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

Abstract

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 𝐼_mp resulted in the most consistent MML reductions. Sorting by 𝑉_mp yielded insignificant results. Sorting by 𝑃_mp 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 both manufacturers had already done would make 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.

1. 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 [1–5] 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.

2. 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] 𝐼^′=𝐼_ph−𝐼_𝑜𝑒^{[𝐸(𝑉′+𝐼′𝑅𝑠)/𝑘𝐴𝑇𝑐]}−𝑉−1^′+𝐼^′𝑅_𝑠𝑅_sh.(1) Dividing (1) by the average max-power current 𝐼_mp, introducing the average max-power voltage 𝑉_mp as a unity fraction, and setting 𝑉=𝑉^′+𝐼^′𝑅_𝑠,(2)𝐼=𝐼^′+𝑉𝑅_sh,𝐼(3)𝐼_mp=𝐼_ph+𝐼_𝑜𝐼_mp−𝐼_𝑜𝐼_mp𝑒^{[(𝐸𝑉/𝑘𝐴𝑇𝑐)∗(𝑉mp/𝑉mp)]}.(4) Substituting (4a), (4b), and (4c) into (4) results in (5): 𝛼=𝐼_ph+𝐼_ph𝐼_mp,()𝐼4a𝛽=^𝑜𝐼_mp,()𝑒4b𝐶=𝑉_mp𝑘𝐴𝑇_𝑐,()𝐼4c𝐼_mp=𝛼−𝛽𝑒^{𝐶(𝑉/𝑉mp)}.(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 max-power point [1, 2, 5]. 𝐶 is commonly referred to as the cell characteristic factor and can be expressed in terms of the fill factor (FF) defined as [1] 𝐼FF=^mp𝑉_mp𝐼_sc𝑉_oc=𝐶²((.1+𝐶)[𝐶+ln1+𝐶)](6) The relationship between the fill factor and 𝐶 can be seen in Figure 1.

Figure 1 

Cell characteristic factor versus fill factor.

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 𝐶^′, 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): 𝐶FF=^′21+𝐶^′𝐶^′+ln1+𝐶^′.(7)

2.2. Power Loss due to Mismatch

From Bucciarelli [1], 𝑇 total PV modules arranged into a network of 𝑀 parallel series strings of 𝐿 modules; each has a mismatch loss Δ𝑃 estimated by 𝐶Δ𝑃=^′+22𝜎²_𝜂11−𝐿+𝜎²_𝜉𝐿11−𝑀,(8) where 𝜎²_𝜂 and 𝜎²_𝜉 represent the variance of max-power current and voltage, respectively, in our network of PV cells. Figure 2 shows a typical 𝐿×𝑀 network of modules: 𝜎²_𝜂=𝜎_𝐼mp𝐼_mp²,𝜎²_𝜉=𝜎_𝑉mp𝑉_mp².(9) Expressions (9) are derived on the assumption that variations in max-power current and max-power voltage in the 𝑠th PV cell in our network take the nondimensional form [1]: 𝐼_mp,𝑠𝐼_mp=1+𝜖_𝜂𝜂_𝑠,𝜖_𝜂=𝐼_mp,MAX−𝐼_mp,MIN,𝑉_mp,𝑠𝑉_mp=1+𝜖_𝜉𝜉_𝑠,𝜖_𝜉=𝑉_mp,MAX−𝑉_mp,MIN.(10)𝜂_𝑠 and 𝜉_𝑠 are respective measurements of variations in max-power current and max-power voltage and take on values between −1 and 1. 𝜖_𝜂 and 𝜖_𝜉 are small numbers, respectively, measuring the percentage range of variation in 𝐼_mp and 𝑉_mp. In order for (8) to yield acceptably accurate results, the product of 𝜖 and 𝐶^′, known as the weighted variation of max-power 𝜖𝐶^′, 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 𝐼_mp and 𝑉_mp is small with respect to ±1 [1].

Figure 2 

Network of 𝐿 modules in series and 𝑀 modules in parallel.

In prior MML studies, the term 𝜎_𝜂 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 𝜎_𝜂, 𝐶^′, 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 𝜔 based upon the ideal power output 𝑃_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 𝑀 strings, the series MML term in (8) becomes Δ𝑃_series=1𝑀^𝑀_𝑞=1𝜔_𝑞Δ𝑃_𝑞,Δ𝑃_𝑞=𝐶^′_𝑞+22𝜎²_𝜂,𝑞11−𝐿,𝜔_𝑞=𝑃_ideal,𝑞𝑃_ideal,(11) where 𝑞 refers to the string in question. The final, modified MML equation is found to be 1Δ𝑃=1−𝐿1𝑀^𝑀_𝑞=1𝜔_𝑞𝐶^′_𝑞+22𝜎²_𝜂,𝑞+𝐶^′+22𝜎²_𝜉𝐿11−𝑀.(12)

3. Sorting Procedure and PV Array Simulation

Most PV manufacturers sort their modules by power output at 1,000 W/m² 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: 𝑃_mp,𝐼_mp, and 𝑉_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 𝜎²_𝜂 and maximum power voltage 𝜎²_𝜉 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 (𝐿 in series, 𝑀 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).

3.1. Model Verification

This model was validated by reevaluating a problem treated by Bucciarelli [1] for estimating MML in a single series string of PV cells with no 𝑉_mp variation and an 𝐼_mp variation defined by a Gaussian probability density function. Bucciarelli [1] assigns the following values: 𝐼_mp = 39.2 mA,  𝜎_𝐼mp = 2.86 mA, FF = 0.67.

The 1/𝐿 term is dropped with respect to 1, and (8) results in Δ𝑃=2.35%.

Starting with the same parameters, a Gaussian probability density function was used to generate 𝐼_mp’s for a distribution of 204 PV cells with 𝐼_mp and 𝜎_𝐼mp approximately equal to those listed above. The PV module 𝐼_mp distribution can be 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/𝐿 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%.

Figure 3 

𝐼mp 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].

Figure 4 

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

4. Data

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 Figures 5(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).

(a)

(b)

(c)

Figure 5 

𝑉mp,𝐼mp, and 𝑃mp distributions of the anonymous module manufacturer’s 285 Wp module.

(a)

(b)

(c)

Figure 6 

𝑉mp,𝐼mp, and 𝑃mp distributions of the Helios Solar Works 250 Wp module.

As stated in Section 2, it is a requirement that the weighted variation of max-power 𝜖𝐶^′ is less than 1 and the cross-correlation 𝜚 for 𝐼_mp and 𝑉_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.

Table 1 

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.

5. 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 max-power parameters: 𝑃_mp, 𝐼_mp, and 𝑉_mp. The effect of sorting upon average MML, represented in terms of a reduction in the standard deviations of each respective max-power parameter distribution, can be seen in Figures 7, 8(a), and 8(b).

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 𝐼mp, as represented by the reduction in 𝐼mp standard deviations displayed above.

(a)

(b)

Figure 8 

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.

It is immediately clear that sorting by 𝐼_mp has the most consistent, immediate, and greatest reduction on MML. Each module works at its own 𝐼_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 𝐼_mp restriction, a greater net reduction in MML is observed.

Compared to 𝐼_mp, sorting modules by 𝑉_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 𝑉_mp sorting. This phenomenon can be inferred by the MML estimate (8) in that parallel MML is inversely proportional to string length (𝐿). 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 𝐼_mp over 𝑉_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 𝑃_mp yielded the most unpredictable impact on MML reduction. This is not surprising given that a PV module’s 𝑃_mp is the product of both 𝑉_mp and 𝐼_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 𝐼_mp (such as 𝑃_mp) is not going to result in consistent, reliable MML reduction.

Both module populations were subjected to some degree of sorting with respect to 𝑃_mp prior to use by this study and reflect each manufacturer’s 𝑃_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.

6. 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 𝜈 $/year. A reduction in MML will result in a yearly net cash flow increase given by 𝐵_𝑗=𝜈ΔMML(1+EER)^𝑗,𝜈=𝑥𝑦𝑧,(13) where ΔMML = reduction in MML due to sorting (%), 𝑥= array power output per kWp (kWh/kWp), 𝑦= array capacity (kWp), 𝑧= energy cost set by PPA ($/kWh), EER = Energy Escalation Rate set by PPA (%), 𝑗= year.

The net present value NPV of this series of yearly cash flows 𝐵_𝑗 with a discount rate 𝑑 over an array lifetime of 𝑛 years is given by NPV=^𝑛_𝑗=1𝐵_𝑗(1+𝑑)^𝑗,𝑑=𝑟−𝑖,1+𝑖(14) 𝑟= opportunity cost of capital (%), 𝑖= inflation rate (%),

resulting in the expression NPV=𝜈ΔMML^𝑛_𝑗=11+EER1+𝑑^𝑗.(15) The cost 𝐶_𝑜 to the owner of the PV array to presort the modules is given by 𝐶_𝑜=𝐶_𝑠(1+𝛿),(16) where 𝐶_𝑠 is the manufacturer’s cost of sorting and 𝛿 is their desired profit margin. In order to yield a net economic gain, NPV > 𝐶_𝑜, resulting in an expression for the minimum required decrease in MML from sorting: 𝐶ΔMML>^𝑠(1+𝛿)𝜈^𝑛_𝑗=11+EER1+𝑑^𝑗−1.(17)

6.1. Example

We take as an example the Helios 250 Wp module assembled into the 400 kWp PV array simulated previously place it in a region of high irradiance such as the American South-West, and assume a specific yield of 2,000 kWh/kWp. For simplicity we assume an array lifetime and PPA of 20 years each. Using recent data for $/kWh, EER, and discount rates [9–12] and assuming a modest profit margin of 20%, the following parameters take the values 𝑥=2,000 kWh/kWp, 𝑦=400 kWp, 𝑧= $0.16/kWh,  𝑛=20 years, EER = 2.4%, 𝑟=2.72%, 𝑖=2.42%, 𝛿=20%.

Substituting into (17) yields 𝐶_𝑠$2,673,000<ΔMML.(18) 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 𝐼_mp and reducing the 𝐼_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 𝐼_mp will yield the most consistent and reliable reduction in MML.

(a)

(b)

Figure 9 

Allowed manufacturer sorting cost for a given MML reduction resulting from module sorting to be considered a good economic decision. Labelled data points correspond to the strictest sorting criteria for each parameter.

7. 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. Bucciarelli’s [1] equation for estimating MML in a PV networks was modified to take into account individual module placement within the network. Two different PV modules (250 Wp and 285 Wp) were used to populate a realistically dimensioned, large-scale PV array (700 V., 400 kWp) [8].

The analysis indicated that sorting PV modules by max-power current 𝐼_mp yielded the most consistent and greatest overall reduction in MML when compared to sorting by 𝑃_mp, which yielded inconsistent MML reduction, and 𝑉_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.

Nomenclature

𝐼ph:	Light generated current
𝐼0:	Diode saturation current
𝐸:	Electron charge
𝑉′:	Cell voltage
𝐼′:	Cell current
𝑅𝑠:	Cell series resistance
𝑅sh:	Cell shunt resistance
𝐴:	Diode ideality factor
𝑘:	Boltzmann’s constant
𝑇𝑐:	Cell temperature
Δ𝑃:	Fractional power loss due to electrical mismatch
ΔMML:	Reduction in mismatch loss
𝐶:	Cell characteristic factor
𝐶′:	Modified cell characteristic factor
FF:	Fill factor
𝑃mp:	Module max-power output
𝑉mp:	Module max-power voltage
𝑉mp:	Average module max-power voltage
𝐼mp:	Module max-power current
𝐼mp:	Average module max-power current
𝑉oc:	Module open circuit voltage
𝐼sc:	Module short circuit voltage
𝜎𝑉mp:	Standard deviation of max-power voltage
𝜎𝐼mp:	Standard deviation of max-power current
𝜎𝜉:	Coefficient of variation of max-power voltage
𝜎𝜂:	Coefficient of variation of max-power current
𝜎2𝜉:	Variance of max-power voltage
𝜎2𝜂:	Variance of max-power current
𝐿:	Number of modules connected in series
𝑀:	Number of series strings connected in parallel
𝑇:	Total number of modules in the PV network
𝜖𝐶′:	Weighted variation of max-power current/voltage
𝜚∶	Pearson’s correlation
NPV:	Net present value
PPA:	Power purchase agreement
EER:	Energy escalation rate
𝑑:	Discount rate
𝐶𝑠:	Cost to manufacturer to sort modules
𝛿:	Manufacturer profit margin.