2.1 Cell-crack formation in a PV module with nonuniform mechanical loading

A sequential mechanical loading test was conducted on a commercially available PV module(1970 × 993 × 35 mm) assembled with 72 mono-c-Si PV cells (156 × 156 mm², four busbars) to form cell cracks reflecting non-uniform wind loads during a strong typhoon, as described in our previous report [37]. The I–V parameters—maximum power (P_max), I_sc, open-circuit voltage (V_oc), and FF—in the PV module before this sequential mechanical loading test were assessed as 359.8 W, 9.88 A, 45.6 V, and 76.5%, respectively. Thus, the mean P_max of the individual PV cells was 4.997 W/cell, and the mean V_oc was 0.66 V/cell. In this module, no cell cracks were detected in the EL image prior to the mechanical loading test. After the non-uniform static mechanical loads with a combination of pressure load (1 h) and suction load (1 h) were applied seven times, non-uniform dynamic mechanical loads (10 cycles/min for 6 h) were applied to the PV module. The uneven mean surface pressure pattern (MSPP) used in the non-uniform static/dynamic mechanical loading can be found in the S1 Fig. The EL image was obtained at a forward bias current of 8 A. In accordance with IEC standards [15], the EL images of the respective PV cells were evaluated by the human eye with respect to the presence and severity of cell cracks.

2.2 Electrical isolation of the individual PV cells within the PV module

To assess the electrical characteristics of the individual PV cells within the PV module, the interconnector ribbons located in the intercell spaces of each cell string were exposed to the backsheet peeling of the corresponding portions (Fig 1), as reported in our previous article [38]. Briefly, the backsheet and encapsulant layers of the PV modules were peeled off with a micro grinder and a small wire brush to expose the interconnector ribbons between the PV cells, as can be found in S2A Fig. Copper solder leads were soldered to the exposed interconnector ribbons (S2B Fig) and connected with another set of wide copper solder ribbons at each side of the respective PV cells to evaluate the electrical characteristics of the individual PV cells, as shown in Fig 1B and 1C.

2.3 Electrical characterization of isolated PV cells

The I–V characteristics of the PV cells were individually assessed using a surface mask with an aperture area on an object PV cell, through the leads of the respective PV cells, under the standard test conditions of 1,000 W/m², 25°C, and AM 1.5G [38]. Dark I–V data were collected at 23.3–23.9°C in the dark, by the connection of the leads of an object PV cell to a source measurement unit (Keysight B2901a). The I–V parameters, including R_s, R_sh, saturation current densities (J₀₁ and J₀₂), and ideality factors of diodes (n₁ and n₂), were extracted from the respective I–V data obtained under illuminated or dark conditions by fitting to two-diode equivalent circuit models as presented in (1) and (2) according to [39, 40]: (1) (2) where I_L is the current under illumination, I_ph is the photo current, q is the elementary charge, k_B is Boltzmann’s constant, T is the temperature, A is the cell area, and I_dark is the current in the dark. The median root mean square error (RMSE) in the illuminated I–V fittings was 0.015, and that of the root mean square logarithmic error (RMSLE) in the dark I–V fittings was 0.040, as shown in Fig 2. These results indicate that these I–V parameters can be estimated with sufficient accuracy. In the dark I–V fitting, n₁ values in the individual PV cells fell within the range of 1.18 ± 0.05, whereas the n₂ values fell within the range of 1.97 ± 0.04. These values indicated that each J₀₂ in the dark I–V parameters estimated from the individual PV cells corresponds to the current density, owing to the recombination current in the respective PV cells. To accurately quantify the total series resistance of each PV cell within the PV module, the lumped series resistance was calculated as R_s-ld from the respective illuminated and dark I–V data (3) in accordance with the procedure described by Spataru et al. [26]:

(3) where I_mp and V_mp are the current and voltage at the maximum power point under the illuminated conditions, respectively. V_d-mp is the voltage corresponding to I_dark = I_sc − I_mp on the dark I–V curve.

The AC impedance data of the individual PV cells were acquired by connecting the four-point probes of an LCR meter (Keysight 4284a with 001 DC bias option) at 22.3–24.0°C in the dark, under the condition that a 10-mV-amplitude AC voltage with various intensities of DC bias voltage was applied. According to [41], the voltage across the p–n junction (V_j) of the PV cell is defined as V_j = V–IR_s, where V, I, and R_s are the applied DC bias voltage, applied DC current, and series resistance, respectively (cf. R_s or R₁ in Fig 5A-1–5A-3). From the AC impedance (amplitude |Z| and argument θ) collected at each frequency in the range of 20 Hz to 10 kHz, the real (Z’) and imaginary (Z”) impedances were calculated, and the AC impedance parameters in the postulated AC equivalent circuits (cf., Fig 5A-1–5A-3) were determined using a Web-based impedance-fitting application (Elchemea Analytical) [42].

2.4 Statistical evaluation

The Steel–Dwass test is a statistical procedure for evaluating the stochastic equality among multiple sample groups, utilizing the pairwise ranking non-parametric method. Because the distribution of the respective parameters adopted in this study is not corresponding to the normal distribution, we applied this non-parametric test to assess the difference among the distributions of these parameters, using the pSDCFlig function (Monte Carlo method with 10,000 iterations) in the NSM3 package [43] embedded into the R software [44]. Statistically significant differences were judged by the p-value limit (p < 0.05). Particularly, the multi-endpoints issue arising from post hoc definitions were not considered, since this study aimed to evaluate distinguished electrical characteristics attributed to cell cracks. The discretization of the cumulative frequency distribution to a multimodal distribution was performed using the normalmixEM function in the mixtools package [45] built into the R software.

3.1 Power losses in the cracked PV cells

As presented in our previous report [46], various types of cell cracks were observed in the PV module after the nonuniform mechanical loading test, which included Mode B/C cracks (with inactive cell area(s) detectable in the EL image) and Mode A cracks (without any inactive cell area) [15, 19], as shown in Fig 3A. Based on the rating criteria, the individual PV cells with cell cracks were divided into two groups, particularly, the cracked cells with or without the inactive cell area were categorized as hard-cracked (HC) or minorly cracked (MC) cells, respectively. In these HC cells, the inactive areas were identified in the central region of the respective PV cells (e.g., C08 cell), as well as at the edges of the PV cells (e.g., C07 cell). The PV cells without cracks were referred to as non-cracked (NC) cells. The spatial distributions of these cell groups in the PV module are shown in Fig 3B, accompanied by the cell address, defined by the location of the respective PV cells within the PV module. Although the distribution of these cracked cells (HC and MC cells) within the PV module did not sufficiently coincide with that of the MSPP applied to the PV module, these cells are likely to be located in steep regions in the applied MSPP [46]. To quantitatively assess the extent of power loss attributed to the cell cracks, the respective maximum powers of the individual PV cells were measured and indicated in the cell matrix of the PV module (Fig 3C) as values normalized with the pristine P_max in individual PV cells (4.997 W/cell). It is recognized that obvious power loss occurred the HC cells, although that in the MC cells was not detectable at a glance, as reported in the PV module with only Mode A cracks [19].

A histogram of the actual P_max values in each PV cell is shown in Fig 4, with the mean P_max in the pristine PV cells (4.997 W/cell). A broad distribution with a long tail in the low P_max range is observed, and the median of the entire PV cell is 4.586 W/cell. Almost all HC cells were in this long tail with a low P_max, although those in the MC cells were completely overlaid with those in the NC cells. This power-loss behavior was statistically confirmed by nonparametric multiple comparison analysis (see the inset figure of Fig 4); particularly, the significant decrease in P_max of HC cells from those in other cell groups was clearly demonstrated, although a substantial difference between the respective power losses in the MC cells was not proven.

3.2 AC impedance characteristics of the cracked PV cells

As a potential electrical signature for cell cracks, it has been reported that the intensity of the parallel resistance (R_p: cf. Fig 5A-1) in the AC equivalent circuit of a PV cell/module is reduced by cell cracks that are intentionally introduced in the PV cell/module [33–35]. We then examined whether this parameter could be crucial for cell cracks (particularly, MC cells). Generally, the locus of AC impedance for a PV cell/module is expressed as a semicircle in the Nyquist plot when the AC equivalent circuit for a PV cell with a single p–n junction is assumed to be composed of a series resistance (R_s) and a parallel circuit including a resistor and capacitor (Fig 5A-1: R_s-R_p||C_p model). Fig 5B shows the AC impedance locus observed in the PV cell without any cell cracks (F08 cell) under the conditions that the bias voltage is not applied. Although the AC impedance spectra could not be sufficiently fitted to the R_s-R_p||C_p model with a perfect semicircle curve (dashed curve), a satisfactory value (1.32 × 10⁻⁴) for the goodness of fit (also known as the reduced χ²) was obtained in the R₁-R₂||CPE₂ model (Fig 5A-2), as can be observed in the depressed semicircle (solid curve) of Fig 5B. In this R₁-R₂||CPE₂ model, because the impedance of the constant-phase element (CPE) is given by Z_CPE = 1/[Q(jω) ^φ] [47], that of the R||CPE component can be described as Z_R||CPE = 1/[(1/R) + Q(jω) ^φ], and the effective capacitance (C*) for the R||CPE component can be defined as C* = [(QR)^1/φ]/R [48]. Where j and ω are the imaginary operator and angular frequency, respectively. Moreover, φ is the capacitor factor for the phase angle [θ = −(90 × φ), where θ is expressed in degrees unit] of the CPE impedance, with a value of 0 to 1. If φ = 1, the capacitance parameter (Q) indicates the capacitance of an ideal capacitor, that is, C* = Q. When V_j was applied at less than 0.3 V, a similar behavior in the AC impedance spectra was observed (Fig 5C), and φ estimated under these conditions was almost always over 0.95 (Fig 5E). Therefore, we deduce that the R₁-R₂||CPE₂ model can be effectively applied into the estimation of AC impedance parameters in PV cells under these DC-bias conditions. At V_j > 0.3 V, however, we were not able to obtain a sufficient value of the reduced χ² by fitting the measured AC impedance data to the R₁-R₂||CPE₂ model (Fig 5F), and φ fell below 0.95 (Fig 5E). Because the effects of the low–high interface (p–p⁺ junction in the back surface field) observed in the high forward bias voltage range (at V_j > 0.3 V) should be considered, these AC impedance data were fitted to the R₁-Z₂-Z₃ model (Fig 5A-3 and 5D), according to previous reports [49, 50]. The entire AC impedance curve (indicated by the thick curve in orange) calculated from the parameters for this model had very small values in the reduced χ² (indicated by the orange triangles in Fig 5F) and was clearly divided into two curves comprising Z₂ (AC impedance derived from the p−n junction) and Z₃ (that from the p−p⁺ junction) in Fig 5D. Thus, to identify the critical parameters attributed to the cell cracks in individual PV cells, we used the R₁-R₂||CPE₂ or R₁-Z₂-Z₃ models, depending on the applied bias voltage. In all the estimated parameters of these models, the standard errors were almost always within 10%.

The dependency of the parallel resistances (R₂) extracted from the AC impedance spectra of the individual PV cells on the bias voltage is shown in Fig 6A. As reported previously [51], a nearly constant resistance was estimated in the reverse bias voltage range, although drastic reductions were observed when the forward bias was applied, irrespective of the cell groups categorized as the crack mode. The extent of R_sh derived from R₂ (hereafter, this R_sh is referred as R_sh-R2) in the individual PV cells was numerically calculated as the mean value of R₂ at −0.6. to −0.3 V in the bias voltage range [51], and their distributions in the respective PV cell categories were not significantly segregated, as shown in Fig 6B and Table 1. Although these extents are widely spread in all categories, R_sh−R2 was over 1 × 10⁴ Ω·cm² even in the HC cells (Fig 6B), and the R_sh-R2 level was considerably higher than that proposed as the industrial R_sh criteria of the PV cell (0.8–2.4 × 10³ Ω·cm²) [30, 52–54]. Furthermore, because it is well known that the power loss attributed to potential-induced degradation (PID) with an obvious reduction in V_oc occurs when R_sh in the degraded cells/modules is below a critical value (2–3 × 10³ Ω·cm²) [55], we can conclude that the R_sh extents indicated in Fig 6B would not be a crucial cause of V_oc reduction. In previous studies [34, 35], the extent of parallel resistance (R_p) was estimated from the AC impedance data collected at V_j = 0, and those in the cracked cells and modules with cracked cells were maintained at 4.86 × 10⁴ Ω·cm² and ca. 6 × 10⁷ Ω·cm², respectively; however these extents were decreased by approximately 40–50% of those in the pristine PV cell/module. A similar reduction in R₂ measured at V_j = 0 was observed in the HC cells in our results (Fig 6A); however, a significant decrease in the MC cells was not observed compared to that in the NC cells, judging from the statistical analysis (p = 0.933). From these results, we conclude that the reduction in the parallel resistance (R_p/R₂) does not directly correlate with the occurrence of cell cracks, including microcracks.

3.3 Elevation of minority-carrier recombination in cracked PV cells

To identify a specific electrical signature of the cracked PV cells (particularly, for the MC cells), we compared the time constants for minority-carrier recombination among these cell categories on cracks through the evaluation of the R₂ and C*/C₂ parameters that reflected the electrically dynamic behavior in the p–n junction of a PV cell. In the V_j range over 0.2 V, the intensity of R₂ changed in a logarithmic linear manner, depending on the applied bias voltage (Fig 7A), and these data could be completely traced by the proposed relationship on the diffusion resistance in the p–n junction [41, 56]. For C*/C₂, their intensities were drastically elevated in the same range as V_j (> 0.2 V), and the theoretical curve for the diffusion capacitance entirely coincided with the trends of these data [41, 56, 57], as shown in Fig 7B. For both parameters (R₂ and C*/C₂), we deduced that the parameters in the PV cell with complex cell cracks (in C08 and C03 cells) have a different dependency on the bias voltage from those in other PV cells. In fact, the intensities of R₂ and C*/C₂ were lower than those measured in other PV cells, whereas those at a higher forward bias voltage (> 0.45 V) could not be measured. This is because their comparably lower R_sh contributes to the apparent reduction of the R₂ value in the forward bias voltage range, and the loading of a higher forward bias voltage (> 0.45 V) to these PV cells was difficult with the impedance spectrometer used in this study, as also suggested by Yeow et al. [58]. However, for the bias voltage dependencies of R₂ and C*/C₂ in other cells, there were small but definite differences among the crack categories. Because the time constant of the carrier recombination in the p-n junction can be simply calculated as a multiplication of the parallel resistance (R₂) and capacitance (C*/C₂), their bias voltage dependency in the representative cells of the respective crack categories is shown in Fig 7C. Reflecting the difference in R₂ and C*/C₂ among the crack categories, the time constant over the 0.3 V forward bias voltage range was distinguished between the PV cells with and without cell cracks. That is, the minimum time constants in all the cracked PV cells [including the MC cells (F03 and F01 cells)] were considerably smaller than those in the NC cells (F08 and F10 cells) at approximately 0.4 V forward bias voltage. To confirm the significant differences among these time constants observed in the respective crack categories, we statistically compared these values estimated at 0.4 V forward DC bias voltage using the nonparametric multiple comparison test (Fig 7D). A nearly significant reduction in the time constant was identified between the NC and MC cells (p = 0.072), which was not confirmed in other electrical signatures examined in this study, as discussed in Section 4. Simultaneously, critical reductions in the HC cells from other crack categories were also observed. These results suggest that the enhancement of the carrier recombination, which is observed through the reduction of the time constant measured by AC impedance spectroscopy, is common to cracked PV cells, regardless of the crack categories.

To clarify whether the reduction in the time constant of the MC cells compared to that of the NC cells is meaningful, the time constants of all the cells were plotted as a cumulative frequency profile (Fig 8). In this profile, two major breakpoints were observed at approximately 5 × 10⁻⁵ s (−4.3 in logarithmic scale) and 1 × 10⁻⁴ s (−4.0 in logarithmic scale), with a nearly flat leaning between them. This inflected profile comprises at least two distributions of the time constant. That is, there was one distribution below 1 × 10⁻⁴ s (lower time−constant range) and the other spread at the higher part of the time constant (higher time-constant range). This cumulative frequency profile can be almost entirely fitted to the bimodal distribution consisting of two lognormal distributions (RMSE = 3.93), as also shown in Fig 8. The parameters of the lognormal distributions are listed in Table 2. The significant difference between the population means of these distributions was confirmed by the unequal variances t-test (Welch’s t-test) with p < 0.001. Remarkably, the time constants of almost all cracked cells (HC- and MC-cells) located in the lower time-constant range, especially those of the HC cells were concentrated near the lower end (Fig 8). Meanwhile, a major portion of the time constants in the NC cells spread in the higher time-constant range, although some of them were in the lower time-constant range. When the statistical multiplicity arising from the post hoc analysis was considered, we confirmed that the median value (−4.308) of the time-constant distribution in the cracked cells (HC- and MC-cells) was significantly lower than that (−3.968) in the NC cells (Table 1: footnote 3) using a nonparametric statistical test (Brunner–Munzel test) for stochastic differences between two samples [59]. The mean/median values in both distributions (lognormal and real data distribution) were comparable in the respective time-constant ranges. These results suggest that the distributions located in the lower and higher time-constant ranges correspond to the electrical characteristics of the cracked (HC and MC) and non-cracked (NC) cells, respectively. Moreover, the sizes of the PV cells with cracks (including microcracks) were estimated to be 25 out of 72 (cells) when all the PV cells with cell cracks were assumed to be distributed in the lower time-constant range (Table 2). This indicates that the recall rate in our rating was 64% [= 16 cells (consisting of nine HC cells and seven MC cells) / 25 cells]. Because this rate is comparable to those (67 ± 15%) reported in a previous article [60], there is no contradiction between the cracked cell sizes estimated from the bimodal distribution of the time constants and those presumed by the actual inspection. Consequently, we deduce that the remarkable decrease in the time constant of the carrier recombination should be recognized as a specific electrical signature for all cracked PV cells (including MC cells).

3.4 I–V characteristics of the cracked PV cells

The I–V parameters of the individual PV cells within the PV module subjected to the nonuniform mechanical loading test (Fig 3) were extracted from their I–V curves obtained under the illuminated and dark conditions (hereafter referred to as flash I–V and dark I–V, respectively); these are summarized in Fig 9, and all p-values estimated by the Steel–Dwass test are added to Table 1. As potential electrical signature for cell cracks, the critical reduction I_mp in the Mode B and C cells laminated in the mini-PV modules has been recently reported [61]; however the authors did not mention the evolution in the Mode A cells. In this study (Table 1), a significant decrease in the I_mp was confirmed in the HC cells (Mode B and C cells), although no significant difference between the MC (Mode A cells) and NC cells was verified. There was also a significant difference in the I_mp between the MC and HC cells. Del Prado Santamaria et al. demonstrated that the decrease in the I_sc was a specific feature in Mode C cells [61], and it was also confirmed in our study that I_sc was decreased in the HC cells (Fig 9A and Table 1), but not in the MC cells. Based on the combination of two independent observations (in [61] and this study), we infer that I_mp reduction is a common signature in HC cells, and the crucial reduction of I_sc could be attributed to the occurrence of an electrically inactive area within the HC cells. Furthermore, I_mp reduction cannot be regarded as a distinguishing feature of MC cells. Similar to the I_sc case, obvious decreases in V_oc, V_mp, and FF were statistically identified in their distributions in the HC and NC cells. For the MC cells, substantial reductions against the NC cells were not statistically detected. Thus, for the I–V parameters extracted from the flash I–V curves, we did not identify any common feature(s) caused by the microcracks (3rd column in Table 1).

Among the flash I–V parameters indicated in Fig 9A–9C, the most intense decrease in the HC cells was observed in the FF case. Because FF reduction depends on the evolution of R_s and/or R_sh, we statistically analyzed the correlation between the crack behavior and the extent estimated from the dark I–V data. No significant deviation was observed among all cell groups in the R_sh case (d-R_sh in Table 1), as well as those estimated from the AC impedance data (R_sh-R2 in Table 1). The significant elevation of R_s (p = 0.049) in the HC cells was validated, as shown in Fig 9D; however, the obvious evolution of R_s was not confirmed in the MC cells (p = 0.765). To critically assess the contribution of R_s elevation in the MC cells, we estimated the total R_s value (R_s-ld) in these cell groups (Figs 10 and 11), which was suggested as the lumped R_s around the maximum power point [26, 62].

In Fig 10A–10F, the (I_L−I_dark)–V_dark curves are drawn with the flash I–V curves for the respective PV cells, in accordance with a previous report [26]. Here, I_L, I_dark, and V_dark are the maximum currents determined from I_sc in the flash I–V data, and the current and voltage on the dark I–V curve, respectively. The R_s-ld was calculated using (3). Although the values of I_mp, V_mp, and V_d-mp were nearly equal between the NC cells (F08 and F10 cells) and MC cells (F03 and F01 cells), I_mp and V_mp decreased in the HC cells (C07 and C08 cells). Hence, because an increase in V_d-mp was also observed in the HC cells, R_s-ld was drastically elevated only in the HC cells (Fig 11). A significant elevation in R_s-ld was not detected in the MC cells with obvious microcracks. As shown in Fig 10E and 10F, the divergence between the flash and dark I–V curves at the low voltage range (0 to 0.4 V) is clearly demonstrated in the HC cells with high R_s-ld; this phenomenon is known as “Fake Shunt” which is caused by the spatial inhomogeneous distribution of R_s in a PV cell/module [63, 64]. Because this “fake shunt” was not recognized in the MC cells, it was confirmed that a noticeable non-uniform elevation in the R_s did not occur in the individual MC cells.

The elevation of J₀₂ was reported as a crucial electrical signature for a PV cell with microcracks, which can be observed in the global I–V curve, as well as in the local I–V image obtained in the microcrack region [29]. The values of J₀₁ and J₀₂ were estimated from the dark I–V data of the individual cells, and are indicated as cumulative frequency profiles (Fig 12). Both frequencies monotonically proliferated with an increase in the respective saturation current densities, and no obvious flection was observed. Additionally, an uneven distribution owing to the difference in the crack categories was not observed in the J₀₁ and J₀₂ cases. Accordingly, J₀₂-elevation in both the HC and MC cells was not significantly identified in our data (Table 1), and J₀₁-elevation in both cell groups was not statistically evident. In this subsection, when PV cells are encapsulated as a PV module, we demonstrate that the peculiar electrical signatures of the individual PV cells with microcracks cannot be practically identified by the electrical parameters derived from the conventional I–V curve analyses, but those of the PV cells with an electrically inactive area can be recognized easily.