Quality Control of Wide Collections of PV Modules: Lessons ...

PROGRESS IN PHOTOVOLTAICS: RESEARCH AND APPLICATIONS

Prog. Photovolt: Res. Appl. 7, 137±149 (1999)

Applications

Quality Control of Wide Collections of PV Modules: Lessons Learned from the IES Experience E. CaamanÄo,1,* E. Lorenzo1 and R. Zilles2 1Instituto 2

de EnergõÂa Solar, Ciudad Universitaria s/n, 28040 Madrid, Spain Instituto de ElectroteÂcnica e EnergõÂa, Av. Prof. Luciano Gualberto 1289, 05508-900 SaÄo Paulo, Brazil

This paper presents the Quality Control method for PV modules developed by the Instituto de EnergõÂa Solar (IES), an institution with almost 10 years of experience in the ®eld of Quality Control of PV modules associated with supply procedures of PV projects. The method is easy and fast to implement. It consists of Technical and Contractual procedures, both closely related. Concerning the ®rst procedure, detailed description is oered of the type of measurements performed, equipments used, data processing and steadiness control of the method. Regarding the Contractual procedure, after its detailed description an example coming from the Toledo PV plant is oered and commented. The paper then summarises the IES experience on Quality Control processes, together with additional information about time requirements and costs of the IES method. In addition, some re¯ections upon the possible adoption of the method by countries involved in PV Rural Electri®cation programmes are ®nally included. Copyright # 1999 John Wiley & Sons, Ltd.

INTRODUCTION

I

n a commercial product such as a PV module, it is essential that the purchaser can receive some assurance that the high capital cost will be oset by the expected performance in terms of both rated power and long trouble-free service life. Current PV modules are largely able to withstand even hard environmental conditions without signi®cant degradation, and they generally honour their reliability reputation.1±3 However, it is also true that, unfortunately, under-rated PV modules remain largely present in real PV projects. A representative example is found in the German 1000-Roofs PV Programme, where operational experience has shown that the module peak-power speci®ed by the manufacturers cannot be generally con®rmed.4 Other experiences5±7 also reveal that actual power below 15% of the nominal value is often found, and the situation is probably worse in non-monitored experiences, as is typically the case in PV Rural Electri®cation projects. Quality Control of PV modules associated with supply procedures has demonstrated its usefulness in the struggle against this problem.8±10 However, such practice has remained, up to now, limited to the frame of `big' projects involving MWp.11 PV modules for terrestrial use are commonly rated at the so-called Standard Test Conditions (STC) or Standard Reporting Conditions (SRC), namely, normal irradiance of 1000 W m ÿ2, spectral distribution AM 1.5 and cell operation temperature of 258C. The main problem in PV performance measurements * Correspondence to: E. CaamanÄo, IES, Ciudad Universitaria s/n, E-28040 Madrid, Spain CCC 1062±7995/99/020137±13$17.50 Copyright # 1999 John Wiley & Sons, Ltd.

Received 2 August 1998 Revised 22 October 1998

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E. CAAMANÄO, E. LORENZO AND R. ZILLES

arises from the fact that solar cells have a highly selective spectral response and are, therefore, very sensitive to the spectral composition of the incident irradiance. Outdoors, this aspect varies considerably with location, weather, season and time of the day; indoors, it depends on the type of simulator used and may also change as the equipment ages. Unless measurement procedures take account of these variations and other diculties, such as the marked temperature dependence of solar cells, the results can be grossly erroneous. In the past, performance ratings of the same module by dierent laboratories reached as much as 20%.12 Fortunately, well established standards for measurement procedures are today available,13,14 so that performance ratings disagreements can be smaller than 2%.15 However, there are yet no international standards for production acceptance and quality assurance: it is entirely left to the manufacturer to specify his own requirements in these matters. This fact, together with the widely extended practice of tolerating a +10% deviation from the nominal power values, has paved the way for the proliferation of under-rated PV modules. As far as we know, most of the PV modules Quality Control experiences have been based on indoor absolute I±V measurements carried out in rather sophisticated equipment, such as solar simulators and traceable irradiance sensors.8,10 At the same time, the use of outdoor I±V measurements extrapolated to STC for determining adherence to contractual performance has been questioned,16 due to the possible uncertainties associated with the spectral mismatch and cell temperature determination. In this aspect, the IES experience has demonstrated that Quality Control of PV modules can eectively lie on relative and rather simple outdoor I±V measurements. The central idea consists of using a reference PV module of the same technology, previously calibrated in an independent laboratory having traceable sensors. Tested PV modules and the reference unit are quasi-simultaneously measured Ð their characteristic I±V curve Ð in real operation conditions. Both measurements are then extrapolated to STC and results (main electrical parameters) are correlated by comparison with the calibration values of the reference module. The IES experience on Quality Control of wide collections of modules started in 1989. In 1992, a particular procedure was developed for the 1 MW Toledo PV Plant.9 Since then, it has been adopted in other signi®cantly smaller projects.17±19 The full procedure is easy and fast to implement. Therefore, we believe it may be worth to consider its use even within the frame of rather small projects, let us say, a few tens of kWp. This paper describes the technical and contractual parts of the IES PV modules Quality Control procedure, gives some recommendations for its practical implementation and presents the statistical properties of the collections of PV modules already tested at the IES facility. Finally, some ideas for further simpli®cation of the procedure are discussed, having in mind the possibility of local implementation on the particular frame or PV Rural Electri®cation programmes.

QUALITY CONTROL PROCEDURE Technical procedure The procedure is as follows: (a) One (any) PV module from the collection submitted to the Quality Control, e.g. all modules involved in the purchase, is rated (calibrated) at STC in an independent laboratory recognised by both manufacturer and customer. In what follows, the maximum power of this reference PV module is called PMRC . (b) A representative set of PV modules, typically 4% of the total collection, with a minimum of 15 modules, is selected for testing. (c) For each tested PV module, its characteristic curve is measured outdoors, quasi-simultaneously with the measurement of the reference unit I±V curve. This means that the following conditions are imposed: Ð Global in-plane irradiance, G 4 600 W m ÿ2 Ð Diuse fraction of G, D/G 5 20% (D is diuse in-plane irradiance) Copyright # 1999 John Wiley & Sons, Ltd.


QUALITY CONTROL OF PV MODULES

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Ð Wind speed at the modules surface 55 ms ÿ1 Ð Time interval between tested and reference module measurements 53 min. The working temperature of the solar cells, Tc , is estimated as: NOCT8C ÿ 208C ÿ2 1 Tc 8C Ta 8C GW m 800 W mÿ2 where Ta is the ambient temperature and NOCT the Nominal Operating Cell Temperature, as given by the manufacturer. (d) Both I±V curves are extrapolated to STC, according with the usual procedure described in IEC 60891.13 This calculation provides two `tested' maximum power values, PMTT and PMRT , corresponding respectively to the tested and reference PV modules. (e) The àcceptance' maximum power value of the tested PV module, PMTA , is calculated as: PMTA PMTT

PMRC PMRT

2

where PMTT is the `tested' maximum power of the PV module under test, PMRT is the `tested' maximum power of the reference PV module, PMRC is the calibrated maximum power of the reference module. Observe that the described procedure is, in fact, based on the assumption that errors associated with the extrapolation from real to STC (spectrum and cell temperature eects) and with the determination of the real operating conditions (irradiance and ambient temperature sensors calibration) are the same in both modules. In this way, the simultaneous use of a reference PV module allows to eliminate the uncertainties related to STC values obtained from outdoor measurements. It is important to note that this Quality Control procedure exclusively focuses on the comparison of an object with a reference of similar nature and constitution. Meanwhile, the absolute rating of the reference module is entrusted to another ®eld of activities. Consequently, the essential procedure requirement is steadiness, not accuracy. Fortunately, PV module performance is extremely steady, which means that the previously mentioned procedure does not require other precaution than the use of a single instrumental set-up facility (irradiance, ambient temperature, current and voltage sensors) for all the tested and reference PV modules. These must be obviously placed coplanarily and without signi®cant hindrances to thermal dissipation, in order to ensure they operate on the same conditions. In practice, we implemented this procedure by means of the very simple capacity load presented in Figure 1. Being initially discharged, the capacitor is charged by the PV module current when I1 is switched ON (I2 remains OFF); R1 is a very small resistance, just acting as a sensor current. Ideally, the charging process starts at the PV module short-circuit and ends when the voltage at the capacitor equals the open circuit voltage of the PV module. This way, the charging process takes over the full characteristic curve in

Figure 1. Schematic diagram of the capacitive load Copyright # 1999 John Wiley & Sons, Ltd.



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Figure 2. Operating principle of the capacitive load

the ®rst quadrant, as Figure 2 shows. Later, the energy stored into the capacitor is dissipated through R2 (I2 ON, I1 OFF), in order to fully discharge the capacitor and prepare the load for the next measurement. Capacitor values should be selected so that the charging time, tc , extends from 20 to 100 ms and, therefore, transitory and unstable phenomena are avoided. As a practical rule, tc can be estimated as C Voc/Isc : for example, for a PV module with nominal STC parameters Isc 3 A and Voc 17 V, the capacitor value should be selected within the range 3500±18 000 mF. I1 can be easily implemented with a Mosfet transistor, which is switched ON by biasing the gate through a simple manual switch ( pushbutton) and a dry cell. Because of the short charging times, the transistor can be mounted without heat dissipation; furthermore, its rated current (catalogue value) can be substantially exceeded without damage risks. At the IES, we have used the Motorola IRF 540. I2 can be another switch, similar to I1 , and R2 any resistor assuring that the capacitor discharging time is large enough to limit the involved power below reasonable values. The IES load uses a value of 47 O (2.5 W), leading to discharging times of a few seconds for most of the current PV modules. On the other hand, it is highly recommended that the acquisition of each characteristic curve consists of, at least, 30 I±V points. This means that the sampling frequency should be larger than 1.5 kHz, a rather low value in the context of the current oer of data acquisition equipments. At the IES, we use a portable digital oscilloscope (Fluke Scopemeter) able to register 512 I±V points, associated to an optoisolated voltage probe for current measurements, as Figure 3 shows. This probe serves to fully isolate the I and V channels Ð both have their reference point interconnected inside the oscilloscope Ð thus eliminating wiring in¯uences on the PV module characteristic curve. Providing the capacitor is initially discharged, at the very ®rst instant of the I±V curve measurement (I1 ON), the voltage of the PV module, Vg , is close to the value: Vg t 0 Isc R1 Rw RI1

3

where Rw and RI1 are, respectively, the wiring and Il resistances. This means that the precise short-circuit condition is never attained. However, this can be easily solved by pre-charging the capacitor with a negative voltage, using an independent voltage source or a simple dry cell. It has been previously mentioned that the measured I±V points are translated to STC following the procedure described in IEC 60891, which leads to the calculation of the main electrical parameters of the tested PV module. At the IES, we have implemented a further digital ®ltering process, based on the Fourier Analysis Theory (see the Appendix), which improves the accuracy of the global testing procedure. Figure 4 shows an example of our method. On the left side, the I±V curves of both tested and reference modules (labels ``MoÂdulo medido'' and ``MoÂdulo patroÂn'', respectively) are depicted separately. For each curve there can be clearly seen the I±V points experimentally measured, the same points extrapolated to STC and the resulting characteristic curve after digital ®ltering has been applied. The table on the upper Copyright # 1999 John Wiley & Sons, Ltd.



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Figure 3. Measurement procedure used by IES

Figure 4. Measured I±V curves of reference and tested PV modules

right corner, entitled ``Condiciones de medida'', shows the operation conditions of the measurements. The table below, with the title ``CaracterõÂ sticas a STC'', summarises the modules electrical parameters at STC (column with heading ``336071'' corresponding to the tested module and column with heading ``patroÂn'' to the reference module). Finally, the table on the lower right corner, entitled ``CalibracioÂn'', shows the ®nal accepted value for the tested module, after the correction given by Equation (2) has been applied. Numerical values of the maximum power for both modules are, according to the ®gure: Calibration value of reference module: PMRC 70.9 W `Tested' values: reference module, PMRT 72.41 W; tested module, PMTT 73.43 W Àcceptance' value of tested module: PMTA 71.9 W Copyright # 1999 John Wiley & Sons, Ltd.



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Figure 5. Steadiness control of the testing method

Steadiness control of the testing method is done by periodical measurements of three modules (reference 2 testing units, all of the same technology), whose calibrated parameter values are known. Figure 5 shows the PMTA values of the testing units, measured on dierent days over 6 years, which means very dierent operation conditions. It should be observed that values keep within a narrow range, proving excellent steadiness of the measurement procedure. Meanwhile, PMRT values of the reference module varied considerably more, between ÿ8% and 2% of the calibrated value. This demonstrates the fact that the use of a reference PV module eectively compensates spectral, irradiance and cell temperature eects. On the other hand, it is worth commenting that the same measuring method has also been applied to complete PV generators. Another capacity load that uses thyristors (Semikron SKKT 91/08D) as I1 and I2 switches has been constructed at the IES for this purpose. With such equipment, up to Isc 80 A and Voc 1200 V have been simultaneously measured at the Toledo PV plant. Contractual procedure Contractual procedures should be designed in such a way that any controversies concerning the validity of test results are avoided. Otherwise, their application would lead to embarrassing and con¯ictive situations, specially if PV modules acceptance and price are contractually linked to them. Once steadiness is assured by the use of a reference PV module, as described in the preceding section, the main reason for discussions remains at the absolute rating of the last. To rely on independent, external to the project and widely recognised calibration institutions is the best way to overcome this problem. The IES contractual procedure is as follows: (a) All PV modules must be indelible marked. (b) Prior to the tests, four PV modules must be delivered by the manufacturer and calibrated in a widely recognised laboratory (CIEMAT, Madrid; JRC-ESTI, Ispra; Sandia, USA, etc.). (c) One of the four reference PV modules is returned to the manufacturer. Another two will be used in the acceptance tests, acting respectively as reference module and irradiance sensor. The last module is kept reserved, in the event of accidental damage to any of the others. (d) The PV manufacturer must provide a list with the reference or series numbers of all modules submitted to the Quality Control, together with their main electrical characteristic parameters (at least, Voc , Isc , Vm and Im at STC) measured at the production factory. All modules submitted must comply with the following requirement: dierence between the value claimed by the manufacturer, PMTF , and catalogue value must fall within a given percentage, to be speci®ed in contract ( for example, +10%). (e) The IES selects a set of PV modules to be tested (see (b) under `Technical procedure'). Copyright # 1999 John Wiley & Sons, Ltd.



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( f) After the experimental measurements and extrapolation calculations have been done, the acceptance values obtained, PMTA , and the values claimed by the manufacturer, PMTF , are plotted and ®tted to a linear regression equation of the form: PMTA a PMTF

4

The regression coecient, a, is speci®ed using the least squares criterion. Furthermore, the standard error of the estimate, se , is also calculated in order to know how well the line ®ts the data. (g) The linear regression is applied to all PMTF values given by the manufacturer. Final acceptance or rejection of the full set of delivered PV modules follows classical statistical methods.20 A real example, corresponding to one set of PV modules of the Toledo PV plant, will help to clarify this procedure: Delivered PV modules Reference: BP495 Number: n 1257 Power at STC (manufacturer): mean value, PMTF 86.61 W; standard deviation, s 2.12 W Contractual acceptance criteria The bill, issued by the total power as given the manufacturer, was to be paid as initially stated if the total acceptance power was not below 5% of the value claimed by the manufacturer. Otherwise, the bill amount was to be reduced. Test: (see Figure 6) Modules tested: nT 24 Regression coecient: a 0.962 Standard error of estimate: se 1.10 W Interpretation Two important inferences about population characteristics can be extracted, by making use of sample data and simply remembering that the probability of an observation being within three standard deviations from the mean is 0.9974. This means that, with a con®dence degree of 99%, the acceptance power value of any individual PV module is given by PMTA aPMTF + 4% (Note: the con®dence margin is equal to three times the standard error of estimate, 3 1.10 W, which is 3.8% 4% of the mean PMTF value given by the manufacturer). It also means that the acceptance value of the mean power for the . total collection of delivered PV modules lies between 83.32 W+0 p 01% 8332 W 8661 W 0962; p 001% 100 3 se =PMTF n 100 3 110=8661 1257.

Figure 6. PMTA versus PMTF for the tested modules Copyright # 1999 John Wiley & Sons, Ltd.



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It can be observed that the error of the estimate when using the regression line for an individual PV module is relatively important ( 4%), which is consistent with the results presented by other authors.8 Meanwhile, the error of the estimate for the total population ( 0.01%) is negligible. Obviously, more speci®c approaches to the testing of statistical hypotheses (null hypothesis, type I and II errors, etc.)20 can be adopted if needed.

THE IES EXPERIENCE Table I summarises the results of the PV module Quality Control processes already performed by the IES, following the procedure described above. Some aspects should be outlined: (a) In all cases, the values of the regression coecient indicate low deviations between tests and PV manufacturer claims. However, this optimistic scene should be further commented, because PMTF values were given by the manufacturers after they received one of the reference PV modules mentioned in (c) under `Contractual procedures'. As a matter of fact, in some cases the mean value of PMTF was signi®cantly below the value suggested on the commercial information (catalogues). (b) The main trouble arose, years ago, when a PV manufacturer questioned the calibration values of the reference PV modules. The problem ended when it was con®rmed that rating disagreements between the CIEMAT-Madrid and the JRC-Ispra were smaller than +1%. Today, the IES Quality Control procedure is well understood and accepted among the Spanish PV manufacturers. (c) Typical time requirements can be described as follows: Ð Rating of reference PV modules: 20 days Ð Selection of modules for testing: 2 days Ð Testing of selected modules: 20 modules/day Ð Reporting: 3 days Ð Others (transport, etc.): 4 days Assuming proper meteorological conditions and excluding the rating of the reference PV modules, it can be said that the whole procedure can be done in about 2 weeks. Such rating should be separately considered, because it can be done out of the production process of the PV modules to be submitted to the Quality Control. In fact, the same reference modules can be used in several consecutive Quality Control processes of the same PV module type. (d) Excluding the rating of the reference PV modules, the estimated costs for the Quality Control of a collection of 1000 modules (about 20 tested modules) is equivalent to 10±20 days/man. At the current Table I. Main results of Quality Control processes carried out at the IES Project Name Toledo PV *BP *NUKEM IES-BP17 Cies Isl.18 UNIVER19

PV modules

PMTF

Regression

Power* (kWp)

Name

Number Total Tested

Mean (W) Total Tested

s{ (W) Tested

a (mean)

se{ (W)

507.5 462.3 5.8 12.9 67.1

BP495 PP204 BP495 BP275 I106

5826 2118 66 175 660

87.10 218.30 88.01 73.65 101.64

2.302 10.354 1.218 0.933 1.122

0.976 0.992 0.985 0.986 0.989

2.03 7.53 1.27 1.08 2.64

100 84 66 18 80

86.05 215.51 88.01 73.87 101.65

*From manufacturer measurements. {Standard deviation. {Standard error of estimate. Copyright # 1999 John Wiley & Sons, Ltd.



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Table II. Defects leading to PV module rejection (detection by visual inspection) Defect

Rejection criteria

Broken or cracked cells Cells out of line Front surface of cells Inclusions in lamination Bubbles in the encapsulant Front glass Connecting tape Labels Dirty modules Tedlar Connection box

Breaking or spreading of a crack, causing a piece of more than 10% of a cell area to be separated Cells touching each other Very noticeable metal stains Coverage of more than 1% of a cell area If they allow a path between the cells and the frame or edge of the module Broken Torn apart Indelible labels missing Extraneous silicone or encapsulant Damaged or holed Broken or worked loose

IES-UPM fees this represents about 5000±10 000 ECU; CIEMAT-Madrid calibration fees are about 500 ECU per module. (e) In addition to the I±V curve measurements, all modules selected for experimental test are visually inspected. Table II lists the defects leading to a PV module rejection. Up to now, this has never been the case. Finally, Figure 7 shows the measurements phase of a Quality Control process carried out by the IES in July 1997. The picture shows ®ve modules placed in the measurement facility: one for irradiance measurements, the reference module and three modules to be tested (these have been previously exposed to outdoor conditions for at least 1 h, in order to assure they reach thermal equilibrium before measurements are performed). The above-mentioned experience has led us to believe that eective Quality Control of PV modules can be performed in a rather simple way, so that it can be locally implemented in most of the countries currently undertaking PV Rural Electri®cation programmes, as it has already been demonstrated in the case of Brazil.21 Quality Control costs are fully justi®ed for projects up to some tens of kWp, that is, from 1000 systems in the case of Solar Home Systems. Moreover, we think that there are some possibilities of further simpli®cation associated, on the one hand, to the use of the same reference module as irradiance and temperature sensor (through the measurements of its short-circuit current and open-circuit voltage, respectively) and, on the other hand, to the classical one-exponential modelling of the PV module I±V curve. The ®rst reduces hardware requirements, while the second should lead to reductions in the number of the measured I±V points that are necessary to reconstruct the characteristic curve accurately. This way, even automatic data acquisition systems could be probably avoided.

CONCLUSIONS Since 1989, the IES performs Quality Controls of the electrical characteristics of whole collections of PV modules. The particular procedure is based on outdoor I±V measurements. Tested PV modules and a reference unit are quasi-simultaneously measured in real operation conditions. The acquired I±V curves are extrapolated to Standard Test Conditions and results further corrected through comparison with the calibration values of the reference module, previously determined in an independent laboratory having traceable sensors and recognised by both the PV manufacturer and the customer. The IES procedure assumes that uncertainties associated with the extrapolation from real to STC (spectrum and cell temperature eects) and the determination of the real operation conditions (irradiance and ambient temperature) are the same in the tested and reference PV modules, because they are all of the same technology. The steadiness of the method has been experimentally demonstrated. It has been Copyright # 1999 John Wiley & Sons, Ltd.


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Figure 7. Outdoor exposure of PV modules before experimental I±V curves measurements (Quality Control process carried out at the IES in July 1997)

implemented by means of a very simple capacity load and a portable digital oscilloscope with data recording facilities. This paper has described the technical and contractual parts of the procedure, and has given some recommendations for its practical implementation. A real example, corresponding to one set of PV modules of the Toledo PV plant has been presented and discussed, giving particular attention to the statistical interpretation of the results in terms of acceptance or rejection of the PV modules. Finally, the IES experience in Quality Control processes has been presented, together with some comments about time requirements and costs. We believe that this procedure is very eective to assure whether the actual maximum power of the PV modules corresponds to the rated values speci®ed by the manufacturers.

APPENDIX The charging process of a capacitor by means of a PV module (see Figure 8) can be described by the following expressions: Z 1 it dt 5 vt C vt ÿ Voc it Rs it Isc 1 ÿ exp 6 Ns VT Copyright # 1999 John Wiley & Sons, Ltd.



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Figure 8. Capacitor charging process

where Isc , Voc , Rs and Ns are, respectively, the short-circuit current, open-circuit voltage, series resistance and number of series connected cells of the PV module, and VT is the thermal voltage (25 mV at ambient temperature). For simplicity, eects of the parallel resistance of the module have been neglected. Both equations can be combined to generate, through a straightforward numerical simulation exercise, ®nite sequences of N values corresponding to the samples measured during an hypothetically ideal (noisefree) charging process: vn ÿ Voc in Rs 7 in Isc 1 ÿ exp Ns VT vn 1

1 in Dt C

8

Dt being the sampling interval, i.e. the inverse of the sampling frequency, fs . According to the Fourier Series Theory,22 each sequence can be described by a Discrete Fourier Transform, DFT, also with N coecients. This way, the signal frequency distribution of i[n] is given by: 8 N 1 > : 0 rest and a similar equation is valid for v[n]. Unfortunately, noise coming from dierent sources (`white', analog-to-digital conversion, thermal, `shot', etc.) is always present in measurements carried out under outdoor conditions, as it is the case of PV module electrical characterisation. Bearing this in mind, the IES has developed a method of digital processing23,24 that reduces signi®cantly the presence of noise, while improving the accuracy in the determination of the maximum power point of a PV module. It is based on digital ®ltering of real ir[n] and vr[n] sample sequences. As Figure 9 shows, assuming that the sampling frequency enables to capture the most relevant information contained in the analog signals ir(t) and vr(t) Ð see Figure 9(a) and (b), sequences without processing Ð the power of both signals is mainly concentrated in the very low frequency range: as Figure 9(c) and (d) show, more than 99% of accumulated energy falls within the range f 4 01fs . The previous fact, direct consequence of the oversampling of ir(t) and vr(t), also implies that only a few coecients of the DFT of both sequences ir[n] and vr[n] are necessary to reconstruct, with reasonable accuracy, the corresponding analog signals. The novel feature of the IES method lies precisely on this. A special low-pass digital ®lter is applied to both DFTs, which ®lters the measured noise but lets the signal information content pass, thus allowing a more accurate determination of the module maximum power point. The ®ltering criterion is to limit the relative distortion of the maximum power region. To guarantee this independently from the sampled signals, the ®lter is adaptative, that is, its cuto frequency is selected according to the speci®c sequences measured. It can be shown that this cuto frequency, which represents the number of ìnformation' coecients, is very much related to the capacitor charging time which is, in its Copyright # 1999 John Wiley & Sons, Ltd.



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Figure 9. Current and voltage analysis in the time and frequency domain

turn, regulated by the ratio C Voc/Isc . The shorter this ratio, the faster the charge and the larger the number of information coecients. Hence, ®ltered DFTs will give the reconstructed signals after applying them the inverse DFT. The described procedure has demonstrated its usefulness in ®ltering all kind of I±V curves, even anomalous ones. Figure 10 shows an example of this.

Figure 10. I±V curve of a PV module severely shadowed Copyright # 1999 John Wiley & Sons, Ltd.



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