research article mismatch loss reduction in photovoltaic ...variations in photovoltaic (pv) module...

Hindawi Publishing CorporationISRN Renewable EnergyVolume 2013 Article ID 327835 9 pageshttpdxdoiorg1011552013327835

Research ArticleMismatch Loss Reduction in Photovoltaic Arrays as a Result ofSorting Photovoltaic Modules by Max-Power Parameters

J Webber1 and E Riley2

1 European Solar Engineering School 78170 Borlange Sweden2 Black amp Veatch Energy San Francisco CA 94111 USA

Correspondence should be addressed to J Webber mrjeffreywebbergmailcom

Received 8 April 2013 Accepted 12 May 2013

Academic Editors A Bosio and A Stoppato

Copyright copy 2013 J Webber and E Riley This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

Variations in photovoltaic (PV) module current-voltage curves result in a power loss in PV arrays often referred to as mismatchloss (MML) As a means of reducing MML newly fabricated PV modules are sorted to meet a set tolerance for variation inoverall maximum power output with respect to a given modulersquos rated power Starting with flash test data sets for two differentpolycrystalline PV modules and a simulated sorting procedure Monte Carlo techniques were used to generate a large numberof artificial PV arrays The MMLs for each of these arrays were then calculated to assess the sorting procedurersquos ability to reduceMML Overall MMLs were quite small (0001ndash001) Sorting by 119868mp resulted in the most consistent MML reductions Sorting by119881mp yielded insignificant results Sorting by 119875mp yielded significant MML reduction in only one of the two PV module data setsAnalysis was conducted to quantify if additional sorting on top of what bothmanufacturers had already donewouldmake economicsense Based on high level economic analysis it appears that additional sorting yields little economic gain however this is highlydependent upon manufacturer sorting cost

1 Introduction

Differences in the current-voltage characteristics of photo-voltaic (PV)modules give rise to a type of power loss referredto as ldquoelectrical mismatchrdquo once modules are connected tonetworks of series and parallel strings called arrays Theconsequence of ldquomismatch lossrdquo (MML) is that the totalpower output of a PV array will be less than the sum ofthe power outputs of the modules as if they were actingindependently [1]

This phenomenon has been investigated by a number ofauthors [1ndash5] primarily by using one of two methods (1) thecomparison between a PV arrayrsquos ideal max-power and theactual max-power computed by progressively synthesizingthe I-V curves of modules series strings and finally thecomplete PV array and (2) MML estimates of PV arrayscomposed of modules with known or statistically generatedI-V characteristics This second method was made possibleby an equation developed by Bucciarelli [1] that estimatesMML inPVarrays composed ofmoduleswith relatively small

variations in their I-V characteristics Bucciarellirsquos [1] modelwas found to adequately estimate MMLs in 100 kWp arraysby Iannone et al [2] combining and expanding on the workof Bishop [3] who used random numbers to generate PVmodules arranged into arrays and Chamberlin et al [4] whoestimated small MML values in randomly arranged arraysof four 48Wp PV modules After a brief outline of some ofthe causative mechanisms of MML in PV arrays Kaushikaand Rai [5] used Bucciarellirsquos [1] model to explore theconsequences of aging on MML values showing extremelysmall MMLs (lt001) and relatively large MMLs (gt10) inPV arrays composed of newly fabricated cells and aging cellsrespectively

A common practice employed bymanufacturers is to sortnewly fabricated PVmodules prior to installation as a meansof reducing MMLs in assembled PV arrays Previous studieshave focused primarily on predicting and estimatingMML inexisting or simulated PV arrays however very little researchexists on the effectiveness of sorting as a means of MMLreduction The purpose of this paper is to quantify to what

2 ISRN Renewable Energy

degree manufacturer sorting has upon MML reduction andwhether it remains a good economic decision

This investigation was carried out in three steps First asimulated sorting procedure was applied to a population ofreal PV modules Sorting methods included sorting modulesby reducing variances in a population of modules maximumpower current maximum power voltage and maximumpower Second Monte Carlo techniques were used to ran-domly select and arrange the sorted PV modules into PVarrays Bucciarellirsquos [1] method was then used to calculatethe MML in each randomly generated PV array producinga MML distribution over a series of simulations This wassystematically repeated over ever stricter sorting proceduresthe results of which showcase the effect of sorting uponMMLreduction Finally the impacts of these results were discussedby means of a high level economic analysis

2 Mismatch Loss Estimation Model

21 Current-Voltage Relationship in a PV Cell In terms oflight generated current the I-V curve of a PV cell can beexpressed as [5]

1198681015840= 119868ph minus 119868

119900(119890[119864(1198811015840+1198681015840119877119904)119896119860119879

119888]minus 1) minus

1198811015840+ 1198681015840119877119904

119877sh (1)

Dividing (1) by the averagemax-power current 119868mp introduc-ing the average max-power voltage 119881mp as a unity fractionand setting

119881 = 1198811015840+ 1198681015840119877119904 (2)

119868 = 1198681015840+

119881

119877sh (3)

119868

119868mp=

(119868ph + 119868119900)

119868mpminus

119868119900

119868mp119890[(119864119881119896119860119879

119888)lowast(119881mp119881mp)] (4)

Substituting (4a) (4b) and (4c) into (4) results in (5)

120572 = 119868ph +

119868ph

119868mp (4a)

120573 =119868119900

119868mp (4b)

119862 =

119890119881mp

119896119860119879119888

(4c)

119868

119868mp= 120572 minus 120573119890

119862(119881119881mp) (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 curveof a photovoltaic cell (or network of cells) near their max-power point [1 2 5] 119862 is commonly referred to as the cell

400

600

800

1000

1200

1400

1600

1800

060 062 064 066 068 070 072 074 076 078 080FF

119862

Figure 1 Cell characteristic factor versus fill factor

characteristic factor and can be expressed in terms of the fillfactor (FF) defined as [1]

FF =

119868mp119881mp

119868sc119881oc=

1198622

(1 + 119862) [119862 + ln (1 + 119862)] (6)

The relationship between the fill factor and 119862 can be seen inFigure 1

This study takes into consideration the placement ofindividual modules within a PV array To account for thiswe introduce themodified cell characteristic factor 1198621015840 whichis based upon the average fill factor of only the PV modulesused within the assembled PV array (as opposed to the entireavailable population)

FF =

(1198621015840)2

(1 + 1198621015840) [1198621015840 + ln (1 + 1198621015840)] (7)

22 Power Loss due toMismatch From Bucciarelli [1]119879 totalPV modules arranged into a network of 119872 parallel seriesstrings of 119871modules each has a mismatch loss Δ119875 estimatedby

Δ119875 =

(1198621015840+ 2)

2[1205902

120578(1 minus

1

119871) +

1205902

120585

119871(1 minus

1

119872)] (8)

where 1205902120578and 120590

2

120585represent the variance ofmax-power current

and voltage respectively in our network of PV cells Figure 2shows a typical 119871 times 119872 network of modules

1205902

120578= (

120590119868mp

119868mp)

2

1205902

120585= (

120590119881mp

119881mp)

2

(9)

ISRN Renewable Energy 3

Expressions (9) are derived on the assumption that variationsin max-power current and max-power voltage in the 119904th PVcell in our network take the nondimensional form [1]

119868mp119904

119868mp= 1 + 120598

120578120578119904

120598120578= 119868mpMAX minus 119868mpMIN

119881mp119904

119881mp= 1 + 120598

120585120585119904


(10)

120578119904and 120585119904are respective measurements of variations in max-

power current and max-power voltage and take on valuesbetween minus1 and 1 120598

120578and 120598120585are small numbers respectively

measuring the percentage range of variation in 119868mp and 119881mpIn order for (8) to yield acceptably accurate results theproduct of 120598 and 119862

1015840 known as the weighted variation ofmax-power 120598119862

1015840 must be less than 1 [1] In practice thislimits the standard deviations of max-power parameters in apopulation of PV modules to within a few percentage pointsof their mean values [5] It is also assumed that max-powercurrent and voltage in each PV module are uncorrelatedspecifically that the Pearsonrsquos Correlation between 119868mp and119881mp is small with respect to plusmn1 [1]

In prior MML studies the term 120590120578is calculated based

upon the entire sample population ofmodules used in a givenarray without taking into consideration module placementThis neglects the effect that module placement has on seriesMML in that different configurations of the same moduleswithin the same array will yield significantly different seriesMML values In order to account for module placement aunique 120590

120578 1198621015840 and series MML value are calculated for each

string based solely upon themoduleswithin that stringTheseMML values are then given a weight 120596 based upon the idealpower output 119875ideal of that string relative to the average idealpower output of every string in the array The final seriesMML is taken as the average of the weighted series MML ofeach string in the array Thus for an array of 119872 strings theseries MML term in (8) becomes

Δ119875series =1

119872

119872

sum

119902=1

120596119902Δ119875119902

Δ119875119902=

(1198621015840

119902+ 2)

21205902

120578119902(1 minus

1

119871)

120596119902=

119875ideal119902

119875ideal

(11)

where 119902 refers to the string in question The final modifiedMML equation is found to be

Δ119875 = (1 minus1

119871)

1

119872

[

[

119872

sum

119902=1

120596119902

(1198621015840

119902+ 2)

21205902

120578119902

]

]

+

(1198621015840+ 2)

2

1205902

120585

119871[1 minus

1

119872]

(12)

1 2 3

1

2

119872

119871

Figure 2Network of119871modules in series and119872modules in parallel

3 Sorting Procedure and PV Array Simulation

Most PV manufacturers sort their modules by power outputat 1000Wm2 and 25 degree Celcius cell temperature andAtmosphericMass of 15 guaranteeing some small amount ofvariation from their rated power Current industry tolerancesbetween the modules nameplate rating and measured powerrating are plusmn3ndash5

Sorting for the purpose of this paper was conductedby imposing a maximum allowed deviation from the meanfor each max-power parameter 119875mp 119868mp and 119881mp on theentire population of modules This maximum deviation wasimposed by removing outlier modules and thus reducingthe variance of max-power current 1205902

120578and maximum power

voltage 1205902

120585within the remaining population of PV modules

This maximum allowed deviation from the mean was thenreduced to examine the sensitivity of MML specific to eachsorting parameter and maximum allowed deviation

Monte Carlo techniques for randomly selecting andarranging available remaining PV modules into arrays werecarried out using Microsoft Excelrsquos random number genera-tor which has been shown to be suitable for generating smallbatches of random numbers [6 7] After specifying PV arraydimensions (119871 in series 119872 in parallel) the required numberof modules was selected from sorted groups and placed atrandom into the array Finally a MML value was calculatedusing (12)

31 Model Verification This model was validated by reevalu-ating a problem treated by Bucciarelli [1] for estimatingMMLin a single series string of PV cells with no 119881mp variation andan 119868mp variation defined by a Gaussian probability densityfunction Bucciarelli [1] assigns the following values

119868mp = 392mA120590119868mp

= 286mAFF = 067

The 1119871 term is dropped with respect to 1 and (8) results inΔ119875 = 235

Startingwith the same parameters a Gaussian probabilitydensity function was used to generate 119868mprsquos for a distributionof 204 PV cells with 119868mp and 120590

119868mpapproximately equal to

those listed above The PV module 119868mp distribution can be


Table 1 Statistical measures for the Helios Solar Works 250Wp module data set

Helios Solar Works mono-Cx 250Wp119868sc (A) 119881oc (V) 119881mp (V) 119868mp (I) 119875mp (W) FF

Number of modules 2132 Avg 885 376 305 827 2523 0758984858 (119868mp 119881mp) minus035 Max 901 385 311 846 2604 07721205981198621015840 (119881mp) 051 Min 859 370 298 805 2470 0741

1205981198621015840 (119868mp) 058 120590 0083 021 024 0076 247 00055

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

Anonymous Module Manufacturer 285Wp119868sc (A) 119881oc (V) 119881mp (V) 119868mp (I) 119875mp (W) FF


1205981198621015840 (119868mp) 029 120590 0041 013 015 0030 151 00037

012345678

204 modulesBin width = 05 mA

120590 = 287mAAvg = 392 mA

Max = 4645 mA

Min = 3195 mA

314

532

45

334

534

45

354

536

45

374

538

45

394

540

45

414

542

45

434

544

45

454

546

45

()

119868mp (mA)

Figure 3 119868mp frequency distribution for generated data set usedto verify the MML model A Gaussian pdf was used to createthe distribution based upon the example carried out originally inBucciarelli [1]

seen in Figure 3 10000 simulations were run calculating theMML in a series string composed of 100 randomly selectedmodules (so as to neglect the 1119871 term with respect to 1) Asseen in Figure 4 the average value of the resulting seriesMMLdistribution is nearly identical to that found by Bucciarelli [1]only differing by 001

4 Data

Flash test data was provided by an anonymous modulemanufacturer andHelios SolarWorksThe anonymousmod-ule manufacturer provided flash test data for 3850 285Wpmodules the details of which are shown in Figures 5(a)5(b) and 5(c) Helios Solar Works is a Milwaukee WI-basedmanufacturer of mono-Cx PV modules and provided flashtest data for 2132 250Wp PV modules upon request thedetails of which are shown in Figures 6(a) 6(b) and 6(c)

As stated in Section 2 it is a requirement that theweightedvariation of max-power 120598119862

1015840 is less than 1 and the cross-correlation 984858 for 119868mp and 119881mp is small with respect to plusmn1 [1]Statistical analysis shows that both measures are within

0

1

2

3

4

5

6

160

1

68

176

1

85

193

2

02

210

2

18

227

2

35

244

2

52

260

2

69

277

2

86

294

3

03

311

3

19

Series MML

Average = 236Max = 322Min = 162

Bin width = 0028

120590 = 022

()

119899 = 10000 iterations

Figure 4 Series MML frequency distribution using Monte Carlotechniques for module selection and placement Results are nearlyidentical to those of Bucciarelli [1]

acceptable ranges for each module type the values of whichare shown in Tables 1 and 2

5 Results and Analysis

Repetition of the process outlined in Section 3 yields adistribution of MMLs specific to the dimensions of thatparticular PV array and module sorting criteria Keeping PVarray dimensions constantwhile systematically restricting thesorting criteria upon the population of available PV modulesyields different results dependent upon which particularparameter one sorts by A realistically dimensioned PV array(400 kWp 700V) [8] was considered For this array MMLswere calculated by sorting based on each of the three max-power parameters 119875mp 119868mp and 119881mp The effect of sortingupon average MML represented in terms of a reductionin the standard deviations of each respective max-power


0

1

2

3

4

5

6

7

0985 0989 0993 0997 1000 1004 1008 1012 1016

Max 1013Min 0986Range 272

3850 modules120590 = 00042

Anon 285 Wp

()

119881mp average 119881mp

(a)

0

2

4

6

8

10

12

14

0986 0991 0996 1000 1005 1010 1014


3850 modules120590 = 00038

Anon 285 Wp

()


(b)

0

1

2

3

4

5

0992 0995 0997 1000 1002 1005 1008 1010 1013


3850 modules120590 = 00053

Anon 285 Wp

()


(c)

Figure 5 119881mp 119868mp and 119875mp distributions of the anonymous module manufacturerrsquos 285Wp module

parameter distribution can be seen in Figures 7 8(a) and8(b)

It is immediately clear that sorting by 119868mp has the mostconsistent immediate and greatest reduction onMML Eachmodule works at its own 119868mp under max-power conditionsand in a series string the worst performingmodule will lowerthe current output of every module in that string As a resultindividual module placement heavily influences series MMLin a PV array As outlier modules are removed with respect tothe imposed 119868mp restriction a greater net reduction in MMLis observed

Compared to 119868mp sorting modules by 119881mp results innegligible or even counterproductive MML reduction (as inthe case of the 250Wpmodule) Parallel MML represents theoverall voltage drop in a PV array as a result of the seriesstring with the lowest voltage In a realistically dimensionedarray such as the one considered in this study each stringin the array is composed of twenty or more PV modulesThis effectively reduces the impact that individual moduleplacement has upon string voltage variance thus reducingthe importance of 119881mp sorting This phenomenon can beinferred by the MML estimate (8) in that parallel MML isinversely proportional to string length (119871) The greater thestring length the less string voltage variation onewill find andthe less parallelMMLonewillmeasure Evidence of the heavy

dependence of overall MML upon 119868mp over 119881mp variance hasbeen demonstrated by Kaushika and Rai [5] in which parallelMMLs are found on average to be a full order of magnitudeless than series MMLs even in arrays with a string length ofone

Sorting by 119875mp yielded the most unpredictable impacton MML reduction This is not surprising given that a PVmodulersquos119875mp is the product of both119881mp and 119868mpWith overallMMLbeing shown to heavily favour seriesMMLover parallelMML sorting by any parameter that does not directly reducethe variation of 119868mp (such as 119875mp) is not going to result inconsistent reliable MML reduction

Both module populations were subjected to some degreeof sorting with respect to 119875mp prior to use by this study andreflect each manufacturerrsquos 119875mp variation tolerance Thereis a significant difference between the two module setswith the Helios module showing a modest plusmn3 variationfrom the mean whereas the modules from the anonymousmanufacturer show plusmn1 The consequences of this can beseen in averageMMLs found before we applied the simulatedsorting procedure with the modules from the anonymousmanufacturer showing a MML of 0009 compared to theHelios module MML of 0055mdashroughly 6 times morebut still orders of magnitude smaller than MML estimatesreported in prior studies [1 2 5]


0

1

2

3

4

5

6

7

098 099 100 101

Max1020Min 0977Range 433

2132 modules120590 = 00077

Helios 250 Wp

()


(a)

0

1

2

3

4

5

6

7

097 098 099 100 101 102

Max 1023

Range 489

2132 modules120590 = 00092

Helios 250 Wp

Min 0974

()


(b)

0

1

2

3

4

5

098 099 100 101 102 103


2132 modules120590 = 00098

Helios 250 Wp

()


(c)

Figure 6 119881mp 119868mp and 119875mp distributions of the Helios Solar Works 250Wp module

0

20

40

60

80

100

MML ()

120590 = 038

120590 = 054

120590 = 062

120590 = 07

120590 = 079

120590 = 086 120590 = 094

120590 = 1

0008

0016

0024

0032

0040

0048

0055

()

119871 = 23119872 = 70

Helios 250 Wp

Figure 7 MML frequency distributions for a 700V 400 kWp PVarray populated with the Helios 250Wp module Each distributionis the result of ever stricter sortingwith respect to 119868mp as representedby the reduction in 119868mp standard deviations displayed above

Any reduction in MML resulting from sorting modulessimilar in fashion to this study would only be reliably seenin the short term The work by Kaushika and Rai [5] showsdramatic increases in MMLs as PV cells age Outside the

scope of this study but clearly the next step would be toinvestigate the long-term reduction in PV array MML as afunction of sorting modules prior to installation

6 Economic Benefit of Sorting PV Modules

Reducing max-power parameter standard deviations of agiven set of PV modules results in a ΔMML reductioncorresponding to a net economic gain over the lifetime ofthe PV array For a set Power Purchase Agreement (PPA) agiven PV array will produce on average ] $year A reductionin MML will result in a yearly net cash flow increase given by

119861119895= ]ΔMML(1 + EER)119895

] = 119909119910119911

(13)

where

ΔMML = reduction in MML due to sorting ()119909 = array power output per kWp (kWhkWp)119910 = array capacity (kWp)119911 = energy cost set by PPA ($kWh)EER = Energy Escalation Rate set by PPA ()119895 = year


0000

0002

0004

0006

0008

0010

1 09 08 07 06 05 04

MM

L (

)Anon 285 Wp119871 = 20119872 = 70

119868mp119881mp119875mp

120590 (sorted)120590 (unsorted)

(a)

000

001

002

003

004

005

006

007

MM

L (

)

Helios 250 Wp

1 09 08 07 06 05 04

119871 = 23119872 = 70

119868mp119881mp119875mp


(b)

Figure 8 Average MML as a function of sorting by each max-power parameter for each module Sorting is represented by a percentagereduction in the standard deviation of each max-power parameterrsquos distribution with respect to their unsorted standard deviations

000

001

002

003

004

005

006

0 200 400 600 800 1000 1200 1400 1600Manufacturer sorting cost ($)

Helios 250 Wp

ΔM

ML

()

119871 = 23119872 = 70

120590 = 1

120590119868mp = 04

120590119875mp = 04

120590119881mp = 04

(a)

00000

00015

00030

00045

00060

00075

00090


ΔM

ML

()

Anon 285 Wp119871 = 20119872 = 70

120590 = 1

120590119868mp = 04

120590119875mp = 04

120590119881mp = 04

(b)

Figure 9 Allowed manufacturer sorting cost for a given MML reduction resulting from module sorting to be considered a good economicdecision Labelled data points correspond to the strictest sorting criteria for each parameter

The net present value NPV of this series of yearly cash flows119861119895with a discount rate 119889 over an array lifetime of 119899 years is

given byNPV =

119899

sum

119895=1

119861119895

(1 + 119889)119895

119889 =119903 minus 119894

1 + 119894

(14)

119903 = opportunity cost of capital ()

119894 = inflation rate ()

resulting in the expression

NPV = ]ΔMML119899

sum

119895=1

(1 + EER1 + 119889

)

119895

(15)

The cost 119862119900to the owner of the PV array to presort the

modules is given by

119862119900= 119862119904(1 + 120575) (16)

where 119862119904is the manufacturerrsquos cost of sorting and 120575 is their

desired profit margin In order to yield a net economic gain


NPV gt 119862119900 resulting in an expression for the minimum

required decrease in MML from sorting

ΔMML gt119862119904(1 + 120575)

][

[

119899

sum

119895=1

(1 + EER1 + 119889

)

119895

]

]

minus1

(17)

61 Example We take as an example the Helios 250Wpmodule assembled into the 400 kWp PV array simulatedpreviously place it in a region of high irradiance such asthe American South-West and assume a specific yield of2000 kWhkWp For simplicity we assume an array lifetimeand PPA of 20 years each Using recent data for $kWhEER and discount rates [9ndash12] and assuming a modest profitmargin of 20 the following parameters take the values

119909 = 2000 kWhkWp119910 = 400 kWp119911 = $016kWh119899 = 20 yearsEER = 24119903 = 272119894 = 242120575 = 20

Substituting into (17) yields

119862119904

$2673000lt ΔMML (18)

The net economic benefit of sorting modules can be seen inFigures 9(a) and 9(b) Even in the most significant case thatof sorting by 119868mp and reducing the 119868mp standard deviation to40 in the Helios 250Wp module the net economic gainsover the lifetime of the PVarray areminimalThis is howeverhighly dependent upon manufacturer sorting cost Despitethis all indications suggest that sorting modules by 119868mp willyield the most consistent and reliable reduction in MML

7 Summary and Conclusions

A procedure for estimating average MMLs of PV arraysbuilt from a given set of real PV modules was developedby simulating preinstallation sorting followed by generatingrandom artificial PV arrays using Monte Carlo techniquesBucciarellirsquos [1] equation for estimating MML in a PV net-works was modified to take into account individual moduleplacement within the network Two different PV modules(250Wp and 285Wp) were used to populate a realisticallydimensioned large-scale PV array (700V 400 kWp) [8]

The analysis indicated that sorting PV modules by max-power current 119868mp yielded the most consistent and greatestoverall reduction inMML when compared to sorting by 119875mpwhich yielded inconsistent MML reduction and 119881mp whichyielded insignificant MML reduction These results wereexpressed in terms of a reduction in the standard deviationsof each max-power parameter distribution

A brief economic analysis was carried out describing thenet economic benefit over the lifetime of the PV array ofMML reduction by means of sorting Even in the best casescenario it appears that sorting yields little or no economicgain however this is highly dependent upon manufacturersorting cost

This study was limited by the lack of data pertaining tothe time sensitivity of PV module current-voltage variationAs modules degrade over time their electrical characteristicschange which can impact MML

Nomenclature

119868ph Light generated current1198680 Diode saturation current119864 Electron charge1198811015840 Cell voltage

1198681015840 Cell current

119877119904 Cell series resistance

119877sh Cell shunt resistance119860 Diode ideality factor119896 Boltzmannrsquos constant119879119888 Cell temperature

Δ119875 Fractional power loss due to electrical mismatchΔMML Reduction in mismatch loss119862 Cell characteristic factor1198621015840 Modified cell characteristic factor

FF Fill factor119875mp Module max-power output119881mp Module max-power voltage119881mp Average module max-power voltage119868mp Module max-power current119868mp Average module max-power current119881oc Module open circuit voltage119868sc Module short circuit voltage120590119881mp

Standard deviation of max-power voltage120590119868mp

Standard deviation of max-power current120590120585 Coefficient of variation of max-power voltage

120590120578 Coefficient of variation of max-power current

1205902

120585 Variance of max-power voltage

1205902

120578 Variance of max-power current

119871 Number of modules connected in series119872 Number of series strings connected in parallel119879 Total number of modules in the PV network1205981198621015840 Weighted variation of max-power currentvoltage

984858 Pearsonrsquos correlationNPV Net present valuePPA Power purchase agreementEER Energy escalation rate119889 Discount rate119862119904 Cost to manufacturer to sort modules

120575 Manufacturer profit margin

References

[1] L L Bucciarelli Jr ldquoPower loss in photovoltaic arrays due tomismatch in cell characteristicsrdquo Solar Energy vol 23 no 4 pp277ndash288 1979


[2] F Iannone G Noviello and A Sarno ldquoMonte carlo techniquesto analyse the electrical mismatch losses in large-scale photo-voltaic generatorsrdquo Solar Energy vol 62 no 2 pp 85ndash92 1998

[3] J W Bishop ldquoComputer simulation of the effects of electricalmismatches in photovoltaic cell interconnection circuitsrdquo SolarCells vol 25 no 1 pp 73ndash89 1988

[4] C E Chamberlin P Lehman J Zoellick and G Pauletto ldquoEf-fects of mismatch losses in photovoltaic arraysrdquo Solar Energyvol 54 no 3 pp 165ndash171 1995

[5] N D Kaushika and A K Rai ldquoAn investigation of mismatchlosses in solar photovoltaic cell networksrdquo Energy vol 32 no 5pp 755ndash759 2007

[6] J E Gentle Random Number Generation and Monte CarloMethods Springer New York NY USA 2nd edition 2003

[7] B D McCullough ldquoMicrosoft Excelrsquos lsquoNotTheWichmann-Hillrsquorandom number generatorsrdquo Computational Statistics and DataAnalysis vol 52 no 10 pp 4587ndash4593 2008

[8] National Fire Protection Association National Electric Code2008 Edition NFPA Quincy Mass USA 2008

[9] Barronrsquos ldquoAdjustable mortgage base ratesrdquo Barronrsquos MarketWeek M57 May 2012

[10] US Energy Information Administration ldquoAnnual Energy Out-look 2011rdquo httpwwweiagov

[11] US Department of Energy ldquoFY 2011 Field Budget Call Escala-tion Ratesrdquo httpscienceenergygov

[12] Bureau of Labor Statistics ldquoTable containing history of CPI-UUS All items indexes and annual percent changes from 1913to presentrdquo httpwwwblsgov

TribologyAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FuelsJournal of


Journal ofPetroleum Engineering


Industrial EngineeringJournal of


Power ElectronicsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in

CombustionJournal of


Journal of


Renewable Energy

Submit your manuscripts athttpwwwhindawicom


StructuresJournal of


RotatingMachinery


EnergyJournal of


Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

International Journal ofPhotoenergy


Nuclear InstallationsScience and Technology of


Solar EnergyJournal of


Wind EnergyJournal of


Nuclear EnergyInternational Journal of


High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


degree manufacturer sorting has upon MML reduction andwhether it remains a good economic decision

This investigation was carried out in three steps First asimulated sorting procedure was applied to a population ofreal PV modules Sorting methods included sorting modulesby reducing variances in a population of modules maximumpower current maximum power voltage and maximumpower Second Monte Carlo techniques were used to ran-domly select and arrange the sorted PV modules into PVarrays Bucciarellirsquos [1] method was then used to calculatethe MML in each randomly generated PV array producinga MML distribution over a series of simulations This wassystematically repeated over ever stricter sorting proceduresthe results of which showcase the effect of sorting uponMMLreduction Finally the impacts of these results were discussedby means of a high level economic analysis

2 Mismatch Loss Estimation Model

21 Current-Voltage Relationship in a PV Cell In terms oflight generated current the I-V curve of a PV cell can beexpressed as [5]

1198681015840= 119868ph minus 119868

119900(119890[119864(1198811015840+1198681015840119877119904)119896119860119879

119888]minus 1) minus

1198811015840+ 1198681015840119877119904

119877sh (1)

Dividing (1) by the averagemax-power current 119868mp introduc-ing the average max-power voltage 119881mp as a unity fractionand setting

119881 = 1198811015840+ 1198681015840119877119904 (2)

119868 = 1198681015840+

119881

119877sh (3)

119868

119868mp=

(119868ph + 119868119900)

119868mpminus

119868119900

119868mp119890[(119864119881119896119860119879

119888)lowast(119881mp119881mp)] (4)

Substituting (4a) (4b) and (4c) into (4) results in (5)

120572 = 119868ph +

119868ph

119868mp (4a)

120573 =119868119900

119868mp (4b)

119862 =

119890119881mp

119896119860119879119888

(4c)

119868

119868mp= 120572 minus 120573119890

119862(119881119881mp) (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 curveof a photovoltaic cell (or network of cells) near their max-power point [1 2 5] 119862 is commonly referred to as the cell

400

600

800

1000

1200

1400

1600

1800

060 062 064 066 068 070 072 074 076 078 080FF

119862

Figure 1 Cell characteristic factor versus fill factor

characteristic factor and can be expressed in terms of the fillfactor (FF) defined as [1]

FF =

119868mp119881mp

119868sc119881oc=

1198622

(1 + 119862) [119862 + ln (1 + 119862)] (6)

The relationship between the fill factor and 119862 can be seen inFigure 1

This study takes into consideration the placement ofindividual modules within a PV array To account for thiswe introduce themodified cell characteristic factor 1198621015840 whichis based upon the average fill factor of only the PV modulesused within the assembled PV array (as opposed to the entireavailable population)

FF =

(1198621015840)2

(1 + 1198621015840) [1198621015840 + ln (1 + 1198621015840)] (7)

22 Power Loss due toMismatch From Bucciarelli [1]119879 totalPV modules arranged into a network of 119872 parallel seriesstrings of 119871modules each has a mismatch loss Δ119875 estimatedby

Δ119875 =

(1198621015840+ 2)

2[1205902

120578(1 minus

1

119871) +

1205902

120585

119871(1 minus

1

119872)] (8)

where 1205902120578and 120590

2

120585represent the variance ofmax-power current

and voltage respectively in our network of PV cells Figure 2shows a typical 119871 times 119872 network of modules

1205902

120578= (

120590119868mp

119868mp)

2

1205902

120585= (

120590119881mp

119881mp)

2

(9)



119868mp119904

119868mp= 1 + 120598

120578120578119904


119881mp119904

119881mp= 1 + 120598

120585120585119904


(10)











Δ119875series =1

119872

119872

sum

119902=1

120596119902Δ119875119902

Δ119875119902=

(1198621015840

119902+ 2)

21205902

120578119902(1 minus

1

119871)

120596119902=

119875ideal119902

119875ideal

(11)


Δ119875 = (1 minus1

119871)

1

119872

[

[

119872

sum

119902=1

120596119902

(1198621015840

119902+ 2)

21205902

120578119902

]

]

+

(1198621015840+ 2)

2

1205902

120585

119871[1 minus

1

119872]

(12)

1 2 3

1

2

119872

119871






voltage 1205902





119868mp = 392mA120590119868mp

= 286mAFF = 067









1205981198621015840 (119868mp) 058 120590 0083 021 024 0076 247 00055




1205981198621015840 (119868mp) 029 120590 0041 013 015 0030 151 00037

012345678


120590 = 287mAAvg = 392 mA

Max = 4645 mA

Min = 3195 mA

314

532

45

334

534

45

354

536

45

374

538

45

394

540

45

414

542

45

434

544

45

454

546

45

()

119868mp (mA)



4 Data




0

1

2

3

4

5

6

160

1

68

176

1

85

193

2

02

210

2

18

227

2

35

244

2

52

260

2

69

277

2

86

294

3

03

311

3

19

Series MML


Bin width = 0028

120590 = 022

()







0

1

2

3

4

5

6

7

0985 0989 0993 0997 1000 1004 1008 1012 1016


3850 modules120590 = 00042

Anon 285 Wp

()


(a)

0

2

4

6

8

10

12

14

0986 0991 0996 1000 1005 1010 1014


3850 modules120590 = 00038

Anon 285 Wp

()


(b)

0

1

2

3

4

5

0992 0995 0997 1000 1002 1005 1008 1010 1013


3850 modules120590 = 00053

Anon 285 Wp

()


(c)









0

1

2

3

4

5

6

7

098 099 100 101


2132 modules120590 = 00077

Helios 250 Wp

()


(a)

0

1

2

3

4

5

6

7

097 098 099 100 101 102

Max 1023

Range 489

2132 modules120590 = 00092

Helios 250 Wp

Min 0974

()


(b)

0

1

2

3

4

5

098 099 100 101 102 103


2132 modules120590 = 00098

Helios 250 Wp

()


(c)


0

20

40

60

80

100

MML ()

120590 = 038

120590 = 054

120590 = 062

120590 = 07

120590 = 079

120590 = 086 120590 = 094

120590 = 1

0008

0016

0024

0032

0040

0048

0055

()

119871 = 23119872 = 70

Helios 250 Wp






119861119895= ]ΔMML(1 + EER)119895

] = 119909119910119911

(13)

where



0000

0002

0004

0006

0008

0010

1 09 08 07 06 05 04

MM

L (

)Anon 285 Wp119871 = 20119872 = 70

119868mp119881mp119875mp


(a)

000

001

002

003

004

005

006

007

MM

L (

)

Helios 250 Wp

1 09 08 07 06 05 04

119871 = 23119872 = 70

119868mp119881mp119875mp


(b)


000

001

002

003

004

005

006


Helios 250 Wp

ΔM

ML

()

119871 = 23119872 = 70

120590 = 1

120590119868mp = 04

120590119875mp = 04

120590119881mp = 04

(a)

00000

00015

00030

00045

00060

00075

00090


ΔM

ML

()

Anon 285 Wp119871 = 20119872 = 70

120590 = 1

120590119868mp = 04

120590119875mp = 04

120590119881mp = 04

(b)



given byNPV =

119899

sum

119895=1

119861119895

(1 + 119889)119895

119889 =119903 minus 119894

1 + 119894

(14)




NPV = ]ΔMML119899

sum

119895=1

(1 + EER1 + 119889

)

119895

(15)


modules is given by

119862119900= 119862119904(1 + 120575) (16)






ΔMML gt119862119904(1 + 120575)

][

[

119899

sum

119895=1

(1 + EER1 + 119889

)

119895

]

]

minus1

(17)




119862119904

$2673000lt ΔMML (18)







Nomenclature










1205902


1205902





References


















FuelsJournal of







Advances in



Journal of


Renewable Energy





RotatingMachinery


EnergyJournal of



















119868mp119904

119868mp= 1 + 120598

120578120578119904


119881mp119904

119881mp= 1 + 120598

120585120585119904


(10)











Δ119875series =1

119872

119872

sum

119902=1

120596119902Δ119875119902

Δ119875119902=

(1198621015840

119902+ 2)

21205902

120578119902(1 minus

1

119871)

120596119902=

119875ideal119902

119875ideal

(11)


Δ119875 = (1 minus1

119871)

1

119872

[

[

119872

sum

119902=1

120596119902

(1198621015840

119902+ 2)

21205902

120578119902

]

]

+

(1198621015840+ 2)

2

1205902

120585

119871[1 minus

1

119872]

(12)

1 2 3

1

2

119872

119871






voltage 1205902





119868mp = 392mA120590119868mp

= 286mAFF = 067









1205981198621015840 (119868mp) 058 120590 0083 021 024 0076 247 00055




1205981198621015840 (119868mp) 029 120590 0041 013 015 0030 151 00037

012345678


120590 = 287mAAvg = 392 mA

Max = 4645 mA

Min = 3195 mA

314

532

45

334

534

45

354

536

45

374

538

45

394

540

45

414

542

45

434

544

45

454

546

45

()

119868mp (mA)



4 Data




0

1

2

3

4

5

6

160

1

68

176

1

85

193

2

02

210

2

18

227

2

35

244

2

52

260

2

69

277

2

86

294

3

03

311

3

19

Series MML


Bin width = 0028

120590 = 022

()







0

1

2

3

4

5

6

7

0985 0989 0993 0997 1000 1004 1008 1012 1016


3850 modules120590 = 00042

Anon 285 Wp

()


(a)

0

2

4

6

8

10

12

14

0986 0991 0996 1000 1005 1010 1014


3850 modules120590 = 00038

Anon 285 Wp

()


(b)

0

1

2

3

4

5

0992 0995 0997 1000 1002 1005 1008 1010 1013


3850 modules120590 = 00053

Anon 285 Wp

()


(c)









0

1

2

3

4

5

6

7

098 099 100 101


2132 modules120590 = 00077

Helios 250 Wp

()


(a)

0

1

2

3

4

5

6

7

097 098 099 100 101 102

Max 1023

Range 489

2132 modules120590 = 00092

Helios 250 Wp

Min 0974

()


(b)

0

1

2

3

4

5

098 099 100 101 102 103


2132 modules120590 = 00098

Helios 250 Wp

()


(c)


0

20

40

60

80

100

MML ()

120590 = 038

120590 = 054

120590 = 062

120590 = 07

120590 = 079

120590 = 086 120590 = 094

120590 = 1

0008

0016

0024

0032

0040

0048

0055

()

119871 = 23119872 = 70

Helios 250 Wp






119861119895= ]ΔMML(1 + EER)119895

] = 119909119910119911

(13)

where



0000

0002

0004

0006

0008

0010

1 09 08 07 06 05 04

MM

L (

)Anon 285 Wp119871 = 20119872 = 70

119868mp119881mp119875mp


(a)

000

001

002

003

004

005

006

007

MM

L (

)

Helios 250 Wp

1 09 08 07 06 05 04

119871 = 23119872 = 70

119868mp119881mp119875mp


(b)


000

001

002

003

004

005

006


Helios 250 Wp

ΔM

ML

()

119871 = 23119872 = 70

120590 = 1

120590119868mp = 04

120590119875mp = 04

120590119881mp = 04

(a)

00000

00015

00030

00045

00060

00075

00090


ΔM

ML

()

Anon 285 Wp119871 = 20119872 = 70

120590 = 1

120590119868mp = 04

120590119875mp = 04

120590119881mp = 04

(b)



given byNPV =

119899

sum

119895=1

119861119895

(1 + 119889)119895

119889 =119903 minus 119894

1 + 119894

(14)




NPV = ]ΔMML119899

sum

119895=1

(1 + EER1 + 119889

)

119895

(15)


modules is given by

119862119900= 119862119904(1 + 120575) (16)






ΔMML gt119862119904(1 + 120575)

][

[

119899

sum

119895=1

(1 + EER1 + 119889

)

119895

]

]

minus1

(17)




119862119904

$2673000lt ΔMML (18)







Nomenclature










1205902


1205902





References


















FuelsJournal of







Advances in



Journal of


Renewable Energy





RotatingMachinery


EnergyJournal of





















1205981198621015840 (119868mp) 058 120590 0083 021 024 0076 247 00055




1205981198621015840 (119868mp) 029 120590 0041 013 015 0030 151 00037

012345678


120590 = 287mAAvg = 392 mA

Max = 4645 mA

Min = 3195 mA

314

532

45

334

534

45

354

536

45

374

538

45

394

540

45

414

542

45

434

544

45

454

546

45

()

119868mp (mA)



4 Data




0

1

2

3

4

5

6

160

1

68

176

1

85

193

2

02

210

2

18

227

2

35

244

2

52

260

2

69

277

2

86

294

3

03

311

3

19

Series MML


Bin width = 0028

120590 = 022

()







0

1

2

3

4

5

6

7

0985 0989 0993 0997 1000 1004 1008 1012 1016


3850 modules120590 = 00042

Anon 285 Wp

()


(a)

0

2

4

6

8

10

12

14

0986 0991 0996 1000 1005 1010 1014


3850 modules120590 = 00038

Anon 285 Wp

()


(b)

0

1

2

3

4

5

0992 0995 0997 1000 1002 1005 1008 1010 1013


3850 modules120590 = 00053

Anon 285 Wp

()


(c)









0

1

2

3

4

5

6

7

098 099 100 101


2132 modules120590 = 00077

Helios 250 Wp

()


(a)

0

1

2

3

4

5

6

7

097 098 099 100 101 102

Max 1023

Range 489

2132 modules120590 = 00092

Helios 250 Wp

Min 0974

()


(b)

0

1

2

3

4

5

098 099 100 101 102 103


2132 modules120590 = 00098

Helios 250 Wp

()


(c)


0

20

40

60

80

100

MML ()

120590 = 038

120590 = 054

120590 = 062

120590 = 07

120590 = 079

120590 = 086 120590 = 094

120590 = 1

0008

0016

0024

0032

0040

0048

0055

()

119871 = 23119872 = 70

Helios 250 Wp






119861119895= ]ΔMML(1 + EER)119895

] = 119909119910119911

(13)

where



0000

0002

0004

0006

0008

0010

1 09 08 07 06 05 04

MM

L (

)Anon 285 Wp119871 = 20119872 = 70

119868mp119881mp119875mp


(a)

000

001

002

003

004

005

006

007

MM

L (

)

Helios 250 Wp

1 09 08 07 06 05 04

119871 = 23119872 = 70

119868mp119881mp119875mp


(b)


000

001

002

003

004

005

006


Helios 250 Wp

ΔM

ML

()

119871 = 23119872 = 70

120590 = 1

120590119868mp = 04

120590119875mp = 04

120590119881mp = 04

(a)

00000

00015

00030

00045

00060

00075

00090


ΔM

ML

()

Anon 285 Wp119871 = 20119872 = 70

120590 = 1

120590119868mp = 04

120590119875mp = 04

120590119881mp = 04

(b)



given byNPV =

119899

sum

119895=1

119861119895

(1 + 119889)119895

119889 =119903 minus 119894

1 + 119894

(14)




NPV = ]ΔMML119899

sum

119895=1

(1 + EER1 + 119889

)

119895

(15)


modules is given by

119862119900= 119862119904(1 + 120575) (16)






ΔMML gt119862119904(1 + 120575)

][

[

119899

sum

119895=1

(1 + EER1 + 119889

)

119895

]

]

minus1

(17)




119862119904

$2673000lt ΔMML (18)







Nomenclature










1205902


1205902





References


















FuelsJournal of







Advances in



Journal of


Renewable Energy





RotatingMachinery


EnergyJournal of


















0

1

2

3

4

5

6

7

0985 0989 0993 0997 1000 1004 1008 1012 1016


3850 modules120590 = 00042

Anon 285 Wp

()


(a)

0

2

4

6

8

10

12

14

0986 0991 0996 1000 1005 1010 1014


3850 modules120590 = 00038

Anon 285 Wp

()


(b)

0

1

2

3

4

5

0992 0995 0997 1000 1002 1005 1008 1010 1013


3850 modules120590 = 00053

Anon 285 Wp

()


(c)









0

1

2

3

4

5

6

7

098 099 100 101


2132 modules120590 = 00077

Helios 250 Wp

()


(a)

0

1

2

3

4

5

6

7

097 098 099 100 101 102

Max 1023

Range 489

2132 modules120590 = 00092

Helios 250 Wp

Min 0974

()


(b)

0

1

2

3

4

5

098 099 100 101 102 103


2132 modules120590 = 00098

Helios 250 Wp

()


(c)


0

20

40

60

80

100

MML ()

120590 = 038

120590 = 054

120590 = 062

120590 = 07

120590 = 079

120590 = 086 120590 = 094

120590 = 1

0008

0016

0024

0032

0040

0048

0055

()

119871 = 23119872 = 70

Helios 250 Wp






119861119895= ]ΔMML(1 + EER)119895

] = 119909119910119911

(13)

where



0000

0002

0004

0006

0008

0010

1 09 08 07 06 05 04

MM

L (

)Anon 285 Wp119871 = 20119872 = 70

119868mp119881mp119875mp


(a)

000

001

002

003

004

005

006

007

MM

L (

)

Helios 250 Wp

1 09 08 07 06 05 04

119871 = 23119872 = 70

119868mp119881mp119875mp


(b)


000

001

002

003

004

005

006


Helios 250 Wp

ΔM

ML

()

119871 = 23119872 = 70

120590 = 1

120590119868mp = 04

120590119875mp = 04

120590119881mp = 04

(a)

00000

00015

00030

00045

00060

00075

00090


ΔM

ML

()

Anon 285 Wp119871 = 20119872 = 70

120590 = 1

120590119868mp = 04

120590119875mp = 04

120590119881mp = 04

(b)



given byNPV =

119899

sum

119895=1

119861119895

(1 + 119889)119895

119889 =119903 minus 119894

1 + 119894

(14)




NPV = ]ΔMML119899

sum

119895=1

(1 + EER1 + 119889

)

119895

(15)


modules is given by

119862119900= 119862119904(1 + 120575) (16)






ΔMML gt119862119904(1 + 120575)

][

[

119899

sum

119895=1

(1 + EER1 + 119889

)

119895

]

]

minus1

(17)




119862119904

$2673000lt ΔMML (18)







Nomenclature










1205902


1205902





References


















FuelsJournal of







Advances in



Journal of


Renewable Energy





RotatingMachinery


EnergyJournal of


















0

1

2

3

4

5

6

7

098 099 100 101


2132 modules120590 = 00077

Helios 250 Wp

()


(a)

0

1

2

3

4

5

6

7

097 098 099 100 101 102

Max 1023

Range 489

2132 modules120590 = 00092

Helios 250 Wp

Min 0974

()


(b)

0

1

2

3

4

5

098 099 100 101 102 103


2132 modules120590 = 00098

Helios 250 Wp

()


(c)


0

20

40

60

80

100

MML ()

120590 = 038

120590 = 054

120590 = 062

120590 = 07

120590 = 079

120590 = 086 120590 = 094

120590 = 1

0008

0016

0024

0032

0040

0048

0055

()

119871 = 23119872 = 70

Helios 250 Wp






119861119895= ]ΔMML(1 + EER)119895

] = 119909119910119911

(13)

where



0000

0002

0004

0006

0008

0010

1 09 08 07 06 05 04

MM

L (

)Anon 285 Wp119871 = 20119872 = 70

119868mp119881mp119875mp


(a)

000

001

002

003

004

005

006

007

MM

L (

)

Helios 250 Wp

1 09 08 07 06 05 04

119871 = 23119872 = 70

119868mp119881mp119875mp


(b)


000

001

002

003

004

005

006


Helios 250 Wp

ΔM

ML

()

119871 = 23119872 = 70

120590 = 1

120590119868mp = 04

120590119875mp = 04

120590119881mp = 04

(a)

00000

00015

00030

00045

00060

00075

00090


ΔM

ML

()

Anon 285 Wp119871 = 20119872 = 70

120590 = 1

120590119868mp = 04

120590119875mp = 04

120590119881mp = 04

(b)



given byNPV =

119899

sum

119895=1

119861119895

(1 + 119889)119895

119889 =119903 minus 119894

1 + 119894

(14)




NPV = ]ΔMML119899

sum

119895=1

(1 + EER1 + 119889

)

119895

(15)


modules is given by

119862119900= 119862119904(1 + 120575) (16)






ΔMML gt119862119904(1 + 120575)

][

[

119899

sum

119895=1

(1 + EER1 + 119889

)

119895

]

]

minus1

(17)




119862119904

$2673000lt ΔMML (18)







Nomenclature










1205902


1205902





References


















FuelsJournal of







Advances in



Journal of


Renewable Energy





RotatingMachinery


EnergyJournal of


















0000

0002

0004

0006

0008

0010

1 09 08 07 06 05 04

MM

L (

)Anon 285 Wp119871 = 20119872 = 70

119868mp119881mp119875mp


(a)

000

001

002

003

004

005

006

007

MM

L (

)

Helios 250 Wp

1 09 08 07 06 05 04

119871 = 23119872 = 70

119868mp119881mp119875mp


(b)


000

001

002

003

004

005

006


Helios 250 Wp

ΔM

ML

()

119871 = 23119872 = 70

120590 = 1

120590119868mp = 04

120590119875mp = 04

120590119881mp = 04

(a)

00000

00015

00030

00045

00060

00075

00090


ΔM

ML

()

Anon 285 Wp119871 = 20119872 = 70

120590 = 1

120590119868mp = 04

120590119875mp = 04

120590119881mp = 04

(b)



given byNPV =

119899

sum

119895=1

119861119895

(1 + 119889)119895

119889 =119903 minus 119894

1 + 119894

(14)




NPV = ]ΔMML119899

sum

119895=1

(1 + EER1 + 119889

)

119895

(15)


modules is given by

119862119900= 119862119904(1 + 120575) (16)






ΔMML gt119862119904(1 + 120575)

][

[

119899

sum

119895=1

(1 + EER1 + 119889

)

119895

]

]

minus1

(17)




119862119904

$2673000lt ΔMML (18)







Nomenclature










1205902


1205902





References


















FuelsJournal of







Advances in



Journal of


Renewable Energy





RotatingMachinery


EnergyJournal of




















ΔMML gt119862119904(1 + 120575)

][

[

119899

sum

119895=1

(1 + EER1 + 119889

)

119895

]

]

minus1

(17)




119862119904

$2673000lt ΔMML (18)







Nomenclature










1205902


1205902





References


















FuelsJournal of







Advances in



Journal of


Renewable Energy





RotatingMachinery


EnergyJournal of

































FuelsJournal of







Advances in



Journal of


Renewable Energy





RotatingMachinery


EnergyJournal of

















research article mismatch loss reduction in photovoltaic ...variations in photovoltaic (pv) module...

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