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Simplified technique for calculating mismatch loss in mass production
Solar Energy Materials & Solar Cells 134 (2015) 236–243
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Solar Energy Materials & Solar Cells journal homepage: www.elsevier.com/locate/solmat
Simplified technique for calculating mismatch loss in mass production Rhett Evans a,b,n, Kyung H. Kim c, Xiaoyi Wang e, Adeline Sugianto e, Xiaochun Chen d, Rulong Chen d, Martin A. Green a a
Australian Centre for Advanced Photovoltaics, University of New South Wales, Anzac Parade, Sydney, NSW 2052, Australia Solinno, Sydney Olympic Park, Sydney, NSW 2127, Australia c School of Photovoltaic and Renewable Energy Engineering, University of New South Wales, Anzac Parade, Sydney, NSW 2052, Australia d Wuxi Suntech Power Co Ltd, New District Wuxi, Jiangsu 214028, China e Formerly with Wuxi Suntech Power Co Ltd, China b
art ic l e i nf o
a b s t r a c t
Article history: Received 15 October 2014 Accepted 17 November 2014
This paper describes a generalised approach to estimating mismatch loss for series connected cells, utilising the result that the deviation from maximum power as a function of deviation from maximum power–point–current is a relatively stable relationship for a wide range of cell performances. This Power–Current relationship can be used to verify the strong link between mismatch loss and the variance in the maximum power–point–current values for the cells being mixed together. The mismatch loss in a modern photovoltaic module is low – below 0.1% even when there is no cell sorting whatsoever. Cell-to-module variance effects need to be understood and controlled to ensure mismatch loss remains low in a finished module. Without the constraints of mismatch loss in module design, other motivations for cell sorting should be considered, such as for optimising the manufacturing system or meeting particular product requirements. & 2014 Published by Elsevier B.V.
Keywords: Photovoltaics Solar cells Solar modules Mismatch Manufacturing Cell-to-module
1. Introduction As manufactured, all individual solar cells exhibit some slight differences in their performance. When these cells are series connected in a module and that module is operating at maximum power, the individual cells will typically output some power slightly below their maximum. The difference between the maximum power of a module and the sum of the maximum powers of the individual cells is known as mismatch loss. It is important to understand mismatch loss when optimising module manufacturing strategies. Mismatch loss is a well known effect and calculations of the mismatch loss resultant from the mixing of groups of cells has been a subject of study for over 30 years. The original work in this field by Bucciarelli [1] remains one of the most well cited, thorough and generalizable and it is further discussed throughout this work. Many of the papers [2–7] examine the mismatch loss resulting from different ways to bin and interconnect cells of different properties. The main limitations of these works are that it is difficult to generalise the results and there is no direction for dealing with encapsulation effects. Some of the works phrase the mismatch loss issue in the larger context of manufacturing yield
n
Corresponding author. E-mail address: [email protected] (R. Evans).
http://dx.doi.org/10.1016/j.solmat.2014.11.036 0927-0248/& 2014 Published by Elsevier B.V.
and total Watts produced [8,9]. Total mismatch loss is also dependent on solar insolation level [3] and so the issue has also been explored in terms of its influence on energy yield [10,11]. Many works investigate mismatch loss in the context of stressed or non-standard operating conditions such as partial cell shading ([6,12] to name a few) and many more studies ([13–15] to name a few) deal with the additional subject of mismatch loss at an array level. Both of these topics are important issues to which the results here are applicable, but neither is the immediate focus of this work. Mismatch loss is itself one component of a broader group of changes typically referred to as Cell to Module (CTM) effects. Usually expressed as a loss [16] or a conversion ratio [17], CTM calculations account for all the changes in average performance when cells are encapsulated into modules. For the purpose of this study, the term encapsulation refers to the group of processes by which cells are made into modules, including tabbing, stringing and lamination. Three studies make attempts to quantify mismatch loss at a final module level, rather than just as a consequence of cell mixing [18–20]. One [18] is particularly notable as the only study that attempts to measure mismatch loss at the final module level through direct experimentation. Unfortunately, the resolution of the technique described is insufficient to calculate loss for cells sorted in ways typical of modern manufacturing, and the technique is not directly generalisable to all module making strategies. This is not a weakness in the approach: rather it serves
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to highlight the important fact that mismatch loss is very low for a modern photovoltaic module, a fact that can also be verified by following the derivations of Bucciarelli from over 30 years ago. The second [19] builds a mathematical model for CTM effects that also includes mismatch, and the third [20] uses the expected difference between module I–V and SunsVoc curves to calculate mismatch loss. But the emphasis of both techniques is to deal with nonstandard conditions such as partial cell shading and so neither apparently has sufficient accuracy to detect the relatively small mismatch effect in standard modules. In this study, the new concept of CTM variance (CTMV) is also introduced and changes in variance associated with cell encapsulated are examined. This will highlight a new set of issues that need to be considered to keep mismatch loss to the low levels expected from modern production.
2. Overview of this study The aim of this study is to develop a simple and generalisable method for the calculation of mismatch loss and the monitoring of CTMV in a manufacturing context, so it can be used as an input in the design of optimal photovoltaic products. As a first step, the I–V curves of groups of production cells are summed to form a hypothetical module I–V curve and the maximum power (Pmp) and maximum current (Imp) calculated from this curve. The hypothetical curve includes no additional series resistance due to interconnection or changes in cell current due to encapsulation. Mismatch loss, L, is calculated from the difference between the sum of the maximum powers of the 72 component cells and the maximum power of the hypothetical module. This is the same technique as used in the Evergreen studies of more than 10 years ago [8,9]. A calculation of this form is referred to as Relative or Fractional Power Loss (RPL / FPL) in other studies [1,7,18]: L¼
∑72 i ¼ 1 pi P mp ∑72 i ¼ 1 pi
ð1Þ
where pi is the maximum power of the ith cell and Pmp is the maximum power of the module as previously mentioned. Lower case variables are used here to describe cell properties, upper case variables being used for module properties. L is calculated for four different types of cell sorting arrangements as shown in Table 1. This curve summing exercise is intended to give the most direct estimate of mismatch loss to serve as a validation of the new techniques based on the Power–Current (P–I) relationship. The simplification of using the P–I relationship is only possible when the range of cells being mixed remains very small and when the cells are series connected, as is the case for the majority of modern production. Overall, as is verifiable from the derivations in other studies [1,6,18], the typical loss for even rudimentary sorting schemes is shown to be extremely low for the variance typical of modern manufacturing. In this situation of a small variance, many of the established notions of mismatch loss – such as the module being limited to the lowest performing cell [19,21], simply do not apply: rather the module maximum power current will tend to just be close to an average of the component cell currents. Another established notion that
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mismatch loss is simply reduced by classifying cells into even smaller ranges of performance also becomes less relevant – in the context of low overall variance, process variance associated with cell encapsulation and the limitations of measurement accuracy will also have an impact.
2.1. Materials For the simulations in this study, I–V curves were collected from 3000 cells from a production order in 2012 from a Suntech manufacturing facility in Jiangsu, Wuxi, China. The cells are standard multi-crystalline silicon (mc-Si) with acidic texturing, belt diffusion, SiN ARC, full rear aluminium BSF and front silver screen printed contacts. The cells were measured on an inline cell flash tester at two illumination levels. The I–V curves were assembled from temperature and illumination corrected measurements on 200 points in forward bias and to assist with summing these were recalculated to 400–450 point curves with a consistent set of current coordinates using polynomial interpolation. The histogram of the cell's normalised efficiency is shown in Fig. 1. All of the cell data is normalised to generalise the results and protect data sensitivity, without interfering with the interpretability of the outcomes. The variance is scaled to be a percentage of mean, and then the data is mean centred. The full range of cell efficiencies is a little over 73% relative.
3. Results—calculation of losses from cell mixing 3.1. Loss calculated from summing I–V curves A random number function was used to select groups of 72 cells from the group of 3000 according to the simple sorting restrictions outlined in Table 1, and without regard to the original time order sequence in which the cells were made. The hypothetical module I–V characteristics were calculated and L was determined for each arrangement, as shown in Fig. 2. This shows the loss to be extremely low. Fig. 3 shows L as a function of key summary parameters within the group of cells. Of all these simple statistics, the strongest relationship is with the standard deviation of cell imp. This is shown in detail in Fig. 4. This relationship has been shown in another theoretical study [1] but not directly by curve summing [8,9]. The strength of this relationship is affected by the precise technique used for the curve summing – less points on the original I–V curve and linear interpolation between the points can add significant levels of noise. Fig. 4 can be fit with a quadratic passing through, and with its minimum at the origin. The line of best fit has equation: L ¼ 9:23 σ 2iN
ð2Þ
Table 1 The selection restriction for the four different cell grouping arrangements. Group
Selection restriction
A B C D
Random selection Efficiency range o 0.2% abs. imp range o 0:15 A imp range o 0:05 A
Fig. 1. Histogram of normalise cell efficiency for the 3000 cells used in this study.
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be using too few points on the raw P–I curve and any more and the quadratic fit would deteriorate due to the natural asymmetrical nature of the P–I curve. The implications of this are discussed later. The approximating quadratic in Fig. 5b is of the form:
Δpmp ¼ AΔi2mp
Fig. 2. Calculation mismatch loss L for the 4 different sorting arrangements, A through D, as specified in Table 1. The loss is extremely low, less than 0.1% even for unsorted cells and less then 0.04% for rudimentary sorting.
ð3Þ
where Δpmp and Δimp are respectively the proportional deviations in power and current from their respective maximums, and A is a constant that defines the quadratic. In Fig. 5b, the value of A is estimated to be 9.48 from a least squares fit to all of the P–I data over the 7 2% range. The P–I relationship can be used to calculate loss when the deviation between operating current and cell imp is known using Eq. (3). For the kth cell in a series connection of n cells, let the maximum power point current, voltage and power of that cell be denoted by ik, vk and pk respectively. The data from the 3000 cells used in this study showed that the maximum power point current of the theoretical module formed by the series connection of
Fig. 3. Mismatch loss L as a function of the key summary parameters for the groups of cells that make each module. The discontinuities on some of the graphs result from distinctions between the different sorting groups.
Fig. 4. L as a function of the normalised standard deviation in cell imp, of the average . Fig. 2 more clearly shows for a clearer representation of the difference between the sorting groups.
Fig. 5. Two versions of the Power–Current (P–I) relationship (a) sample curve for a 17.5% cell; (b) the P–I relationship for all 3000 cells normalised to a percentage deviation from their respective maximum powers and maximum power point current. (b) also includes a quadratic fit to the data.
where σ iN is the standard deviation of the imp values of the cells, normalised to the average of the cells. 3.2. Loss calculated from the measured P–I curve of all cells Loss calculations for series interconnections can be made using the P–I characteristic of a cell. Fig. 5 shows the P–I characteristic taken from the original raw I–V curves used in this study. Fig. 5a is an example P–I characteristic for one of the cells from the study, Fig. 5b shows a normalisation of this data in terms of the percentage deviation of current and power from the respective maximums, plotted together for all of the 3000 cells in the study. Fig. 5b also includes the quadratic fit forced to pass through and have its maximum at the origin. The P–I relationship is very consistent even for the large range of cells included in this study. The quadratic approximation by taking the normalised P–I data to 7 2% of cell imp; any less would
Fig. 6. Hypothetical Imp for the module (derived from summing I–V curves) vs I , the average imp for the group cells. The value of I (A) is an excellent estimate to the Hypothetical Imp. All the data is normalised to their respective means as outlined in Section 2.1.
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72 cells (Imp) is very closely approximated by the average current of all the cells in series (see normalised data in Fig. 6), with this latter quantity referred to as I . The “mismatch” loss for an individual cell (lk) is therefore " #2 I ik l k ¼ Δp k ¼ A ð4Þ ik The mismatch loss for the module with n cells in series, can now be calculated using the expression: " #2 I ik n 2 A∑k ¼ 1 ik vk ik Avi k ∑nk ¼ 1 I ik ∑n p lk L ¼ kn¼ 1 k ¼ ¼ k ð5Þ ∑k ¼ 1 pk nIV nIV I and V are the average of the cell's maximum power point current and voltage respectively, hence ½1=n∑nk ¼ 1 pk ¼ IV and so the simplification in the denominator follows in the previous equation. The cell data (from Fig. 1) can be used to show that the quantity vk =ik can be assumed to be a constant (see Fig. 7), because the standard deviation is a low percentage of the mean (less that 1%). Furthermore, if vk =ik is a constant, then vk =ik ¼ V =I is the same constant. Eq. (5) reduces to 2 n 2 1 1 ∑ I ik ð6Þ L¼A n I k¼1 The variance in imp for a group of cells (σ2i ) can be represented by an expression of the form:
σ 2i ¼
2 1 n ∑ I ik nk¼1
ð7Þ
The variance normalised to the average current I becomes an expression of the form: 2 n 2 1 1 σ 2iN ¼ ∑ I ik ð8Þ n I k¼1
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electrical properties of the cell with any reasonable correlation to A is the FF, even when checking for relationships as part of a multi-variate regression. But as can be seen in Fig. 8, the relationship is very noisy, with the FF accounting for only about 25% of the variance in A. It is also different to the relationship derived in Bucciarelli's study [1]. To understand the origin of the variance in A, the 3000 I–V curves were curve fitted to ameliorate the impact of measurement uncertainty in the I–V curve. The equivalent circuit for the curve fitting used two diodes (n ¼1 and n ¼2) in parallel with a resistively limited enhanced recombination term, along with an Rsh and Rs term [22]. Fitting was done to the I–V and m–V curves simultaneously, where m is the value of the local ideality factor. A model such as this does not have a unique solution for any given I–V curve, nor is a justification made here for using such a model in terms of the actual physical recombination mechanisms in the device. But the model allows for a very accurate fit of nearly any I–V curve, and so it serves the purpose in this instance of having a smooth curve unaffected by measurement error. The value of A can now be calculated using the fitted curve and these are shown in Fig. 9. One relationship is derived by fitting A to 7 2% of imp in the P–I relationship as was done for the raw curves, and another by fitting to 7 1%. Ideally, A is fitted by using the smaller range, but for the raw curves the larger range was used due to the data collection interval. Compared to the values of A fitted directly from the curve (Fig. 8), nearly 95% of the variance is removed. A strong relationship can now be seen between A and FF, and more importantly, this can be derived from the relatively simple task of fitting to an equivalent circuit model. The average value of A when fitting to the lower range of imp is 9.28, which is very close to the proportionality constant from Eq. (2).
Combining Eqs. (6) and (8) yields the simplified expression: L ¼ Aσ 2iN
ð9Þ
The value of A ¼9.48 from Fig. 5b, means the relationship of Eq. (9) is nearly identical to the one of Eq. (2) derived from the empirical study. 3.3. Curve fitting In a production context, calculating the value of A from a quadratic fit to the normalised P–I properties of a large range of cells is as unworkable as summing I–V curves directly. In Bucciarelli's work [1], the value of the proportionality constant is estimated from the electrical properties of the cells, and this convenience is adopted by nearly every study which cites the work. In this study, the only
Fig. 8. Constant A derived from the P–I properties for the 3000 cells in this study, plotted as a function of the normalised FF. The solid line shows the same relationship as derived from [1]. The Bucciarelli data has the same normalisation to make it directly comparable.
Fig. 7. Histogram of the values of vmp =imp as calculated from the I–V curves of the 3000 cells used in this study.
Fig. 9. A, derived from a fitted curve, versus normalised FF. Bucciarelli's relationship is included with the same FF normalisation. Scale is the same as Fig. 8.
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4. Results—cell to module effects The expression of mismatch loss as a function of σ2iN, derived so far will still hold when the cells are encapsulated. But the proportionality constant will change due to changes in FF from the added Rs from tabbing. Encapsulation can also change the value of σ2iN for the group of cells. 4.1. Series resistance effects Fig. 9 and to a lesser extent Fig. 8 showed the relationship between FF and A. When cells are encapsulated, the series resistance and hence the FF will change and so a new value of A is required. The relationship identified in Fig. 9 cannot be immediately used as it would involve an extrapolation for the FF values expected post encapsulation. One option for extending the range of this relationship is to use direct measurements of tabbed cells to determine A directly from the P–I curves as was done in Section 3.2. To this end, a group of ten mcSi cells comparable to the original 3000 and of varying efficiency, were measured with and without tabbing. The cells were tabbed with 3 busbars on each side, with the tabs joined together by a highly conductive cross tab about 10 mm outside the active area of the cell. Already, this will return a lower FF than for a standard tabbed and laminated cell as this 10 mm acts like the cell gap in the module case and is a non-trivial source of series resistance. These tabbed cells were loaded onto a cell tester and contacted with a spring-loaded probe card as is standard on in-line production cell testers. The cell tester uses a SAN-EI 3A Solar Simulator with a class B light source, with a detailed I–V curve swept at a resolution of 3-4 points/mV. In this case the probes are contacted to the tabbed cells and not directly to the screen printed grid. With the probe card engaged, the cells were measured directly through the probe card to get the “untabbed” measurement, and then by connecting to the tabbing leads to get the “tabbed” measurements. Not all the differences between untabbed and tabbed cells are captured with this arrangement – most particularly if the tab is only spot soldered at a select number of points, or there are otherwise grid to tab contact resistance issues, this technique will be an underestimation of the difference. However, it is a reasonable approach to capture the most significant differences and establish the wider ranging A vs FF relationship. Another option for estimating the change in A due to encapsulation is to use well established theoretical models. The grid calculator hosted by PVlighthouse [23] shows that three busbars (1.5 mm wide 250 μm high, with 2 μΩ cm resistivity and a 3 mm cell spacing) on a standard 156 mm cell adds an effective 0.7 Ω cm2 of series resistance to each cell. If this series resistance is added to the equivalent circuits calculated in Section 3.3, the new curves can be used to derive an extended version of the A vs FF relationship. The risk with this approach is that no attempt was originally made to validate the physical relevance of the fitted curves. But given the aim is simply to generate I–V curves with an extend FF range, the approach remains meritorious. As well as adding series resistance, the FF range can also be extended for the fitted curve case by extending the range of cells to include higher n ¼2 dark saturation current values (J02). This is not necessarily relevant to the situation of encapsulation, but it does show that the A vs FF relationship differs depending on the reason for the FF variation. Bucciarelli [1] also derives a version of this constant over a wide range of FF. All of these estimates are compared in Fig. 10. The trend in A with respect to FF is very similar between the tabbed cell case and the equivalent circuit case with extended Rs. As the tabbed cells are only adding series resistance, this is not too surprising. The equivalent circuit with extended resistance and J02 more closely matches the Bucciarelli relationship but shifted up. The reason for these discrepancies is discussed more in Section 5.
Fig. 10. Four relationships between A and FF over an extended range. The equivalent circuit model is an n¼ 1 and n¼ 2 diode in parallel with a resistively limited enhanced recombination with shunt and series resistance [22]. Both of the equivalent circuit relationships were derived from a sample of the fitted I–V curves in this study (see Section 3.3). The FF range was extended by extending the Rs range or the J02 range in the model, giving two slightly different relationships.
The most convenient method for calculating A for the encapsulated case is to use an equivalent circuit model with added series resistance from the tabbing. A value can also be derived from direct measurement, but due to noise in the shape of the curve at least ten different curves should be measured. 4.2. Changes in
σ2iN
To understand how σ2iN will change for the group of cells once they are encapsulated, it is necessary to consider all the sources of cell imp variance and then how these sources of variance change during encapsulation. As a framework for discussion, the concept of Cell-To-Module Variance (CTMV) is introduced here. For any given parameter of interest, the CTMV (σ2CTM) can be expressed as
σ 2CTM ¼ σ 2 ðmodulesÞ σ 2 ðcellsÞ
ð10Þ
This is important as a concept as well as a quantitative expression, because it is not always possible to calculate this for every parameter post en capsulation. Encapsulation adds variance to the cell performance and this should be understood and tracked to keep issues such as mismatch loss under control. There are many sources of potential variance in this part of the manufacturing sequence and it is beyond the scope of this study to develop experimental estimates for all of them. But a mixture of simulations and previously published data provide information and trends to understand what is important. A sample of production data is also used to show how these issues can be monitored in a simple and agglomerated way. For the 3000 cells studied here, σ iN for the full set of cells is 0.73%, and as high as 1% for some 72 cell groupings (see Fig. 2). The primary sources of variance are any of the manufacturing processes that affects the cell imp directly (such as material quality, texturing, diffusion depth, SiN, screen print area or size of white perimeter around the encapsulated cells) or indirectly through the shunt resistance (primarily a result of material quality issues or process faults). Changes in cell series resistance have nearly no impact on cell imp, but affect mismatch loss by changing the proportionality constant between the loss and σ2iN. This is dealt with separately in the following section. Variance related to material quality and screen print area will not change during encapsulation. Variance associated with diffusion profiles may theoretically change due to absorption in the encapsulation materials, but the overall effect is very small [24] and so will be ignored here. Shunt resistance should not change in going from cells to modules [25] if the process is under control, so it will also be ignored. However, this assumption may not always be true – at least one other study has assumed that the shunt resistance will change [19]. Equivalent circuit models such as those used in the curve fitting exercise (Section 3.3) can be used to show a 10%
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variance in Rsh in encapsulation can increase the cell imp variance by 10%, so if the assumption of unchanged shunt resistance is incorrect, there is a potential for error in the mismatch loss estimate. Another possible source of variation is the size of the border around the encapsulated cells, when a white backsheet is used. Other studies [26] show that at a cell spacing of around 2 mm (a typical production standard), the gain in cell isc (and hence cell imp) is around 0.8% per mm of spacing. As an indication of how this could affect the variance, if the normal 2 mm cell spacing was always accurate to 7 10% (i.e 73σ ), the resultant standard deviation in the imp gain would be about 0.05%. This would amount to a very small addition to the overall cell imp variance, but it could possibly be significant in some situation where the cell imp groupings are very tight. The final component to consider is the optical properties of the cell. These will change when cells are encapsulated due to variance in the thickness and refractive index of the SiN and the surface features of the texturing. The reflection is less sensitive to the precise SiN properties when the refractive index of the first material in the light path is 1.5 as for the encapsulated case, compared to 1 as for the air case. For texturing the variance is reduced because some component of the light that is reflected when the cell is measured in air will be totally internally reflected in the encapsulated case [27]. A first principles proof of this is nontrivial as the optics of a textured three layer stack (Glass/SiN/Si) is somewhat complex, but the OPAL2 simulator hosted by PVlighthouse [23] is designed to make exactly these calculations. Using the spherical caps texturing model developed for acidic mc-Si texturing [28], the absorption in the film can be calculated for a range of SiN and texturing properties. This is represented in Fig. 11, which shows that the variance in absorption (and hence imp) reduces in the case of encapsulation. The magnitude of the reduction is not representative of production in this case because the demonstration data is chosen for simplicity to represents the extremes of possible production values, rather than being a realistic cell population. The final situation in which σ2iN may increase in encapsulation, somewhat paradoxically, is when the initial cell binning is based on cell imp and the bin width is narrow. If cells are grouped into a narrow-range imp bin without any optical sorting, the bins will contain cells whose imp value is realised through different permutations of material and optical properties and so the variance within this bin, which is very small by design at the cell sorting step, must increase following encapsulation. In practice, the mismatch loss is likely to be so low in this case that the error will be unimportant. In summary, if the cell spacing is well controlled and there is no introduction of shunting in encapsulation, the variance should
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only decrease by a small amount in encapsulation, meaning the estimate made from the cell σ2iN would represent an upper limit to the mismatch loss. If, further to this, optical variance makes up a very small proportion of overall cell variance, as is the case for well controlled PECVD processing and consistent texturing (such as when using c-Si substrates) or if optical sorting metrics are used, then there should be almost no change in cell variance during encapsulation. A way to check that nothing unexpected is happening is to track the CTMV in production. 4.2.1. Example CTMV and mismatch loss data from production When considering an entire population of cells, different cell sorting methods will give a different value for σ iN for the group of cells. The average of the cell imp values for each group will be I. The variance in I, referred to as σ 2 ðIÞ, will add to the pooled variance within the group of cells, σ2iN, to give the total variance of the cell population. Therefore, the choice of cell sorting methodology will impact on how much of the total production variance occurs within a cell packet, and how much occurs across the average performance of the cell packets. In the case of the maximum power point current, the value of σ 2 ðIÞ will be related to the variance of the module Imp values, σ 2 ðI mp Þ, made from this group of cell packets by the CTMV expression of Eq. (10). Table 2 contains values for these quantities from a small production experiment on different cell sorting techniques. If all is going well with production, the effect of encapsulation material changes is accounted for, and no significant variance is added to the cell properties during encapsulation, the value of σ CTM should be consistent and largely representative of measurement variance. This variance will be related intrinsically to the module measurement but also to the differential error with the cell measurement. In the case of the experiment shown, the value of σ CTM looks reasonably consistent across this trial, but these metrics need to be tracked and quantified across a period of time to interpret deviations with confidence. Note that a significant practical barrier exists to this in many instances, because cell and module processing is often done through independent entities, with often incomplete communication of data at the interface between the two.
5. Discussion Bucciarelli's work [1] remains the most well cited and generalisable treatment of mismatch loss when considering the properties of cells only. He first proposes the I–V curve be described in the region of maximum power by an expression of the form: " # I V ¼ α β exp c ð11Þ I mp V mp where I mp and V mp are described as nominal values for the maximum current and voltage respectively. By consideration of boundary conditions, expressions for α and β are both derived in terms of c, leaving a curve described by three parameters, I mp , V mp and c. This I–V curve is then assumed to be applicable out to Voc and Isc to derive a relationship between c and FF that is conveniently used for finding an appropriate value of c in many of the studies that cite Bucciarelli's work: FF ¼
Fig. 11. The absorption in a solar cell measured in air or encapsulated, as measured by the OPAL2 simulator [23], as a function of refractive index of the SiN (RI), thickness of the SiN (t) and the texturing properties ω [28]. The outer panel show the simulation data split by factors. The inner panel show the complete data clusters with a 95% confidence interval on the mean. The absorption data has a lower variance in the glass case.
c2 ð1 þ cÞðc þlnð1 þ cÞÞ
ð12Þ
An expression is then derived by Bucciarelli for the mismatch loss for N-series connected cells, reproduced as below equation using the notation of this paper: cþ2 1 1 σ 2iN L¼ ð13Þ 2 N
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Table 2 Calculations of CTMV of the module Imp properties from a small production experiment examining different sorting methods. Mismatch loss calculations are also included for these groupings. These are based on the average FF for each group and a derivation of A from the equivalent circuit model with extended series resistance in Fig. 10. Group
Number of modules
Pooled σ iN (%)
σðI Þ (%)
σðI mp Þ (%)
Imp σ CTM (%)
Average FF (%)
Average A
Mismatch loss L (%)
Control 1 Control 2 Control 3 Exp 1 Exp 2 Exp 3
200 336 938 699 541 1195
0.37 0.37 0.37 0.55 0.55 0.65
0.14 0.15 0.16 0.22 0.23 0.38
0.40 0.42 0.40 0.49 0.39 0.52
0.37 0.39 0.37 0.44 0.32 0.35
74.9 75.4 75.4 75.6 75.7 76.5
8.1 8.3 8.3 8.3 8.4 8.6
0.01 0.01 0.01 0.03 0.03 0.04
Relationship (13) is of the same form to the ones derived in this study (Eqs. (2) and (9)). The value of the constant ðc þ 2Þ=c 1 þ 1=N is comparable, but not exactly the same, as the constant A as seen in Figs. 8–10. In the Bucciarelli derivation, when Eq. (11) is assumed to be applicable over the whole I–V curve it has the effect of describing the curve with just three parameters, which is equivalent to using a one-diode model with any given (but constant) ideality factor, but no series or shunt resistance components. This is a limitation when modeling mc-Si cells in particular with varying ideality factors [29], series and shunt resistance components. As a result, the FF variation in the Bucciarelli relationship seen in Fig. 10 is happening entirely due to material changes, which is why the trend more closely matches the equivalent circuit model with extended J02. Kaushika and Rai [6] try to extend the Bucciarelli derivation by including resistance terms in the I–V curve equation, but they do not extend it to a modification of Bucciarelli's relationship between c and FF. The technique adopted here for deriving A from the P–I relationship will generate the same result as the Bucciarelli constant in the case of a one diode model with no series or shunt resistance, provided A is fitted within 7 1% of Imp. This in no way discredits or invalidates Bucciarelli's work, the assumptions made are still useful and meaningful in the context of the full algebraic derivation of mismatch loss that is undertaken, and the fixing of c using Eq. (12) is offered as only one of the options for modeling the curvature at maximum power. It is just such a convenient option that it is adopted without question by many of the works that follow. No attempt is made here to extend the analysis to include the effects of low light performance or overall energy yield. Low shunt resistance particularly can have a significant impact on overall module loss at low light and therefore on energy yields [30]. But some studies also look at the change in mismatch loss as a function of illumination level. One early study shows that mismatch loss is far higher at low illumination [3], but this will depend on the shape of the I–V and P–I curves. For the exposition followed here, the mismatch loss at low light could be assessed by studying and changes in A and σ2iN at low light. For the 3000 cells here, the shape of the P–I curve around maximum power is the same at 0.5 suns and 1 sun, but to study the illumination dependence in detail requires investigation at lower illumination levels than 0.5 suns. Others have suggested that improving the cell sorting to included the shunt resistance will improve the total energy yield by 0.12% [10] compared to the case where cells with different shunt resistance are inter-mixed in modules. But this conclusion is highly dependent on the range of shunting and indeed the range of cell performances in the first place – if there is not a large range, the advantage will diminish. The techniques developed here contribute to the body of evidence showing that mismatch loss will be very low for a modern photovoltaic module, even without any cell sorting taking place. This is an important realisation for module designers as it challenges the fundamental motivation for cell sorting in the first place. With mismatch loss no longer a significant issue, sorting can
be used to achieve a variety of aims related to optimising manufacturing systems and meeting targeted and desired product requirements. For high performance cells, as FF increases so too will the value of A and the mismatch loss. There are undoubtedly still situations where it is worthwhile to minimise mismatch loss. But these aims cannot be solely met by using tighter sorting requirements, they also need to be met by monitoring and controlling measurement and calibration issues and CTMV effects.
6. Conclusion A new and simple method has been demonstrated to express mismatch loss directly as a function of the variance of the cells in series interconnection. The relationship has been derived using the Power–Current (P–I) properties of a cell in the region of maximum power. These properties can be derived from measurements on a sample of relevant cells, or more conveniently from an equivalent circuit model. The technique of using the P–I properties is suited to a modern photovoltaic module where the variance in cell performance is very low. The technique could also be extended to the calculation of mismatch loss for module arrays, and that is left as an area of potential future work. In the case of classifying cells into modules, it has been shown that mismatch loss will remain below 0.1% even for the case where no cell sorting whatsoever is done. When even rudimentary cell sorting is used, loss should be well below 0.05% in all cases. The concept of cell-to-module variance (CTMV) was introduced to describe the changes in variance properties that happen as a result of encapsulation. To achieve the low levels of loss predicted by the P–I relationship, consideration needs to be paid to measurement accuracy and CTMV affects. The low loss allows for a re-evaluation of the motivation for cell sorting, and thus for potential improvements to the way manufacturing is organised. This could including something as simple as mixing wider ranges of cells to improve production yields. A more fundamental change would be the use of cell electrical measurement as a quality control, but not necessarily for sorting. Particularly this may be useful to module producers who want to avoid having to reclassify cells, and so also avoid the extra CapEx, handling and breakage associated with cell sorting. In this context, the optimal electrical measurement could be optimised for process control benefits and so it may also look quite different to the standard I–V data that is typically collected. Fundamentally, the overall manufacturing systems should be primarily organised to meet product requirements with a maximum of production efficiency, and results such as these will help to achieve that in a quantifiable and meaningful way.
Acknowledgements The author would like to thank Wuxi Suntech Power Co Ltd and Suntech R&D Australia for their support of this work. The final
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manuscript was also improved by useful comments from the reviewer. The Australian Center for Advanced Photovoltaics is supported by the Australian Government through the Australian Renewable Energy Agency (ARENA). Responsibility for the views, information or advice expressed herein is not accepted by the Australian Government. References [1] L.L. Bucciarelli, Power loss in photovoltaic arrays due to mismatch in cell characteristics, Solar Energy 23 (4) (1979) 277–288. [2] T.J. Lambarski, D.L. Kadlec, C.B. Rogers, Effects of cell sorting and module matching on array output, in: Conference Record of the IEEE Photovoltaic Specialists Conference, 1981, pp. 841–844. [3] R. Janoch, Analysis of solar cell mismatch losses at standard and non-standard conditions, in: Conference Record of the IEEE Photovoltaic Specialists Conference, 1984, pp. 558–562. [4] A.W.Y. Ho, S.R. Wenham, Intelligent strategies for minimising mismatch losses in photovoltaic modules and systems, in: 17th European Photovoltaic Solar Energy Conference, 2001, p. 406. [5] J.W. Bishop, Computer simulation of the effects of electrical mismatches in photovoltaic cell interconnection circuits, Solar Cells 25 (1) (1988) 73–89. [6] N.D. Kaushika, A.K. Rai, An investigation of mismatch losses in solar photovoltaic cell networks, Energy 32 (5) (2007) 755–759. [7] H. Saha, G. Bhattacharya, D. Mukherjee, Mismatch losses in series combinations of silicon solar cell modules, Solar Cells 25 (2) (1988) 143–153. [8] A.M. Gabor, H. Field, Optimization of binning schemes during Si-wafer based cell and module testing, in: 19th European Photovoltaic Solar Energy Conference, 2004, p. 2403. [9] H. Field, A.M. Gabor, Cell binning method analysis to minimize mismatch losses and performance variation in Si-based modules, in: Conference Record of the IEEE Photovoltaic Specialists Conference, 2002, pp. 418–421. [10] U.A. Yusufoglu, B. Min, T.M. Pletzer, J. van Moelken, H. Kurz, Simulation of solar modules with an extended approach to cell sorting considering optical and electrical parameters, in: 27th European Photovoltaic Solar Energy Conference and Exhibition, 2012, p. 3177. [11] U.A. Yusufoglu, B. Min, T.M. Pletzer, J. van Moelken, H. Kurz, Simulation and analysis of PV module performance by innovative sorting methods, Energy Procedia 27 (2012) 685. [12] M.C. Alonso-García, J.M. Ruiz, F. Chenlo, Experimental study of mismatch and shading effects in the I–V characteristic of a photovoltaic module, Solar Energy Mater. Solar Cells 90 (3) (2006) 329–340. [13] C.E. Chamberlin, P. Lehman, J. Zoellick, G. Pauletto, Effects of mismatch losses in photovoltaic arrays, Solar Energy 54 (3) (1995) 165–171.
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Contents lists available at ScienceDirect
Solar Energy Materials & Solar Cells journal homepage: www.elsevier.com/locate/solmat
Simplified technique for calculating mismatch loss in mass production Rhett Evans a,b,n, Kyung H. Kim c, Xiaoyi Wang e, Adeline Sugianto e, Xiaochun Chen d, Rulong Chen d, Martin A. Green a a
Australian Centre for Advanced Photovoltaics, University of New South Wales, Anzac Parade, Sydney, NSW 2052, Australia Solinno, Sydney Olympic Park, Sydney, NSW 2127, Australia c School of Photovoltaic and Renewable Energy Engineering, University of New South Wales, Anzac Parade, Sydney, NSW 2052, Australia d Wuxi Suntech Power Co Ltd, New District Wuxi, Jiangsu 214028, China e Formerly with Wuxi Suntech Power Co Ltd, China b
art ic l e i nf o
a b s t r a c t
Article history: Received 15 October 2014 Accepted 17 November 2014
This paper describes a generalised approach to estimating mismatch loss for series connected cells, utilising the result that the deviation from maximum power as a function of deviation from maximum power–point–current is a relatively stable relationship for a wide range of cell performances. This Power–Current relationship can be used to verify the strong link between mismatch loss and the variance in the maximum power–point–current values for the cells being mixed together. The mismatch loss in a modern photovoltaic module is low – below 0.1% even when there is no cell sorting whatsoever. Cell-to-module variance effects need to be understood and controlled to ensure mismatch loss remains low in a finished module. Without the constraints of mismatch loss in module design, other motivations for cell sorting should be considered, such as for optimising the manufacturing system or meeting particular product requirements. & 2014 Published by Elsevier B.V.
Keywords: Photovoltaics Solar cells Solar modules Mismatch Manufacturing Cell-to-module
1. Introduction As manufactured, all individual solar cells exhibit some slight differences in their performance. When these cells are series connected in a module and that module is operating at maximum power, the individual cells will typically output some power slightly below their maximum. The difference between the maximum power of a module and the sum of the maximum powers of the individual cells is known as mismatch loss. It is important to understand mismatch loss when optimising module manufacturing strategies. Mismatch loss is a well known effect and calculations of the mismatch loss resultant from the mixing of groups of cells has been a subject of study for over 30 years. The original work in this field by Bucciarelli [1] remains one of the most well cited, thorough and generalizable and it is further discussed throughout this work. Many of the papers [2–7] examine the mismatch loss resulting from different ways to bin and interconnect cells of different properties. The main limitations of these works are that it is difficult to generalise the results and there is no direction for dealing with encapsulation effects. Some of the works phrase the mismatch loss issue in the larger context of manufacturing yield
n
Corresponding author. E-mail address: [email protected] (R. Evans).
http://dx.doi.org/10.1016/j.solmat.2014.11.036 0927-0248/& 2014 Published by Elsevier B.V.
and total Watts produced [8,9]. Total mismatch loss is also dependent on solar insolation level [3] and so the issue has also been explored in terms of its influence on energy yield [10,11]. Many works investigate mismatch loss in the context of stressed or non-standard operating conditions such as partial cell shading ([6,12] to name a few) and many more studies ([13–15] to name a few) deal with the additional subject of mismatch loss at an array level. Both of these topics are important issues to which the results here are applicable, but neither is the immediate focus of this work. Mismatch loss is itself one component of a broader group of changes typically referred to as Cell to Module (CTM) effects. Usually expressed as a loss [16] or a conversion ratio [17], CTM calculations account for all the changes in average performance when cells are encapsulated into modules. For the purpose of this study, the term encapsulation refers to the group of processes by which cells are made into modules, including tabbing, stringing and lamination. Three studies make attempts to quantify mismatch loss at a final module level, rather than just as a consequence of cell mixing [18–20]. One [18] is particularly notable as the only study that attempts to measure mismatch loss at the final module level through direct experimentation. Unfortunately, the resolution of the technique described is insufficient to calculate loss for cells sorted in ways typical of modern manufacturing, and the technique is not directly generalisable to all module making strategies. This is not a weakness in the approach: rather it serves
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to highlight the important fact that mismatch loss is very low for a modern photovoltaic module, a fact that can also be verified by following the derivations of Bucciarelli from over 30 years ago. The second [19] builds a mathematical model for CTM effects that also includes mismatch, and the third [20] uses the expected difference between module I–V and SunsVoc curves to calculate mismatch loss. But the emphasis of both techniques is to deal with nonstandard conditions such as partial cell shading and so neither apparently has sufficient accuracy to detect the relatively small mismatch effect in standard modules. In this study, the new concept of CTM variance (CTMV) is also introduced and changes in variance associated with cell encapsulated are examined. This will highlight a new set of issues that need to be considered to keep mismatch loss to the low levels expected from modern production.
2. Overview of this study The aim of this study is to develop a simple and generalisable method for the calculation of mismatch loss and the monitoring of CTMV in a manufacturing context, so it can be used as an input in the design of optimal photovoltaic products. As a first step, the I–V curves of groups of production cells are summed to form a hypothetical module I–V curve and the maximum power (Pmp) and maximum current (Imp) calculated from this curve. The hypothetical curve includes no additional series resistance due to interconnection or changes in cell current due to encapsulation. Mismatch loss, L, is calculated from the difference between the sum of the maximum powers of the 72 component cells and the maximum power of the hypothetical module. This is the same technique as used in the Evergreen studies of more than 10 years ago [8,9]. A calculation of this form is referred to as Relative or Fractional Power Loss (RPL / FPL) in other studies [1,7,18]: L¼
∑72 i ¼ 1 pi P mp ∑72 i ¼ 1 pi
ð1Þ
where pi is the maximum power of the ith cell and Pmp is the maximum power of the module as previously mentioned. Lower case variables are used here to describe cell properties, upper case variables being used for module properties. L is calculated for four different types of cell sorting arrangements as shown in Table 1. This curve summing exercise is intended to give the most direct estimate of mismatch loss to serve as a validation of the new techniques based on the Power–Current (P–I) relationship. The simplification of using the P–I relationship is only possible when the range of cells being mixed remains very small and when the cells are series connected, as is the case for the majority of modern production. Overall, as is verifiable from the derivations in other studies [1,6,18], the typical loss for even rudimentary sorting schemes is shown to be extremely low for the variance typical of modern manufacturing. In this situation of a small variance, many of the established notions of mismatch loss – such as the module being limited to the lowest performing cell [19,21], simply do not apply: rather the module maximum power current will tend to just be close to an average of the component cell currents. Another established notion that
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mismatch loss is simply reduced by classifying cells into even smaller ranges of performance also becomes less relevant – in the context of low overall variance, process variance associated with cell encapsulation and the limitations of measurement accuracy will also have an impact.
2.1. Materials For the simulations in this study, I–V curves were collected from 3000 cells from a production order in 2012 from a Suntech manufacturing facility in Jiangsu, Wuxi, China. The cells are standard multi-crystalline silicon (mc-Si) with acidic texturing, belt diffusion, SiN ARC, full rear aluminium BSF and front silver screen printed contacts. The cells were measured on an inline cell flash tester at two illumination levels. The I–V curves were assembled from temperature and illumination corrected measurements on 200 points in forward bias and to assist with summing these were recalculated to 400–450 point curves with a consistent set of current coordinates using polynomial interpolation. The histogram of the cell's normalised efficiency is shown in Fig. 1. All of the cell data is normalised to generalise the results and protect data sensitivity, without interfering with the interpretability of the outcomes. The variance is scaled to be a percentage of mean, and then the data is mean centred. The full range of cell efficiencies is a little over 73% relative.
3. Results—calculation of losses from cell mixing 3.1. Loss calculated from summing I–V curves A random number function was used to select groups of 72 cells from the group of 3000 according to the simple sorting restrictions outlined in Table 1, and without regard to the original time order sequence in which the cells were made. The hypothetical module I–V characteristics were calculated and L was determined for each arrangement, as shown in Fig. 2. This shows the loss to be extremely low. Fig. 3 shows L as a function of key summary parameters within the group of cells. Of all these simple statistics, the strongest relationship is with the standard deviation of cell imp. This is shown in detail in Fig. 4. This relationship has been shown in another theoretical study [1] but not directly by curve summing [8,9]. The strength of this relationship is affected by the precise technique used for the curve summing – less points on the original I–V curve and linear interpolation between the points can add significant levels of noise. Fig. 4 can be fit with a quadratic passing through, and with its minimum at the origin. The line of best fit has equation: L ¼ 9:23 σ 2iN
ð2Þ
Table 1 The selection restriction for the four different cell grouping arrangements. Group
Selection restriction
A B C D
Random selection Efficiency range o 0.2% abs. imp range o 0:15 A imp range o 0:05 A
Fig. 1. Histogram of normalise cell efficiency for the 3000 cells used in this study.
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be using too few points on the raw P–I curve and any more and the quadratic fit would deteriorate due to the natural asymmetrical nature of the P–I curve. The implications of this are discussed later. The approximating quadratic in Fig. 5b is of the form:
Δpmp ¼ AΔi2mp
Fig. 2. Calculation mismatch loss L for the 4 different sorting arrangements, A through D, as specified in Table 1. The loss is extremely low, less than 0.1% even for unsorted cells and less then 0.04% for rudimentary sorting.
ð3Þ
where Δpmp and Δimp are respectively the proportional deviations in power and current from their respective maximums, and A is a constant that defines the quadratic. In Fig. 5b, the value of A is estimated to be 9.48 from a least squares fit to all of the P–I data over the 7 2% range. The P–I relationship can be used to calculate loss when the deviation between operating current and cell imp is known using Eq. (3). For the kth cell in a series connection of n cells, let the maximum power point current, voltage and power of that cell be denoted by ik, vk and pk respectively. The data from the 3000 cells used in this study showed that the maximum power point current of the theoretical module formed by the series connection of
Fig. 3. Mismatch loss L as a function of the key summary parameters for the groups of cells that make each module. The discontinuities on some of the graphs result from distinctions between the different sorting groups.
Fig. 4. L as a function of the normalised standard deviation in cell imp, of the average . Fig. 2 more clearly shows for a clearer representation of the difference between the sorting groups.
Fig. 5. Two versions of the Power–Current (P–I) relationship (a) sample curve for a 17.5% cell; (b) the P–I relationship for all 3000 cells normalised to a percentage deviation from their respective maximum powers and maximum power point current. (b) also includes a quadratic fit to the data.
where σ iN is the standard deviation of the imp values of the cells, normalised to the average of the cells. 3.2. Loss calculated from the measured P–I curve of all cells Loss calculations for series interconnections can be made using the P–I characteristic of a cell. Fig. 5 shows the P–I characteristic taken from the original raw I–V curves used in this study. Fig. 5a is an example P–I characteristic for one of the cells from the study, Fig. 5b shows a normalisation of this data in terms of the percentage deviation of current and power from the respective maximums, plotted together for all of the 3000 cells in the study. Fig. 5b also includes the quadratic fit forced to pass through and have its maximum at the origin. The P–I relationship is very consistent even for the large range of cells included in this study. The quadratic approximation by taking the normalised P–I data to 7 2% of cell imp; any less would
Fig. 6. Hypothetical Imp for the module (derived from summing I–V curves) vs I , the average imp for the group cells. The value of I (A) is an excellent estimate to the Hypothetical Imp. All the data is normalised to their respective means as outlined in Section 2.1.
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72 cells (Imp) is very closely approximated by the average current of all the cells in series (see normalised data in Fig. 6), with this latter quantity referred to as I . The “mismatch” loss for an individual cell (lk) is therefore " #2 I ik l k ¼ Δp k ¼ A ð4Þ ik The mismatch loss for the module with n cells in series, can now be calculated using the expression: " #2 I ik n 2 A∑k ¼ 1 ik vk ik Avi k ∑nk ¼ 1 I ik ∑n p lk L ¼ kn¼ 1 k ¼ ¼ k ð5Þ ∑k ¼ 1 pk nIV nIV I and V are the average of the cell's maximum power point current and voltage respectively, hence ½1=n∑nk ¼ 1 pk ¼ IV and so the simplification in the denominator follows in the previous equation. The cell data (from Fig. 1) can be used to show that the quantity vk =ik can be assumed to be a constant (see Fig. 7), because the standard deviation is a low percentage of the mean (less that 1%). Furthermore, if vk =ik is a constant, then vk =ik ¼ V =I is the same constant. Eq. (5) reduces to 2 n 2 1 1 ∑ I ik ð6Þ L¼A n I k¼1 The variance in imp for a group of cells (σ2i ) can be represented by an expression of the form:
σ 2i ¼
2 1 n ∑ I ik nk¼1
ð7Þ
The variance normalised to the average current I becomes an expression of the form: 2 n 2 1 1 σ 2iN ¼ ∑ I ik ð8Þ n I k¼1
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electrical properties of the cell with any reasonable correlation to A is the FF, even when checking for relationships as part of a multi-variate regression. But as can be seen in Fig. 8, the relationship is very noisy, with the FF accounting for only about 25% of the variance in A. It is also different to the relationship derived in Bucciarelli's study [1]. To understand the origin of the variance in A, the 3000 I–V curves were curve fitted to ameliorate the impact of measurement uncertainty in the I–V curve. The equivalent circuit for the curve fitting used two diodes (n ¼1 and n ¼2) in parallel with a resistively limited enhanced recombination term, along with an Rsh and Rs term [22]. Fitting was done to the I–V and m–V curves simultaneously, where m is the value of the local ideality factor. A model such as this does not have a unique solution for any given I–V curve, nor is a justification made here for using such a model in terms of the actual physical recombination mechanisms in the device. But the model allows for a very accurate fit of nearly any I–V curve, and so it serves the purpose in this instance of having a smooth curve unaffected by measurement error. The value of A can now be calculated using the fitted curve and these are shown in Fig. 9. One relationship is derived by fitting A to 7 2% of imp in the P–I relationship as was done for the raw curves, and another by fitting to 7 1%. Ideally, A is fitted by using the smaller range, but for the raw curves the larger range was used due to the data collection interval. Compared to the values of A fitted directly from the curve (Fig. 8), nearly 95% of the variance is removed. A strong relationship can now be seen between A and FF, and more importantly, this can be derived from the relatively simple task of fitting to an equivalent circuit model. The average value of A when fitting to the lower range of imp is 9.28, which is very close to the proportionality constant from Eq. (2).
Combining Eqs. (6) and (8) yields the simplified expression: L ¼ Aσ 2iN
ð9Þ
The value of A ¼9.48 from Fig. 5b, means the relationship of Eq. (9) is nearly identical to the one of Eq. (2) derived from the empirical study. 3.3. Curve fitting In a production context, calculating the value of A from a quadratic fit to the normalised P–I properties of a large range of cells is as unworkable as summing I–V curves directly. In Bucciarelli's work [1], the value of the proportionality constant is estimated from the electrical properties of the cells, and this convenience is adopted by nearly every study which cites the work. In this study, the only
Fig. 8. Constant A derived from the P–I properties for the 3000 cells in this study, plotted as a function of the normalised FF. The solid line shows the same relationship as derived from [1]. The Bucciarelli data has the same normalisation to make it directly comparable.
Fig. 7. Histogram of the values of vmp =imp as calculated from the I–V curves of the 3000 cells used in this study.
Fig. 9. A, derived from a fitted curve, versus normalised FF. Bucciarelli's relationship is included with the same FF normalisation. Scale is the same as Fig. 8.
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4. Results—cell to module effects The expression of mismatch loss as a function of σ2iN, derived so far will still hold when the cells are encapsulated. But the proportionality constant will change due to changes in FF from the added Rs from tabbing. Encapsulation can also change the value of σ2iN for the group of cells. 4.1. Series resistance effects Fig. 9 and to a lesser extent Fig. 8 showed the relationship between FF and A. When cells are encapsulated, the series resistance and hence the FF will change and so a new value of A is required. The relationship identified in Fig. 9 cannot be immediately used as it would involve an extrapolation for the FF values expected post encapsulation. One option for extending the range of this relationship is to use direct measurements of tabbed cells to determine A directly from the P–I curves as was done in Section 3.2. To this end, a group of ten mcSi cells comparable to the original 3000 and of varying efficiency, were measured with and without tabbing. The cells were tabbed with 3 busbars on each side, with the tabs joined together by a highly conductive cross tab about 10 mm outside the active area of the cell. Already, this will return a lower FF than for a standard tabbed and laminated cell as this 10 mm acts like the cell gap in the module case and is a non-trivial source of series resistance. These tabbed cells were loaded onto a cell tester and contacted with a spring-loaded probe card as is standard on in-line production cell testers. The cell tester uses a SAN-EI 3A Solar Simulator with a class B light source, with a detailed I–V curve swept at a resolution of 3-4 points/mV. In this case the probes are contacted to the tabbed cells and not directly to the screen printed grid. With the probe card engaged, the cells were measured directly through the probe card to get the “untabbed” measurement, and then by connecting to the tabbing leads to get the “tabbed” measurements. Not all the differences between untabbed and tabbed cells are captured with this arrangement – most particularly if the tab is only spot soldered at a select number of points, or there are otherwise grid to tab contact resistance issues, this technique will be an underestimation of the difference. However, it is a reasonable approach to capture the most significant differences and establish the wider ranging A vs FF relationship. Another option for estimating the change in A due to encapsulation is to use well established theoretical models. The grid calculator hosted by PVlighthouse [23] shows that three busbars (1.5 mm wide 250 μm high, with 2 μΩ cm resistivity and a 3 mm cell spacing) on a standard 156 mm cell adds an effective 0.7 Ω cm2 of series resistance to each cell. If this series resistance is added to the equivalent circuits calculated in Section 3.3, the new curves can be used to derive an extended version of the A vs FF relationship. The risk with this approach is that no attempt was originally made to validate the physical relevance of the fitted curves. But given the aim is simply to generate I–V curves with an extend FF range, the approach remains meritorious. As well as adding series resistance, the FF range can also be extended for the fitted curve case by extending the range of cells to include higher n ¼2 dark saturation current values (J02). This is not necessarily relevant to the situation of encapsulation, but it does show that the A vs FF relationship differs depending on the reason for the FF variation. Bucciarelli [1] also derives a version of this constant over a wide range of FF. All of these estimates are compared in Fig. 10. The trend in A with respect to FF is very similar between the tabbed cell case and the equivalent circuit case with extended Rs. As the tabbed cells are only adding series resistance, this is not too surprising. The equivalent circuit with extended resistance and J02 more closely matches the Bucciarelli relationship but shifted up. The reason for these discrepancies is discussed more in Section 5.
Fig. 10. Four relationships between A and FF over an extended range. The equivalent circuit model is an n¼ 1 and n¼ 2 diode in parallel with a resistively limited enhanced recombination with shunt and series resistance [22]. Both of the equivalent circuit relationships were derived from a sample of the fitted I–V curves in this study (see Section 3.3). The FF range was extended by extending the Rs range or the J02 range in the model, giving two slightly different relationships.
The most convenient method for calculating A for the encapsulated case is to use an equivalent circuit model with added series resistance from the tabbing. A value can also be derived from direct measurement, but due to noise in the shape of the curve at least ten different curves should be measured. 4.2. Changes in
σ2iN
To understand how σ2iN will change for the group of cells once they are encapsulated, it is necessary to consider all the sources of cell imp variance and then how these sources of variance change during encapsulation. As a framework for discussion, the concept of Cell-To-Module Variance (CTMV) is introduced here. For any given parameter of interest, the CTMV (σ2CTM) can be expressed as
σ 2CTM ¼ σ 2 ðmodulesÞ σ 2 ðcellsÞ
ð10Þ
This is important as a concept as well as a quantitative expression, because it is not always possible to calculate this for every parameter post en capsulation. Encapsulation adds variance to the cell performance and this should be understood and tracked to keep issues such as mismatch loss under control. There are many sources of potential variance in this part of the manufacturing sequence and it is beyond the scope of this study to develop experimental estimates for all of them. But a mixture of simulations and previously published data provide information and trends to understand what is important. A sample of production data is also used to show how these issues can be monitored in a simple and agglomerated way. For the 3000 cells studied here, σ iN for the full set of cells is 0.73%, and as high as 1% for some 72 cell groupings (see Fig. 2). The primary sources of variance are any of the manufacturing processes that affects the cell imp directly (such as material quality, texturing, diffusion depth, SiN, screen print area or size of white perimeter around the encapsulated cells) or indirectly through the shunt resistance (primarily a result of material quality issues or process faults). Changes in cell series resistance have nearly no impact on cell imp, but affect mismatch loss by changing the proportionality constant between the loss and σ2iN. This is dealt with separately in the following section. Variance related to material quality and screen print area will not change during encapsulation. Variance associated with diffusion profiles may theoretically change due to absorption in the encapsulation materials, but the overall effect is very small [24] and so will be ignored here. Shunt resistance should not change in going from cells to modules [25] if the process is under control, so it will also be ignored. However, this assumption may not always be true – at least one other study has assumed that the shunt resistance will change [19]. Equivalent circuit models such as those used in the curve fitting exercise (Section 3.3) can be used to show a 10%
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variance in Rsh in encapsulation can increase the cell imp variance by 10%, so if the assumption of unchanged shunt resistance is incorrect, there is a potential for error in the mismatch loss estimate. Another possible source of variation is the size of the border around the encapsulated cells, when a white backsheet is used. Other studies [26] show that at a cell spacing of around 2 mm (a typical production standard), the gain in cell isc (and hence cell imp) is around 0.8% per mm of spacing. As an indication of how this could affect the variance, if the normal 2 mm cell spacing was always accurate to 7 10% (i.e 73σ ), the resultant standard deviation in the imp gain would be about 0.05%. This would amount to a very small addition to the overall cell imp variance, but it could possibly be significant in some situation where the cell imp groupings are very tight. The final component to consider is the optical properties of the cell. These will change when cells are encapsulated due to variance in the thickness and refractive index of the SiN and the surface features of the texturing. The reflection is less sensitive to the precise SiN properties when the refractive index of the first material in the light path is 1.5 as for the encapsulated case, compared to 1 as for the air case. For texturing the variance is reduced because some component of the light that is reflected when the cell is measured in air will be totally internally reflected in the encapsulated case [27]. A first principles proof of this is nontrivial as the optics of a textured three layer stack (Glass/SiN/Si) is somewhat complex, but the OPAL2 simulator hosted by PVlighthouse [23] is designed to make exactly these calculations. Using the spherical caps texturing model developed for acidic mc-Si texturing [28], the absorption in the film can be calculated for a range of SiN and texturing properties. This is represented in Fig. 11, which shows that the variance in absorption (and hence imp) reduces in the case of encapsulation. The magnitude of the reduction is not representative of production in this case because the demonstration data is chosen for simplicity to represents the extremes of possible production values, rather than being a realistic cell population. The final situation in which σ2iN may increase in encapsulation, somewhat paradoxically, is when the initial cell binning is based on cell imp and the bin width is narrow. If cells are grouped into a narrow-range imp bin without any optical sorting, the bins will contain cells whose imp value is realised through different permutations of material and optical properties and so the variance within this bin, which is very small by design at the cell sorting step, must increase following encapsulation. In practice, the mismatch loss is likely to be so low in this case that the error will be unimportant. In summary, if the cell spacing is well controlled and there is no introduction of shunting in encapsulation, the variance should
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only decrease by a small amount in encapsulation, meaning the estimate made from the cell σ2iN would represent an upper limit to the mismatch loss. If, further to this, optical variance makes up a very small proportion of overall cell variance, as is the case for well controlled PECVD processing and consistent texturing (such as when using c-Si substrates) or if optical sorting metrics are used, then there should be almost no change in cell variance during encapsulation. A way to check that nothing unexpected is happening is to track the CTMV in production. 4.2.1. Example CTMV and mismatch loss data from production When considering an entire population of cells, different cell sorting methods will give a different value for σ iN for the group of cells. The average of the cell imp values for each group will be I. The variance in I, referred to as σ 2 ðIÞ, will add to the pooled variance within the group of cells, σ2iN, to give the total variance of the cell population. Therefore, the choice of cell sorting methodology will impact on how much of the total production variance occurs within a cell packet, and how much occurs across the average performance of the cell packets. In the case of the maximum power point current, the value of σ 2 ðIÞ will be related to the variance of the module Imp values, σ 2 ðI mp Þ, made from this group of cell packets by the CTMV expression of Eq. (10). Table 2 contains values for these quantities from a small production experiment on different cell sorting techniques. If all is going well with production, the effect of encapsulation material changes is accounted for, and no significant variance is added to the cell properties during encapsulation, the value of σ CTM should be consistent and largely representative of measurement variance. This variance will be related intrinsically to the module measurement but also to the differential error with the cell measurement. In the case of the experiment shown, the value of σ CTM looks reasonably consistent across this trial, but these metrics need to be tracked and quantified across a period of time to interpret deviations with confidence. Note that a significant practical barrier exists to this in many instances, because cell and module processing is often done through independent entities, with often incomplete communication of data at the interface between the two.
5. Discussion Bucciarelli's work [1] remains the most well cited and generalisable treatment of mismatch loss when considering the properties of cells only. He first proposes the I–V curve be described in the region of maximum power by an expression of the form: " # I V ¼ α β exp c ð11Þ I mp V mp where I mp and V mp are described as nominal values for the maximum current and voltage respectively. By consideration of boundary conditions, expressions for α and β are both derived in terms of c, leaving a curve described by three parameters, I mp , V mp and c. This I–V curve is then assumed to be applicable out to Voc and Isc to derive a relationship between c and FF that is conveniently used for finding an appropriate value of c in many of the studies that cite Bucciarelli's work: FF ¼
Fig. 11. The absorption in a solar cell measured in air or encapsulated, as measured by the OPAL2 simulator [23], as a function of refractive index of the SiN (RI), thickness of the SiN (t) and the texturing properties ω [28]. The outer panel show the simulation data split by factors. The inner panel show the complete data clusters with a 95% confidence interval on the mean. The absorption data has a lower variance in the glass case.
c2 ð1 þ cÞðc þlnð1 þ cÞÞ
ð12Þ
An expression is then derived by Bucciarelli for the mismatch loss for N-series connected cells, reproduced as below equation using the notation of this paper: cþ2 1 1 σ 2iN L¼ ð13Þ 2 N
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Table 2 Calculations of CTMV of the module Imp properties from a small production experiment examining different sorting methods. Mismatch loss calculations are also included for these groupings. These are based on the average FF for each group and a derivation of A from the equivalent circuit model with extended series resistance in Fig. 10. Group
Number of modules
Pooled σ iN (%)
σðI Þ (%)
σðI mp Þ (%)
Imp σ CTM (%)
Average FF (%)
Average A
Mismatch loss L (%)
Control 1 Control 2 Control 3 Exp 1 Exp 2 Exp 3
200 336 938 699 541 1195
0.37 0.37 0.37 0.55 0.55 0.65
0.14 0.15 0.16 0.22 0.23 0.38
0.40 0.42 0.40 0.49 0.39 0.52
0.37 0.39 0.37 0.44 0.32 0.35
74.9 75.4 75.4 75.6 75.7 76.5
8.1 8.3 8.3 8.3 8.4 8.6
0.01 0.01 0.01 0.03 0.03 0.04
Relationship (13) is of the same form to the ones derived in this study (Eqs. (2) and (9)). The value of the constant ðc þ 2Þ=c 1 þ 1=N is comparable, but not exactly the same, as the constant A as seen in Figs. 8–10. In the Bucciarelli derivation, when Eq. (11) is assumed to be applicable over the whole I–V curve it has the effect of describing the curve with just three parameters, which is equivalent to using a one-diode model with any given (but constant) ideality factor, but no series or shunt resistance components. This is a limitation when modeling mc-Si cells in particular with varying ideality factors [29], series and shunt resistance components. As a result, the FF variation in the Bucciarelli relationship seen in Fig. 10 is happening entirely due to material changes, which is why the trend more closely matches the equivalent circuit model with extended J02. Kaushika and Rai [6] try to extend the Bucciarelli derivation by including resistance terms in the I–V curve equation, but they do not extend it to a modification of Bucciarelli's relationship between c and FF. The technique adopted here for deriving A from the P–I relationship will generate the same result as the Bucciarelli constant in the case of a one diode model with no series or shunt resistance, provided A is fitted within 7 1% of Imp. This in no way discredits or invalidates Bucciarelli's work, the assumptions made are still useful and meaningful in the context of the full algebraic derivation of mismatch loss that is undertaken, and the fixing of c using Eq. (12) is offered as only one of the options for modeling the curvature at maximum power. It is just such a convenient option that it is adopted without question by many of the works that follow. No attempt is made here to extend the analysis to include the effects of low light performance or overall energy yield. Low shunt resistance particularly can have a significant impact on overall module loss at low light and therefore on energy yields [30]. But some studies also look at the change in mismatch loss as a function of illumination level. One early study shows that mismatch loss is far higher at low illumination [3], but this will depend on the shape of the I–V and P–I curves. For the exposition followed here, the mismatch loss at low light could be assessed by studying and changes in A and σ2iN at low light. For the 3000 cells here, the shape of the P–I curve around maximum power is the same at 0.5 suns and 1 sun, but to study the illumination dependence in detail requires investigation at lower illumination levels than 0.5 suns. Others have suggested that improving the cell sorting to included the shunt resistance will improve the total energy yield by 0.12% [10] compared to the case where cells with different shunt resistance are inter-mixed in modules. But this conclusion is highly dependent on the range of shunting and indeed the range of cell performances in the first place – if there is not a large range, the advantage will diminish. The techniques developed here contribute to the body of evidence showing that mismatch loss will be very low for a modern photovoltaic module, even without any cell sorting taking place. This is an important realisation for module designers as it challenges the fundamental motivation for cell sorting in the first place. With mismatch loss no longer a significant issue, sorting can
be used to achieve a variety of aims related to optimising manufacturing systems and meeting targeted and desired product requirements. For high performance cells, as FF increases so too will the value of A and the mismatch loss. There are undoubtedly still situations where it is worthwhile to minimise mismatch loss. But these aims cannot be solely met by using tighter sorting requirements, they also need to be met by monitoring and controlling measurement and calibration issues and CTMV effects.
6. Conclusion A new and simple method has been demonstrated to express mismatch loss directly as a function of the variance of the cells in series interconnection. The relationship has been derived using the Power–Current (P–I) properties of a cell in the region of maximum power. These properties can be derived from measurements on a sample of relevant cells, or more conveniently from an equivalent circuit model. The technique of using the P–I properties is suited to a modern photovoltaic module where the variance in cell performance is very low. The technique could also be extended to the calculation of mismatch loss for module arrays, and that is left as an area of potential future work. In the case of classifying cells into modules, it has been shown that mismatch loss will remain below 0.1% even for the case where no cell sorting whatsoever is done. When even rudimentary cell sorting is used, loss should be well below 0.05% in all cases. The concept of cell-to-module variance (CTMV) was introduced to describe the changes in variance properties that happen as a result of encapsulation. To achieve the low levels of loss predicted by the P–I relationship, consideration needs to be paid to measurement accuracy and CTMV affects. The low loss allows for a re-evaluation of the motivation for cell sorting, and thus for potential improvements to the way manufacturing is organised. This could including something as simple as mixing wider ranges of cells to improve production yields. A more fundamental change would be the use of cell electrical measurement as a quality control, but not necessarily for sorting. Particularly this may be useful to module producers who want to avoid having to reclassify cells, and so also avoid the extra CapEx, handling and breakage associated with cell sorting. In this context, the optimal electrical measurement could be optimised for process control benefits and so it may also look quite different to the standard I–V data that is typically collected. Fundamentally, the overall manufacturing systems should be primarily organised to meet product requirements with a maximum of production efficiency, and results such as these will help to achieve that in a quantifiable and meaningful way.
Acknowledgements The author would like to thank Wuxi Suntech Power Co Ltd and Suntech R&D Australia for their support of this work. The final
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