Mismatch Loss Reduction in Photovoltaic Arrays as a Result of Sorting Photovoltaic Modules by Max-Power Parameters

Variations in photovoltaic (PV) module current-voltage curves result in a power loss in PV arrays often referred to as mismatch loss (MML). As a means of reducing MML, newly fabricated PV modules are sorted to meet a set tolerance for variation in overall maximum power output with respect to a given module’s rated power. Starting with flash test data sets for two different polycrystalline PV modules and a simulated sorting procedure, Monte Carlo techniques were used to generate a large number of artificial PV arrays. The MMLs for each of these arrays were then calculated to assess the sorting procedure’s ability to reduce MML. Overall MMLs were quite small (0.001–0.01%). Sorting by Imp resulted in the most consistent MML reductions. Sorting by Vmp yielded insignificant results. Sorting by Pmp yielded significant MML reduction in only one of the two PV module data sets. Analysis was conducted to quantify if additional sorting on top of what bothmanufacturers had already donewouldmake economic sense. Based on high level economic analysis, it appears that additional sorting yields little economic gain; however, this is highly dependent upon manufacturer sorting cost.


Introduction
Differences in the current-voltage characteristics of photovoltaic (PV) modules give rise to a type of power loss referred to as "electrical mismatch" once modules are connected to networks of series and parallel strings called arrays.The consequence of "mismatch loss" (MML) is that the total power output of a PV array will be less than the sum of the power outputs of the modules as if they were acting independently [1].
This phenomenon has been investigated by a number of authors [1][2][3][4][5] primarily by using one of two methods: (1) the comparison between a PV array's ideal max-power and the actual max-power computed by progressively synthesizing the I-V curves of modules, series strings, and finally the complete PV array; and (2) MML estimates of PV arrays composed of modules with known or statistically generated I-V characteristics.This second method was made possible by an equation developed by Bucciarelli [1] that estimates MML in PV arrays composed of modules with relatively small variations in their I-V characteristics.Bucciarelli's [1] model was found to adequately estimate MMLs in 100 kWp arrays by Iannone et al. [2], combining and expanding on the work of Bishop [3] who used random numbers to generate PV modules arranged into arrays, and Chamberlin et al. [4], who estimated small MML values in randomly arranged arrays of four 48 Wp PV modules.After a brief outline of some of the causative mechanisms of MML in PV arrays, Kaushika and Rai [5] used Bucciarelli's [1] model to explore the consequences of aging on MML values, showing extremely small MMLs (<0.01%) and relatively large MMLs (>10%) in PV arrays composed of newly fabricated cells and aging cells, respectively.
A common practice employed by manufacturers is to sort newly fabricated PV modules prior to installation as a means of reducing MMLs in assembled PV arrays.Previous studies have focused primarily on predicting and estimating MML in existing or simulated PV arrays; however very little research exists on the effectiveness of sorting as a means of MML reduction.The purpose of this paper is to quantify to what degree manufacturer sorting has upon MML reduction and whether it remains a good economic decision.
This investigation was carried out in three steps.First, a simulated sorting procedure was applied to a population of real PV modules.Sorting methods included sorting modules by reducing variances in a population of modules: maximum power current, maximum power voltage, and maximum power.Second, Monte Carlo techniques were used to randomly select and arrange the sorted PV modules into PV arrays.Bucciarelli's [1] method was then used to calculate the MML in each randomly generated PV array, producing a MML distribution over a series of simulations.This was systematically repeated over ever stricter sorting procedures, the results of which showcase the effect of sorting upon MML reduction.Finally, the impacts of these results were discussed by means of a high level economic analysis.

Mismatch Loss Estimation Model
2.1.Current-Voltage Relationship in a PV Cell.In terms of light generated current, the I-V curve of a PV cell can be expressed as [5] Dividing ( 1) by the average max-power current  mp , introducing the average max-power voltage  mp as a unity fraction, and setting Substituting (4a), (4b), and (4c) into (4) results in (5): This derivation is similar in method to Kaushika and Rai [5], resulting in an expression identical to that of Bucciarelli [1].Equation ( 5) is assumed to adequately represent the I-V curve of a photovoltaic cell (or network of cells) near their maxpower point [1,2,5]. is commonly referred to as the cell characteristic factor and can be expressed in terms of the fill factor (FF) defined as [1] FF =  mp  mp The relationship between the fill factor and  can be seen in Figure 1.
This study takes into consideration the placement of individual modules within a PV array.To account for this, we introduce the modified cell characteristic factor   , which is based upon the average fill factor of only the PV modules used within the assembled PV array (as opposed to the entire available population): 2.2.Power Loss due to Mismatch.From Bucciarelli [1],  total PV modules arranged into a network of  parallel series strings of  modules; each has a mismatch loss Δ estimated by where  2  and  2  represent the variance of max-power current and voltage, respectively, in our network of PV cells.Figure 2 shows a typical  ×  network of modules: Expressions ( 9) are derived on the assumption that variations in max-power current and max-power voltage in the th PV cell in our network take the nondimensional form [1]: and   are respective measurements of variations in maxpower current and max-power voltage and take on values between −1 and 1.   and   are small numbers, respectively, measuring the percentage range of variation in  mp and  mp .
In order for (8) to yield acceptably accurate results, the product of  and   , known as the weighted variation of max-power   , must be less than 1 [1].In practice, this limits the standard deviations of max-power parameters in a population of PV modules to within a few percentage points of their mean values [5].It is also assumed that max-power current and voltage in each PV module are uncorrelated; specifically that the Pearson's Correlation between  mp and  mp is small with respect to ±1 [1].
In prior MML studies, the term   is calculated based upon the entire sample population of modules used in a given array without taking into consideration module placement.This neglects the effect that module placement has on series MML in that different configurations of the same modules within the same array will yield significantly different series MML values.In order to account for module placement, a unique   ,   , and series MML value are calculated for each string based solely upon the modules within that string.These MML values are then given a weight  based upon the ideal power output  ideal of that string relative to the average ideal power output of every string in the array.The final series MML is taken as the average of the weighted series MML of each string in the array.Thus, for an array of  strings, the series MML term in (8) becomes where  refers to the string in question.The final, modified MML equation is found to be

Sorting Procedure and PV Array Simulation
Most PV manufacturers sort their modules by power output at 1,000 W/m 2 and 25 degree Celcius cell temperature and Atmospheric Mass of 1.5, guaranteeing some small amount of variation from their rated power.Current industry tolerances between the modules nameplate rating and measured power rating are ±3-5%.
Sorting for the purpose of this paper was conducted by imposing a maximum allowed deviation from the mean for each max-power parameter:  mp ,  mp , and  mp on the entire population of modules.This maximum deviation was imposed by removing outlier modules and thus reducing the variance of max-power current  2  and maximum power voltage  2  within the remaining population of PV modules.This maximum allowed deviation from the mean was then reduced to examine the sensitivity of MML specific to each sorting parameter and maximum allowed deviation.
Monte Carlo techniques for randomly selecting and arranging available remaining PV modules into arrays were carried out using Microsoft Excel's random number generator, which has been shown to be suitable for generating small batches of random numbers [6,7].After specifying PV array dimensions ( in series,  in parallel), the required number of modules was selected from sorted groups and placed at random into the array.Finally, a MML value was calculated using (12).[1] for estimating MML in a single series string of PV cells with no  mp variation and an  mp variation defined by a Gaussian probability density function.Bucciarelli [1]   seen in Figure 3. 10,000 simulations were run calculating the MML in a series string composed of 100 randomly selected modules (so as to neglect the 1/ term with respect to 1).As seen in Figure 4, the average value of the resulting series MML distribution is nearly identical to that found by Bucciarelli [1], only differing by 0.01%.As stated in Section 2, it is a requirement that the weighted variation of max-power   is less than 1 and the crosscorrelation  for  mp and  mp is small with respect to ±1 [1].Statistical analysis shows that both measures are within   acceptable ranges for each module type, the values of which are shown in Tables 1 and 2.

Results and Analysis
Repetition of the process outlined in Section 3 yields a distribution of MMLs specific to the dimensions of that particular PV array and module sorting criteria.Keeping PV array dimensions constant while systematically restricting the sorting criteria upon the population of available PV modules yields different results dependent upon which particular parameter one sorts by.A realistically dimensioned PV array (400 kWp, 700 V) [8] was considered.For this array, MMLs were calculated by sorting based on each of the three maxpower parameters:  mp ,  mp , and  mp .The effect of sorting upon average MML, represented in terms of a reduction in the standard deviations of each respective max-power  It is immediately clear that sorting by  mp has the most consistent, immediate, and greatest reduction on MML.Each module works at its own  mp under max-power conditions, and in a series string the worst performing module will lower the current output of every module in that string.As a result, individual module placement heavily influences series MML in a PV array.As outlier modules are removed with respect to the imposed  mp restriction, a greater net reduction in MML is observed.
Compared to  mp , sorting modules by  mp results in negligible or even counterproductive MML reduction (as in the case of the 250 Wp module).Parallel MML represents the overall voltage drop in a PV array as a result of the series string with the lowest voltage.In a realistically dimensioned array such as the one considered in this study, each string in the array is composed of twenty or more PV modules.This effectively reduces the impact that individual module placement has upon string voltage variance, thus reducing the importance of  mp sorting.This phenomenon can be inferred by the MML estimate (8) in that parallel MML is inversely proportional to string length ().The greater the string length, the less string voltage variation one will find and the less parallel MML one will measure.Evidence of the heavy dependence of overall MML upon  mp over  mp variance has been demonstrated by Kaushika and Rai [5], in which parallel MMLs are found on average to be a full order of magnitude less than series MMLs even in arrays with a string length of one.
Sorting by  mp yielded the most unpredictable impact on MML reduction.This is not surprising given that a PV module's  mp is the product of both  mp and  mp .With overall MML being shown to heavily favour series MML over parallel MML, sorting by any parameter that does not directly reduce the variation of  mp (such as  mp ) is not going to result in consistent, reliable MML reduction.
Both module populations were subjected to some degree of sorting with respect to  mp prior to use by this study and reflect each manufacturer's  mp variation tolerance.There is a significant difference between the two module sets, with the Helios module showing a modest ±3% variation from the mean, whereas the modules from the anonymous manufacturer show ±1%.The consequences of this can be seen in average MMLs found before we applied the simulated sorting procedure, with the modules from the anonymous manufacturer showing a MML of 0.009% compared to the Helios module MML of 0.055%-roughly 6 times more but still orders of magnitude smaller than MML estimates reported in prior studies [1,2,5].Any reduction in MML resulting from sorting modules similar in fashion to this study would only be reliably seen in the short term.The work by Kaushika and Rai [5] shows dramatic increases in MMLs as PV cells age.Outside the scope of this study, but clearly the next step, would be to investigate the long-term reduction in PV array MML as a function of sorting modules prior to installation.

Economic Benefit of Sorting PV Modules
Reducing max-power parameter standard deviations of a given set of PV modules results in a ΔMML reduction corresponding to a net economic gain over the lifetime of the PV array.For a set Power Purchase Agreement (PPA), a given PV array will produce on average ] $/year.A reduction in MML will result in a yearly net cash flow increase given by The cost   to the owner of the PV array to presort the modules is given by where   is the manufacturer's cost of sorting and  is their desired profit margin.In order to yield a net economic gain, ISRN Renewable Energy NPV >   , resulting in an expression for the minimum required decrease in MML from sorting: The net economic benefit of sorting modules can be seen in Figures 9(a) and 9(b).Even in the most significant case, that of sorting by  mp and reducing the  mp standard deviation to 40% in the Helios 250 Wp module, the net economic gains over the lifetime of the PV array are minimal.This is, however, highly dependent upon manufacturer sorting cost.Despite this, all indications suggest that sorting modules by  mp will yield the most consistent and reliable reduction in MML.

Summary and Conclusions
A procedure for estimating average MMLs of PV arrays built from a given set of real PV modules was developed by simulating preinstallation sorting, followed by generating random artificial PV arrays using Monte Carlo techniques.The analysis indicated that sorting PV modules by maxpower current  mp yielded the most consistent and greatest overall reduction in MML when compared to sorting by  mp , which yielded inconsistent MML reduction, and  mp , which yielded insignificant MML reduction.These results were expressed in terms of a reduction in the standard deviations of each max-power parameter distribution.
A brief economic analysis was carried out describing the net economic benefit over the lifetime of the PV array of MML reduction by means of sorting.Even in the best case scenario, it appears that sorting yields little or no economic gain; however this is highly dependent upon manufacturer sorting cost.
This study was limited by the lack of data pertaining to the time sensitivity of PV module current-voltage variation.As modules degrade over time, their electrical characteristics change which can impact MML.Cost to manufacturer to sort modules :

Nomenclature
Manufacturer profit margin.

Figure 2 :
Figure 2: Network of  modules in series and  modules in parallel.

Figure 3 :
Figure 3:  mp frequency distribution for generated data set used to verify the MML model.A Gaussian pdf was used to create the distribution based upon the example carried out originally in Bucciarelli [1].

Flash test data was
provided by an anonymous module manufacturer and Helios Solar Works.The anonymous module manufacturer provided flash test data for 3,850 285 Wp modules, the details of which are shown in Figures5(a), 5(b), and 5(c).Helios Solar Works is a Milwaukee, WI-based manufacturer of mono-Cx PV modules and provided flash test data for 2,132 250 Wp PV modules upon request, the details of which are shown in Figures 6(a), 6(b), and 6(c).

Figure 4 :
Figure 4: Series MML frequency distribution using Monte Carlo techniques for module selection and placement.Results are nearly identical to those of Bucciarelli [1].

Figure 5 :
Figure 5:  mp ,  mp , and  mp distributions of the anonymous module manufacturer's 285 Wp module.

Figure 6 :
Figure 6:  mp ,  mp , and  mp distributions of the Helios Solar Works 250 Wp module.

Figure 7 :
Figure 7: MML frequency distributions for a 700 V, 400 kWp PV array populated with the Helios 250 Wp module.Each distribution is the result of ever stricter sorting with respect to  mp , as represented by the reduction in  mp standard deviations displayed above.

Figure 8 :Figure 9 :
Figure8: Average MML as a function of sorting by each max-power parameter for each module.Sorting is represented by a percentage reduction in the standard deviation of each max-power parameter's distribution with respect to their unsorted standard deviations.

Table 1 :
assigns the following values: Starting with the same parameters, a Gaussian probability density function was used to generate  mp 's for a distribution of 204 PV cells with  mp and   mp approximately equal to those listed above.The PV module  mp distribution can be Statistical measures for the Helios Solar Works 250 Wp module data set.

Table 2 :
Statistical measures for the Anonymous Module Manufacturer 285 Wp module data set.