40 years and is the only one characterized by an expression that does not have any logarithmic term, was able to present a competitive performance in the cases of all PV technologies.
Finally, it is pertinent to highlight that the results that were obtained by the empirical model proposed by Durisch et al. [
29] in the conventional application approach were the most effective for all PV technologies, being closely followed by the models that were proposed by Williams et al. [
26], Montgareuil et al. [
31], Reich et al. [
32], and Huld et al. [
33].
4.3. Results on Operating Conditions
In order to explore the representation potentials and limitations of the empirical models a little more,
Figure 6 shows an overview of the uncertainties that are associated with the results obtained by them in the proposed application approach for the different operating conditions that were recorded for the PV modules under study.
As can be seen, the performance of empirical models in the proposed application approach has some well-defined behaviors. For effective irradiance levels that are above 800 W/m², the results that were obtained by all empirical models are of excellent quality, regardless of the operating temperature. For operating conditions outside this effective irradiance range, it is observed that, in general, the uncertainty that is associated with the results obtained by the empirical models increases as the level of effective irradiance and operating temperature of the PV modules decreases concomitantly.
The empirical models that are proposed by Rosell and Ibanez [
28] and Driesse and Stein [
36] deviate somewhat from this rule, because the uncertainty that is associated with their results increases as the level of effective irradiance decreases, even under conditions of elevated temperature operation of the PV modules.
Nevertheless, the worst quality results that were obtained by the empirical models in the proposed application approach were for operating conditions characterized by effective irradiance levels up to 100 W/m² and/or operating temperatures below 0 °C.
In turn,
Figure 7 shows an overview of the uncertainties that are associated with the results that were obtained by the interpolation method that was recommended by IEC 61853-3 for the different operating conditions that were recorded for the PV modules under study.
By comparing the results that are presented in
Figure 6 and
Figure 7, it is possible to notice that the uncertainties that are associated with the results obtained by the interpolation method recommended by IEC 61853-3 are significantly higher than those verified for the empirical models, in practically all of the ranges of operating conditions. The only exception concerning the empirical model proposed by Rosell and Ibanez [
28], which has a lower performance than this method, mainly for low levels of effective irradiance.
On the other hand,
Figure 8 shows an overview of the uncertainties that are associated with the results obtained by each empirical model in the conventional application approach for the different operating conditions that are verified for the PV modules under study.
As can be seen, the general performance behavior that was observed for the empirical models in the conventional application approach is similar to that discussed above for the proposed application approach. Hence, it can be concluded that this general performance behavior is due to the representation abilities of the empirical models themselves and not exactly to the amount or dispersion of the operational records used to determine their coefficients.
Some more prominent variations of this performance behavior are observed only for the empirical models by Heydenreich et al. [
30] and Driesse and Stein [
36]. However, these variations can be attributed to the issues that were previously discussed about determining their coefficients (see
Section 4.2).
Regarding the general performance behavior of the interpolation method that was recommended by IEC 61853-3, a different situation is observed. Because it is a strictly mathematical method and of a linear nature, it is due to its lack of ability to represent the nonlinearities of the behavior of the PV modules, when considering the measurement matrices referring to IEC 61853-1 made available to the PV modules under study. Although covering a wide range of operating conditions, the quantity of measurements of these matrices is presumably scarce to allow a more efficient application of this method, especially at low levels of effective irradiance and/or operating temperature. Accordingly, it is interesting to note that a similar situation can be expected for the measurement matrix that was established by IEC 61853-1.
In addition, it can be seen in
Figure 6,
Figure 7 and
Figure 8 that, for operating conditions that are characterized by effective irradiance levels between 200 and 600 W/m² and operating temperature below 0 °C, all of the empirical models obtained results with a strangely accentuated uncertainty. These operating conditions concern a portion of the atypical measurements that were verified in the operational records of the PV modules, which were previously discussed in the view of
Figure 4. Thus, the magnitude of these uncertainties must be interpreted with some care, depending on the analysis context.
That said, it is interesting to note that these well-defined behaviors that were observed for the empirical models produce some practical effects on their respective applications. For instance,
Table 5 presents an overview of the uncertainties that are associated with the results obtained by each empirical model in the proposed application approach for the PV modules that were installed in each of the three locations considered.
As can be verified, the quality of the results that were obtained by the empirical models for the PV modules installed in Cocoa-FL was significantly superior to the results obtained for the PV modules installed in Eugene-OR. Because these are the same PV modules, being monitored by the same test system, and with the information in their datasheets duly updated, the primary explanation for this discrepancy is the different operating conditions verified by them in each location.
The records made in Cocoa-FL were concentrated in operating conditions that were characterized by high levels of effective irradiance and elevated operating temperature, as shown in
Figure 2. While, in Eugene-OR, these operational records are more dispersed by low levels of effective irradiance and more recurrent at low operating temperatures. Accordingly, the uncertainties that are associated with the empirical models in the proposed application approach were more significant in the latter location than the former.
This same reasoning can be partially extended to Golden-CO since the manufacturers and models of the PV modules were the same as those that were installed in the other two locations. Because the records that were made in Golden-CO were more dispersed concerning the effective irradiance and the operating temperature of the PV modules, the empirical models demonstrated an intermediate performance as compared to the other two locations.
4.4. Additional Comments
The measurement matrices referring to IEC 61853-1 made available for the PV modules under study have some differences concerning the matrix established by this standard, as shown in
Figure 3 and discussed in
Section 3. Despite this, most of the conclusions that were obtained from discussions held in this section can be extended for the IEC 61853-1 matrix. This is because the application of the empirical models considering these available measurement matrices (proposed approach) proved to be able to preserve the fundamental performance characteristics of the empirical models, mainly in relation to the operating conditions that were registered for the PV modules.
In other words, if the measurements for high temperatures of the available matrices made available for the PV modules were 75 °C, as established by IEC 61853-1 and not 65 °C, as they were possible to be performed, the impacts on the quality of the results presented would be minimal. Although this second one is more representative than the first one (see
Figure 2), the empirical models have intrinsically a very high capacity to represent the behavior of the PV modules in both of the operating conditions (see
Figure 6 and
Figure 8).
In contrast, if the measurements for the operating temperature of 15 °C and effective irradiance of 400–1000 W/m² had been made available, as established by IEC 61853-1, the most significant effects would be the reduction of the uncertainties that are associated with the results obtained in the operating conditions around these. Consequently, reducing the total uncertainties that are presented for each PV technology and installation location of the PV modules under study also be checked. However, as empirical models generally perform poorly under these operating conditions (see
Figure 6 and
Figure 8), this reduction is unlikely to be very impactful. Additionally, the occurrence of this reduction would only reinforce the effectiveness of the application of the empirical models considering a matrix of measurements such as that of IEC 61853-1 for the determination of their coefficients, which favors the conclusions obtained from discussions held in this section.
Therefore, it is possible to declare, particularly for those empirical models that have been verified with more regular performance in the results previously presented in this section, i.e., all but the one proposed by Rosell and Ibanez [
28], that their respective applications considering a measurement matrix that is similar to that established by IEC 61853-1 for the determination of their coefficients are indeed effective, because:
it preserves their respective sensitivities and potentials for representing the behavior of PV modules;
doing this allows for empirical models to obtain good quality results easily, especially for modules of xSi, mSi and CdTe technologies; and
additionally, incidentally, surprisingly more accurate and robust than those verified by the interpolation method recommended by IEC 61853-3.
Nevertheless, some of the conclusions reached here still need further evaluation. The first is related to the average tendencies that were verified for the empirical models from
Table 3. By comparing with the results that are presented in
Table 4, it was observed that these average tendencies have a certain dependency on the operational records that are used to determine the coefficients of the empirical models. Thus, as the measurement matrix established by IEC 61853-1 has more measurements than the matrices made available for the PV modules under study and, even for operating conditions to which the performance of the empirical models are notoriously sensitive, different average tendencies can be occasionally observed.
The other is regarding which of the empirical models assessed is the most appropriate for representing each PV technology. As mentioned, the fact that the measurement matrix that was established by IEC 61853-1 has additional measurements for relevant operating conditions can cause a more significant improvement in an empirical model’s performance than in others. Hence, as it was seen that most empirical models presented a comparable performance, one of them may eventually stand out more than that was verified as more effective in the results that are presented here.
Still, the empirical models that were found to be most effective in this paper serve as a first recommendation, such as the empirical model(s) proposed by:
King et al. [
27] for representing xSi and mSi technologies;
Durisch et al. [
29] for representing CdTe technology; and,
Williams et al. [
26] or Reich et al. [
32] for representing CIGS technology.
Concerning these empirical models, some issues are worth noting. Regarding the model by King et al. [
27], it is essential to emphasize that this has proved to be the most effective for silicon technologies under the relaxed procedures of determining its coefficients, and not under the procedures that were initially proposed by its authors, as explained in
Section 3.
With respect to the model that was proposed by Durisch et al. [
29], it is relevant to mention that it was the one that proved to be the most effective for all PV technologies in the conventional application approach. Therefore, it is an excellent option for studies, in which such an application approach is proposed.
In relation to the empirical models that were proposed by Williams et al. [
26] and Reich et al. [
32], it is pertinent to point out that, although the latter was developed based on the former aiming to avoid that results with negative values, this was not verified in practice. In the proposed application approach, both of the empirical models obtained a few negative results for some PV modules operating at extreme conditions, such as very low levels of effective irradiance and operation temperature. Notwithstanding, this deficiency is shared by most of the empirical models and verified in both application approaches. The only empirical models that were observed to be able to obtain only positive results, regardless of the application approach, were those that were proposed by Randall and Jacot [
25], Durisch et al. [
29], Heydenreich et al. [
30], Silva et al. [
34], and Driesse and Stein [
36]. Thus, special attention should be paid to this issue, depending on the nature of the study at hand.