Estimation of the Effective Irradiance and Bifacial Gain for PV Arrays Using the Maximum Power Current

Bifacial photovoltaic modules are able to convert the solar radiation reaching their front and rear sides, which means that more electricity can be produced using the same array area as monofacial modules with similar ratings. In some locations, the cost per power unit for such a technology has already become cost-competitive with conventional monofacial modules. The so-called effective irradiance and the bifacial gain are useful metrics, respectively, to assess the solar resource and the performance of bifacial arrays. To calculate the effective irradiance, studies previously published employ rear-side irradiance measurements, whereas to compute the bifacial gain, other works make use of monofacial modules with rating similar to those of the bifacial modules under analysis. In this article, a straightforward method is presented, allowing to calculate the effective irradiance from the maximum power current, and to calculate the bifacial gain using a power scaling relation. The proposed method was experimentally tested using an outdoor platform with a dual-axis tracking system with bifacial modules. The effective irradiance was calculated using the novel method presented nRMSE of 2.88%, relative to the results obtained using the consolidated method. The bifacial gains obtained were 6.24% and 6.69%, respectively, using the proposed and traditional calculation methods. The procedure presented in this study might be useful for the quantification of the effective irradiance and the bifacial gain for PV installations, which do not have extensive monitoring hardware.

allowing increased solar energy harvesting with the same array area as monofacial modules. The research work on bifacial PV technologies started in the 1960s, however, bifacial modules did not present relevant market share until around 2010, when several companies started producing bifacial modules-with novel technologies-in large scale. Bifacial technologies are among the most promising solutions for PV installations given that they currently present the same price per watt as conventional monofacial devices [1], [2]. The application of bifacial modules along with single-axis tracker systems provides the most cost-attractive solution for PV plants at present time, and bifacial PV devices are also a promising alternative for vertical and floating PV systems [1], [3].
Several works in the literature focused on the comparison between bifacial and monofacial PV systems in terms of the electricity produced in a given time period. The goal of such studies was to determine the energy increase due to bifaciality, which is the so-called bifacial gain (BG). In most cases, the monofacial and bifacial modules chosen by the authors were very similar in respect to their front-side performance specifications, therefore allowing direct comparison. For cases where the module specifications were not the same, a scaling factor based on datasheet power ratings was adopted, to allow a fair comparison in terms of normalized power rating of each PV system, as proposed in [4].
A critical parameter for bifacial PV systems, which affects the BG, is the bifaciality index (ϕ). It is a measure of how similar the rear-side performance is as in comparison to the front side. The bifaciality index is a module-specific parameter, and depends on the technology employed for the bifacial module production.
Stein et al. [4] report a BG of 19.5%, considering the modules installed 1.6-m distant from the ground and with 30°tilt angle. In [4], the value for the bifaciality index ϕ was not reported by the authors.
A study employing two dual-axis trackers with bifacial and monofacial modules is presented in [5]. One of the systems contained bifacial modules with high bifaciality index (ϕ = 0.92), presenting BG of 14%. In contrast, the other system studied by the authors used bifacial modules with lower bifaciality index (ϕ = 0.62), presenting a 4% BG.
Regarding bifacial PV systems installed in snowy environments, Hayibo et al. [6] compared monofacial and bifacial systems. In winter, bifacial modules take advantage of the increased amount of radiation reflected upward due to snow ground coverage, which is an important contributor for increasing BG. The authors considered large fixed bifacial and monofacial PV arrays, consisting of over 4500 modules in total. A BG of 19% was found during winter operation, however, the value of ϕ for the bifacial modules was not provided in [6].
An annual BG of 14.8% was reported in study [7], which compared fixed-tilt monofacial and bifacial modules (ϕ = 0.8). Gu et al. [7] assessed the BG for sunny and cloudy days, which presented average values of 13.1% and 16.5%, respectively.
A comparison of monofacial and bifacial modules presenting different ϕ was carried out by Muehleisen et al. [8]. The module with ϕ = 0.7 presented BG ranging from 5% to 7% (depending on the orientation), whereas the module with ϕ = 0.92 presented a 3% higher yield than the module with lower ϕ. All modules were installed at the same site.
It is therefore observed a wide range for the bifaciality index, which depends on the technology employed for the module manufacture. In turn, the BG also shows great variation since it is a function of the bifaciality and the characteristics of the installation site, where the ground albedo and module height are of great relevance.
A great variability of the BG is shown in publications [10], [11], which include international surveys considering several bifacial PV systems. Such systems present different configurations and are installed in different geographical locations. Both in [10] and [11], the reported BG levels vary widely and do not directly correlate with the bifaciality, given the influence of other factors that are different among all systems studied.
While the studies [4], [5], [6], [7], [8], [9] considered monofacial modules as a comparison resource for the calculation of the BGs, the present study provides a novel method for BG determination relying solely on monitoring of a bifacial array. For that, it becomes necessary to separate the power contributions of the front and rear sides of the bifacial PV array. This way, the front-side contribution can be considered equivalent to a monofacial array, and therefore the BG can be quantified. Such an approach requires knowledge of the effective irradiance (G E ) reaching the bifacial module, taking into account the rear-side irradiance (G R ), whose quantification is not a simple task due to its nonuniform nature [12].
The quantification of G R is of major relevance for design and performance prediction of bifacial PV arrays. While the bifaciality index is a module-specific parameter, which strongly affects BG, the rear-side irradiance is a site-specific parameter and is also a major contributor for BG [4]. In this context, the study described in [12] compares different approaches for the determination of G R , whereas works in [13] and [14] considered the development of mathematical models for the calculation of rear-side irradiance, which were experimentally validated using several back-side irradiance sensors to account for uneven distribution of irradiance. This way, the authors could precisely assess the contribution of the rear side with respect to power and energy supplied by the PV source. In other words, the bifacial modules performance could be evaluated because the solar resource available was known, as well as the amount of it converted into electricity.
The effective irradiance (G E ) is commonly used to express the amount of solar radiation available to a bifacial PV array. The G E can be calculated from front and rear-side irradiance measurements; however, it is possible to use a calibrated bifacial module-a reference bifacial module-as the source for the effective irradiance quantification. Such an approach was explored in the work [15], where the authors compared the effective and rear irradiance measured with reference bifacial modules-operating under short-circuit-with irradiance levels measured using small-area sensors, such as reference cells and pyranometers. It was found that measurements made using reference PV cells presented the best correlation with those made using reference bifacial modules. Moreover, the authors concluded that, regarding rear-side irradiance measurements, small-area reference cells and large-area reference modules provide comparable measurements.
Braid et al. [16] also considered reference modules to monitor the effective irradiance. They found pyranometers to overestimate the effective irradiance by 2.5% -4.5%, even after spectral corrections were made. The authors state that, besides providing a more accurate irradiance measurement, reference modules also present lower cost when compared to pyranometers and require less correction factors.
Recent developments on electronic monitoring devices, as presented in [17] and [18], allow monitoring reference modules, which are also used to produce power within the PV array. Such devices are able to perform tests on the reference modules while keeping a low impact on electricity production.
Due to the difficulty in assessing the rear-side irradiance on bifacial modules, the present work concerns the determination of the BG of PV arrays without the need of measuring G R , and without relying on monofacial PV modules for the calculation of the BG. For the validation, monitoring of a bifacial PV array in real operating conditions was carried out. This study differs from the literature in the following way.
1) All bifacial modules were individually tested before the array was assembled. Even though the modules were new, I-V testing was carried out to ensure that all modules were performing correctly but most importantly, to allow accurate outdoor measurements with each side (covering the other side). This allowed quantifying the bifaciality factor ϕ for each module-necessary for the validation phase-as well as the actual standard test condition (STC) ratings, obtained through modeling methods. In STC, the global irradiance (G F ) on the front side of the PV array is 1000 W/m 2 , whereas the cell temperature (T c ) is 25°C. 2) A novel method to quantify the effective irradiance is introduced; such a method uses the operating current as an input, that is, front and rear-side irradiance, module mismatch and uneven irradiance distribution are automatically taken into account because such factors impact the electrical performance of the PV array. Testing of this method was carried out with the array operating in real-field conditions, that is, using a nonideal setup and under uncontrolled conditions. 3) I-V curves of the entire array were measured before the measurement campaign started. They are representative of all modules operating in series, thus considering the effects of module mismatch, and serve to check that the seriesconnected modules are performing well and presenting a smooth I-V curve. 4) Another advantage of the proposed method is that the BG is calculated without the need of a reference monofacial module. The bifacial module's front-side individual contribution is quantified, thus being representative of a monofacial module with the same ratings as the bifacial. The rest of this article is organized as follows. Section I provides an overview of the context and the relevance of the study. Section II describes the experimental assembly, the proposed method, as well as the validation phases, making reference to supporting literature when applicable. Section III presents detailed results produced using the proposed method, as well as the validation carried out based the International Standard IEC-60891 [19]. Comparative results obtained with the use of a reference PV cell attached to the rear side of the array are also presented in Section III, although it should be highlighted that the method itself does not rely on rear irradiance measurements. Also, the BG calculation based on the yield of a monofacial PV array is also provided in Section III, however, it should be emphasized that the method proposed in this article does not require a monofacial PV system for the BG quantification. Finally, Section IV concludes this article.

A. Experimental Resource for This Study
The experimental assembly used in this study is installed in Ajaccio-Corse-France, precisely at the UMR SPE CNRS 6134 Laboratory. That research center holds several PV technologies and systems, among them, a two-axis tracker manufactured by HeliosLite, model HL-39, as illustrated in Fig. 1. The use of a dual-axis tracker minimizes the angle of incidence of the radiation, thus increasing the amount of direct solar irradiance on the front surface. Moreover, the tracker keeps the bifacial modules over 3 m apart from the ground, therefore increasing the amount of radiation reaching the rear side because of the higher angle of view of the modules. The tracker holds four different types of PV modules; however, this article focuses only on the study of the bifacial devices, which were manufactured by Trina Solar, model TSM-335DEG6MC(II). Relevant datasheet specifications are presented in Table I.
The bifacial PV array is connected to an independent input of a 3000-W SMA Tri-Power inverter, which in turn is connected to the external utility distribution grid. The inverter provides the voltage, current, and power values for the acquisition system through digital communication; this way, no dedicated hardware for these measurements is required. A calibrated reference cell with temperature compensation measures the in-plane global irradiance reaching the front side of the PV array.
The bifacial array temperature is measured by means of a thin PT-100 sensor attached to the rear-side glass of a module, between two cells. Thermal insulation was added to the portion of the sensor, which is not in contact with the glass, to reduce the interference of wind and ambient temperature.

B. Initial Tests With the Bifacial Modules
The first task carried out when the bifacial PV modules were received was to test them outdoors. The modules were positioned on a fixed stand, in a clear-sky day, when the irradiance was close to 1000 W/m 2 . Then, I-V curves of front and rear sides were measured [see Fig. 2(a) and (b)], each at a time, covering the side that was not under test. The irradiance and cell temperature were measured using sensors attached to the I-V curve tracer, model EKO MP-11.
The reasons for this testing are as follows. 1) Ensure that the modules are performing well after transport and handling. 2) Determine the bifaciality index (missing on the datasheet).
3) Measure the I-V curve notable points: open-circuit voltage (V oc ), short-circuit current (I sc ), voltage and current at maximum power (V mp and I mp ), and the maximum power (P mp ). The I-V curve measurements for the six modules refer to irradiance values around 1000 W/m 2 and cell temperature around 56°C.
The I-V curves measured outdoors were corrected to STC (G STC and T c,STC ) using the single-diode model (SDM). For the parametric identification, it was employed the classification method proposed in [20], which was further developed in [21]. In turn, the adjustment method proposed in [22] was used for the adjustment of the SDM parameters according to the operating condition. Having the I-V curve parameters corrected to STC allows checking the STC specifications from the datasheet.
It is worth noting that in this study, the I-V tests and the calculation of the bifaciality index were carried out only for validation purposes. Such experimental assessments are not required for the application of the novel method proposed in Section II-D.
C. Consolidated Bifacial Modules Theory 1) Bifaciality Index: According to the technical specification IEC-60904-1-2 [23], which is devoted to the characterization of bifacial PV modules, the short-circuit bifaciality ϕ I sc is computed using (1), where I sc,F,STC is the short-circuit current of the device when it is illuminated only on the front side. In turn, I sc,R,STC is the short-circuit current of the device when it is illuminated only on the rear side Similarly, the maximum power bifaciality ϕ P mp is calculated from (2), where P mp,F,STC is the maximum power of the device when it is illuminated only on the front side; and P mp,R,STC is the maximum power of the device when it is illuminated only on the rear side The final bifaciality value ϕ is the minimum value between ϕ I sc and ϕ P mp , as described in (3) 2) Effective Irradiance for Bifacial PV Modules: The socalled effective irradiance (G E ) is described in IEC-60904-1-2 [23] and considers a one-Sun irradiance reaching the front face of a bifacial PV module. It accounts for the contribution of the rear-side irradiance, as described in (4), where G E,STC is the effective irradiance for a front-side irradiance of 1000 W/m 2 and G R is the irradiance reaching the rear side of the module In this work, (4) was adapted for front-side irradiance (G F ) assuming any value, so that The effective irradiance provides a measure of the solar resource available for the bifacial module, and it is a useful parameter for system performance assessment.
3) BG Calculation: The classical expression for the BG is expressed by (6). It has been employed in several works in the literature and relies on the performance of a monofacial PV system as a reference to calculate BG BG = P mp,bifacial P mp,monofacial P mp,STC,monofacial P mp,STC,bifacial − 1.
D. Proposed Alternative Method to Calculate G E and BG 1) Calculating G E From the I mp : It is known that, given the strong correlation between I sc and G parameters, a shortcircuited PV module can be employed as an irradiance sensor. In fact, Razongles et al. [24] studied the validity of such a concept for bifacial modules, and concluded that considering I sc , the behavior of a bifacial module is the same when the current is produced by the front or rear side. However, for the case of a bifacial PV array in real operating condition, it is undesirable to shift the operating point (V, I) from (V mp , I mp ) to (0, I sc ) to calculate the irradiance since this fatally results in a yield loss, given that the output power of a PV module at I sc is zero.
The approach proposed in the following method for G E calculation is based on the dependence of I mp on the irradiance, as described in [25] and experimentally tested in [26]. The method is based on the fact that the temperature coefficient for I mp is small, as discussed in [25] and [27]. This way, being I mp mostly a function of the irradiance, I mp of a bifacial module can be used as a reference to calculate the equivalent irradiance G E by means of (7), which is newly introduced in this article.
The equivalent irradiance for a bifacial module, as calculated by (7), takes into account the irradiance effectively being converted by the PV device, thus, it intrinsically considers the effect of nonuniform irradiance distribution on the rear side Equation (7) is a linear relation of the instantaneous I mp value with I mp,F,STC , which refers to the front surface of the bifacial module for STC condition. In a similar way, the front irradiance G F is expressed by (8), where I mp,F is the portion of I mp referring to the front-side contribution Following the same idea, G R is calculated using (9), where I mp,R is the portion of I mp owing to the rear-side contribution, whereas I mp,R,STC is the current of maximum power for STC referring to the rear side Considering the ratio I sc /I mp constant for both sides of the bifacial module, the expression for ϕ is rewritten as (10) Then, from (10) in (9), G R is expressed in (11) in terms of I mp,F,STC to keep the same denominator as for G E (7) and G F (8) Substituting (7), (8), and (11) into (5) results in (12) 1000 I mp I mp,F,STC = 1000 I mp,F I mp,F,STC + ϕ 1000 I mp,R ϕ I mp,F,STC .
(12) The expression (12) can be simplified to yield (13), which is the relation between the operating currents in a bifacial module. Equation (13) is precisely the relation, which allows extending the proposal described in [25] for bifacial modules

2) Separating the Power Contributions of Each Side of a Bifacial Module:
The previous sections presented the assumptions and calculations to relate the irradiance (G E , G R , G F ) and the current contributions (I mp , I mp,F , I mp,R ). The present section, in turn, presents expressions for computing the power contributions referring to the front and rear sides, respectively, P mp,F and P mp,R . The operating current I mp can be easily measured during the operation of a PV array. This can be accomplished using built-in metering from the inverter (for example, via digital communication as in this study) or using dedicated measuring hardware. The inverter positions the operating point at the maximum power point (MPP), that is, voltage V mp and current I mp . Once I mp value is known, the equivalent irradiance (G E ) can be computed using (7).
The power contributions of the front and rear sides are expressed by scaling P mp and using a power balance, as shown in (14) and (15). It is worth noting that the relations in (14) and (15) already account for the temperature effect because P mp value comes from an onsite measurement. G F , which is an input for (14), should be measured with a device presenting reasonable spectral match with the bifacial array P mp,R = P mp − P mp,F .
3) Alternative BG Calculation: Differently from the classic approach of (6), the BG calculation proposed in this article (16) relies on G F , I mp , and P mp measurements referring to the bifacial device, allowing the contributions of the front and rear

E. Error Metric
The error metric used in the results section is the normalized root-mean-squared error (nRMSE), calculated using (17 In (17), x c and x r are, respectively, the calculated and reference parameters, whereas x r represents the average of the reference parameter within the whole dataset, and n is the number of observations contained in the dataset.

F. Validation of the Proposed Method for Computing G E , P mp,F , and BG
The validation phase for the method proposed in this study consists of three steps. First, the calculation of G E using (7) is compared to G E computed via (5), which takes into account G F and G R measurements made using temperature-compensated monocrystalline reference cells. Then, a reference P mp,F is determined from G F and T C measurements applied to the international standard IEC-60891, and compared with P mp,F obtained from (14). Finally, the BG is computed taking into account a monofacial PV array installed on the same tracker as the bifacial array, and compared with the BG computed using (16). A schematic illustration of the validation process is provided in Fig. 3.

III. EXPERIMENTAL APPLICATION OF THE PROPOSED METHOD
TO CALCULATE G E , P MP,F , AND BG

A. Initial Tests
The initial tests, as described in Section II-B, provided means to calculate ϕ I sc and ϕ P mp . The resulting values, per PV module, are organized in Table II.
Bifaciality values shown in Table II calculated using P mp are lower than those computed using I sc . The average ϕ P mp for the six modules is 0.64, and this is the value taken for ϕ.
From the I-V curves measured outdoors, which were corrected to STC, as explained in Section II-B, slight differences were observed in comparison to the datasheet specifications, mainly with respect to V mp , and consequently, P mp . The actual   Table III.

B. Experimental Dataset
The dataset obtained from the experimental platform described in Section II-A contains over 63 000 measurement points with the following parameters: front-side global irradiance, module temperature, bifacial array current, and voltage.
The front-side irradiance levels within the dataset are plotted as a function of the date and time and shown in Fig. 4. The measurement interruption in April was caused by a maintenance service on the acquisition system.

C. Estimation of G E and Decomposition Into G F and G R
The present section concerns the estimation of the effective irradiance G E being converted by the bifacial PV array.
Given that the array operating point is determined by the inverter's MPP tracker (MPPT), some oscillations in current and voltage occur because the MPPT is constantly shifting the operating point, seeking for the maximum power. Since I mp presents noise due to this constant variation, G E calculated using (7) inevitably presents noise as well, as illustrated in Fig. 5.
An undesirable effect of the noise in G E is the fact that, in some cases, its value becomes lower than the front irradiance G F . As a consequence, the rear irradiance G R calculated using (5) presents negative magnitude, which is inconsistent with the physical meaning of G R .
To reduce the MPPT-induced noise in I mp , an averaging filter was applied. In this study, a ten-period moving average was employed as a noise-reduction measure. The resulting curves are presented in Fig. 6, where a significant improvement in noise is observed.
Despite the filtering, there are still situations where the resulting G R is a negative value, for instance, during steep variations in front-side irradiance, where the MPPT cannot quickly track the optimal I mp level. To avoid negative G R values, the rearside irradiance is only calculated for cases where G E > G F , otherwise, G R is set as zero.   Considering the irradiance curves depicted in Fig. 6, the magnitude balance follows (5), that is, if G R is scaled by a factor ϕ and then added to G F , the result is G E .
Correlating the temperature-corrected array P mp with the G E calculated using (7) results in the plot shown in Fig. 7. A linear fit presents angular coefficient equal to 1.958, that is, for an irradiance level of 1000 W/m 2 , the power is 1958 W. Such a power value is close to the array P mp,STC found during the initial tests: 6 × 324.4 W = 1946.4 W, that is, 0.6% relative difference. This shows that the estimation of G E by means of (7) provides a suitable measure for the total irradiance reaching the bifacial module.

D. Determination of I mp,F From G F and I mp,R From G R
The decomposition of I mp into I mp,F and I mp,R presented in this section is for verification purpose only, since this step is not necessary for the calculation of the power contribution of each face of the bifacial module.
The front-side irradiance is a measured quantity, therefore the respective value of current produced by the front side individually is computed using (8) solved for I mp,F . Similarly, I mp,R is obtained from (11) using G R calculated in the previous steps, from (5).
In Fig. 8, it is shown that I mp = I mp,F + I mp,R , and such a relation is valid for every time step contained in the dataset.

E. Decomposition of P mp Into P mp,F and P mp,R
The power fractions of P mp owing to the front and rear sides of the bifacial module were calculated using (14) and (15). The resulting curves for a given day are illustrated in Fig. 9, where it can be observed that P mp = P mp,F + P mp,R .

F. BG Calculation
From the novel method proposed in this study, the BG for the modules under study could be computed without a monofacial PV array and without measurements of the rear-side irradiance. The BG value found using (16) is 6.24%.
The study conducted by Burnham et al. [5] also considered a dual-axis tracker and bifacial modules with ϕ. = 0.62 (in this study, ϕ = 0.64). Despite the similarities, Burnham et al. [5] found BG = 4%, which is 35% smaller than the BG found in this work. This fact illustrates how similar bifacial systems can perform differently because of site-specific parameters.

G. Validation, Step 1: Using a Back-Side Irradiance Sensor for the Bifacial PV Array to Calculate G E
This section provides an assessment of how the results presented in Section III-C compare with performance metrics calculated using actual rear-side irradiance (G R,meas ) measurements. The values for G R,meas were measured using a small calibrated PV irradiance sensor positioned at the center of the rear side of the PV array. The sensor model is identical to that used to measure the front-side global irradiance. It is worth emphasizing that the use of a rear-side irradiance sensor is considered in this section of this article only for checking purposes since the proposed method-which is the core of the study-does not require on G R measurements for the calculation of G E , P mp,F , and BG using (7), (14), and (16).
The two vectors for G E -calculated using (7) and (5)-are presented in Fig. 10 as a function of time for four days, each with a different sky condition. Fig. 10 shows a good agreement between G E calculated as a function of I mp (7) and G E computed using G F and G R (5), even in transient and low-irradiance conditions. The plots in Fig. 10 illustrate the suitability of the calculation of G E using I mp determined by the inverter's MPPT.
Plotting the levels of G E obtained from (7) and (5) results in Fig. 11. Fig. 11. Correlation for G E from (7) × G E from (5).
In Fig. 11, a correlation factor greater than 0.999 was found using the MATLAB function corrcoef (Pearson correlation coefficient). The nRMSE level for the data is 2.88%, whereas the correlation slope is 1.0062. Factors that contribute to the spread shown in Fig. 11 are the MPPT, which introduces fluctuations in the operating current and the temperature, which is not compensated in (7).

H. Validation, Step 2: Calculating the Front-Face Power Using IEC-60891
This part of the validation process deals with P mp decomposition method proposed in Section III-E. Given that P mp and G F are measured quantities, the quantification of P mp,F by means of another method allows indirect assessment of the correctness of G E estimation via (7). For the determination of a reference P mp,F , the second procedure of the International Standard IEC-60891 was used [19] in the modified form described in [28]. The advantage of the approach presented in [28] is that fewer experimental data are needed to identify the IEC-60891 model parameters.
IEC-60891 was applied to model the electric behavior of the module's front side only, as if the bifacial module was a monofacial device. For that, two I-V curves referring to the bifacial module (with the rear side covered) were used to identify the three unknown parameters for the IEC-60891 voltage correction equation (a = 0.013, R s = 0.43 Ω, and k = 0.002 Ω/K). This way, using T c and G F as inputs, the voltage and current provided by IEC-60891 refer to a monofacial module, and the product between voltage and current results in the front-face power contribution.
The comparison between P mp,F calculated using (14) and P mp,F calculated using IEC-60891 was carried out both as a function of time (see Fig. 12, considering four days with different sky condition) and as a correlation plot (see Fig. 13), considering all measurement points within the dataset.
The plots of P mp,F illustrated in Fig. 12 show a good fit: the general form of the curves agree; however, some low-amplitude noise is observed in the curve of P mp,F calculated using (14). The reason for the noise is the propagation of the noise on G E , which in turn is caused by the oscillations on I mp introduced by the MPPT.  Constructing a plot of P mp,F from (14) and P mp,F (IEC-60891) results in Fig. 13, where there is a correlation coefficient greater than 0.999. This way, it is shown that the decomposition of P mp into P mp,F + P mp,R is consistent with the front-side irradiance and module temperature registered in the dataset, which were used to specify the operating condition for the application of the IEC-60891 correction method. Consequently, the calculation of G E via (7) is also consistent, given that G E is the only input parameter in (14).
Calculating the nRMSE for the data points of Fig. 13 results in 2.68%, whereas the correlation slope is 0.980.

I. Validation, Step 3: Considering a Monofacial PV Array Installed on the Same Tracker to Calculate the BG
The BG calculated using the method proposed in this article using (16) resulted in 6.24%. In turn, calculating the BG based on the performance of a monofacial PV array added for comparison, as carried out in numerous studies in the literature using (6), provides BG = 6.69%, which corresponds to a relative difference of 7.2%. Such a deviation results from the coupled errors associated to the calculations via (6) and (16).
It is worth noting that (6) takes the STC power ratings into account. When this experiment began, both bifacial and monofacial modules were new, and the STC ratings were identified outdoors during the initial I-V tests. Therefore, irradiance and temperature measurement errors are possible sources for deviations between the real and calculated STC ratings, which affects the calculation of the BG via (6).
On the other hand, the BG calculated using (16) relies on the bifacial module's operating current, which is not temperaturecompensated and presents fluctuations due to the MPPT action. These factors are error sources for the calculation of BG via (16).

J. Limitations of the Proposed Method
Since the method to compute G E relies on the maximum power current, it is essential that the PV array operates at the maximum theoretical power point, according to the operating condition defined by G E and T c . However, in practice, there are a number of factors that can potentially shift or prevent the operating point to be at the MPP.
The MPPT might force the PV array to operate far from the MPP during fast-changing irradiance levels, for example, in windy, cloudy days, where the irradiance profile shows steep variations. Given that it is not possible for the MPPT action to be instantaneous, the array's operating point is likely to be out of the MPP short after abrupt irradiance changes.
Another reason for a PV array to operate far from the MPP is the so-called clipping. Such an effect occurs when the dc power is greater than the inverter's rated power. Unusually high dc power levels can be found in cold, clear-sky days, in which the low temperature leads to high voltage levels, whereas high irradiance leads to high current levels. As a result, it is possible for the dc power to be significantly higher than usual, and if such a level is greater than the inverter's rating, the operating point of the PV array will be intentionally shifted. This will reduce the PV array power output to a level within the inverter's capacity, and as a result, the dc power will not be coherent with the operating condition defined by G E and T c , and (7) will not produce reliable results.
As for the clipping, a condition termed curtailment also might force the PV array to operate outside the MPP. Inverters are able to sense the grid operating condition, and in cases where the ac voltage is above the rated limit-for example due to a local excess of reactive power-the inverter would reduce the supply of electricity to the grid, forcing the PV array to operate outside the MPP. Another parameter monitored by the inverters is the grid ac frequency. Frequency levels above the nominal reference are an indication of an imbalance between instantaneous demand and supply. If a system presents more supply than demand, the grid frequency is likely to rise, and the PV inverters will respond reducing the power supplied to the grid. As a result, the operating power of the PV array will not be coherent with the current levels of G E and T c ; therefore, (7) will not provide valid results for G E .
One way to overcome the method's dependence on the inverter's MPPT is through the use of module-level I-V trace equipment, as proposed in [17] and [18]. This would allow (7) to be used in all operating conditions, regardless of the inverter's state.
Finally, the proposed method considers that the bifacial PV array is not faulty, shaded, or soiled.

IV. CONCLUSION
This article considered the problem of determining the effective irradiance for bifacial PV modules from the operating current, without requiring rear-side irradiance measurements. Measuring the rear-irradiance of PV arrays in real operating conditions is a challenging task given the nonuniformity of radiation along the rear surface of the PV modules.
The greatest advantage of the method proposed in this article is that the total irradiance effectively being converted by the PV array is quantified, since the operating current I mp is used as the source for G E calculation. The technique thus allows splitting P mp into two fractions, P mp,F and P mp,R , which are the power contributions of each side of the bifacial PV array, allowing the BG to be computed without the need of a reference monofacial module.
The main limitation of the proposed method is the strong dependence on the inverter's MPPT. Thus, the method will not provide reliable estimations of G E for cases where the PV array is not operating at the MPP. However, the method can still be implemented using I-V tracers, which are able to determine the MPP without relying on the inverter's MPPT.
The method was validated in the following respects.
1) The G E calculation using I mp via (7) was checked against the consolidated approach based on the bifaciality index and rear-side irradiance measurements using (5), presenting nRMSE of 2.88%. 2) The P mp,F calculated using (14) was compared with the results from IEC-60891, showing nRMSE of 2.68%. 3) Finally, the BG calculated with (16) was compared to the BG provided by (6), resulting in 6.24% and 6.69%, respectively.