OPTIMAL WIRE SIZE FOR PHOTOVOLTAIC SYSTEMS OPERATING AT MAXIMUM POWER POINT: A CLOSED FORM APPROACH

Transcription

1 aer rom: OTIMAL WIRE SIZE FOR HOTOOLTAIC SYSTEMS OERATING AT MAXIMUM OWER OINT: A CLOSED FORM AROACH Mihael M.D. Ross RER Renewable Energy Researh 180 alois Ave, Montréal (Q) H1W 3M5 (514) mihael.ross@rerino.a SESCI 005 Conerene British Columbia Institute o Tehnology Burnaby, British Columbia, Canada August 0-4, 005 ABSTRACT While the urhase o able reresents an admittedly minor art o the ost o installing a hotovoltai system, it is a ost nonetheless, and one that should be examined with rigor by a maturing hotovoltai industry. Using larger ables between and oint o onnetion to the battery or gridtied inverter redues ower es, but inreases the osts o the ables. In the ast, little attention has been aid to the otimization o the sizing o the ables; the eort o doing a simulation study did not aear to justiy the otential gains. This study shows that, in at, a very simle relation an be used to determine the inanially otimum able size. This simle relation is derived rom basi onsiderations o the osts assoiated with wiring es and the wiring itsel. The hallenge is to determine average level o es, sine these will deend on the varying level o sunshine over the ourse o a year. This study uses a simle exonential exression or the requeny distribution o ower outut; this distribution is haraterized by three arameters: the number o daylight hours in the year, the annual outut o the, and the imum ower outut o the. This simle distribution has the advantage o giving rise to an integrable exression or the requeny distribution o ower es in the ables. The result an be inororated into an exression or the total ost o the ables (osts assoiated with urhase and es), whih an then be minimized, yielding a simle exression or the otimum able size. This losed orm solution demonstrates that the length o the able run is irrelevant to the otimal able size. This somewhat ounterintuitive result is exlained in the develoment o the losed orm solution itsel. The reditions o this losed orm solution are omared to simulations to establish its validity. The aliation o the lose-orm relation to the roblem o able seletion is then illustrated with an examle. The method is also used to omment on the Canadian Eletrial Code stiulations or wiring. NOMENCLATURE C Constant in the assumed requeny distribution o outut The resent value o the energy dissiated as heat in the abling v The er unit ost o hotovoltai aaity E Annual energy outut o the Cabling ator, the tyial ower es in the able as a ration o es ourring at STC i Current in the able onneting the to the ower onverter i Average squared urrent outut o the squared l ˆ Length o the able onneting the to the ower onverter Nominal ower at STC 1

2 ower outut o the at a given oint in time ower outut o the at a given time, normalized with reset to nominal ower Average ower outut o the, normalized with reset to nominal ower ˆ ower es in the able when the is oerating at STC Average ower in the ables Instantaneous ower dissiated in the able onneting the to the ower onverter Maximum ower outut o the, normalized with reset to nominal ower R Total resistane o able onneting the to the ower onverter r er unit length resistane o the able onneting the to the ower onverter rˆ Otimal er unit length resistane o the able onneting the to the ower onverter STC Standard test onditions or hotovoltai modules t day Total time during year when urnishes ower, i.e., total time o daylight in year Array voltage γ Constant in the assumed requeny distribution o outut INTRODUCTION Cabling or a hotovoltai system reresents a small, but not insigniiant, art o the ost o a hotovoltai system; its relative imortane tends to inrease with dereasing nominal voltage and when the and the oint o interonnetion are distant rom one another. Cable size is normally seleted by rules o thumb related to the voltage dro in the ables. The Canadian eletrial ode stiulates that, in relation to the system voltage, the voltage dro in the ables must not exeed 5% between the and the oint o interonnetion and 3% between any two juntion boxes in that interonnetion (Canadian Standards Assoiation, 00a); an installer is ree to redue this voltage dro to, say, 1%, i es are to be minimized. The hoie o the otimal able size has reeived little attention. As a onsequene o this lak o analysis, a number o questions requently arise onerning hoie o able, suh as: What is the monetary beneit o shiting to a higher voltage, e.g., 48 instead o 1? I the is very ar rom the oint o interonnetion, should the voltage dro be minimized, e.g., 1%, or ermitted to rise to the imum allowable, 5%? What is the otimal able size or a given installation? These questions an be answered relatively easily using a losed orm solution or hoosing the otimal able size. While the derivation o this solution is somewhat involved, the ultimate result is very simle. OWER LOSSES IN CABLING ower Losses at a Given oint in Time: The ower dissiated in the able onneting the to the ower onverter (i.e., inverter or battery harger with imum ower oint traking),, an be ound rom: i Eq. 1 R Sine ower is the rodut o urrent and voltage, this leads to R Eq. Here is assumed onstant; the validity o this assumtion will be established later. In the meantime, it is suiient to onsider that oeration at an aroriately hosen ixed voltage results in es o only to 4% omared with imum ower oint traking (Jantsh et al., 199), (Freilih and Gordon,

3 1991), so oerationally the two are not that dierent. Equation an be reast using a normalized orm or the outut: ( ˆ ) R Eq. 3 where ˆ is the nominal ower at STC, suh that: ˆ. Eq. 4 Tyial ower Losses in the Cable: While Equation 3 an be used to give the ower es in the able at a given ower outut, it gives no inormation about the ower levels that will be tyial and that should be used when alulating otimal able sizes. That is, the ower will vary ontinuously with the irradiane on the and the temerature, and the requeny distribution o irradiane must be used to determine an average ower. A simle aroah or estimating the requeny distribution o outut is resented by eio and Lund (1994) based on (Bendt et al., 1981). In this aroah, the requeny distribution is assumed to be: γ Ce Eq. 5 ( ) The onstants C and γ are ound rom solving the ollowing equations, exressed in terms o the normalized imum and average ower outut o the : 1 γ 1 e + γ γ γ e 1 Eqs. 6 & 7 γ C γ e 1 The normalized average ower outut o the an be ound rom the annual outut o the and the time during a year when the urnishes ower, i.e., total daylight time: E. Eq. 8 t ˆ day Combining the requeny distribution and the relation or ower, the tyial ower an be ound rom integration: 3

4 0 CR ( ˆ ) R ( ) ˆ 0 γ e ˆ γ CR e γ ˆ R where d γ d γ e C γ + 3 γ γ γ + 3 γ γ Eq. 9 Note that the ower es in the able when it is oerating at STC are given by ˆ ˆ R Eq. 10 and thereore ˆ. Eq. 11 Thus,, the abling ator, indiates the magnitude o average abling ower es as a ration o the es that would our i the always ut out its rated ower. This is interesting and useul. For examle, onsider a where the annual outut o the is 1000 kwh/kw, the outut o the does not exeed its nominal ower, and there are 4000 daylight hours. Then the normalized average ower outut o the,, is 0.5. From this, the onstants C3.7 and γ-3.6 an be alulated and is ound to be 0.11: the average es in the abling are only 11% o those ourring when the is oerating at STC. Thereore, the rules normally used to size the abling essentially assume abling ower es an order o magnitude larger than they are, on average, or this installation! MINIMIZATION OF TOTAL WIRING COSTS Formulation: The objetive o otimizing the wiring is to selet R suh that total wiring osts are minimized. Total wiring osts onsist o 1) the installed ost o the ables as a untion o the resistane, (R ), and ) the ost o the energy dissiated as heat in the s. Fortunately, the ost o the energy dissiated as heat in the s an be exressed in terms o the ost o hotovoltai aaity: ˆ v Eq. 1 In essene, or the resent value o the stream o uture energy beneits oreited as heat in the s we are substituting the resent value o the additional aaity that would generate this same stream o uture beneits. That this is the ase an be shown by onsidering an that on average urnishes x amount o ower to the onverter. Then x ˆ ( ) Eq. 13 so x ˆ Eq. 14 4

5 We an then determine how muh the must be enlarged due to wiring es, by alulating the dierene in size between an with es and an without es: ˆ ˆ ˆ with able es x ˆ without x able es Multilying this by the er unit ost o hotovoltai aaity yields, as in Eq. 1. tot Thus total wiring osts are given by + v R ˆ + v + ( R ) ( R ) Eq. 15 ˆ Eq. 16 This an be exressed in terms o the er unit length resistane o the, r : l r ˆ tot v + ( r ) l Eq. 17 The minimum total wiring osts will be ound by setting to zero the derivative o this exression with reset to r : ˆ dtot d + 0 v Eq. 18 dr dr Note that dividing by l has eliminated it rom Equation 18. Cable Costs: In general, it an be exeted that the er unit ost o abling will have a ixed ortion and a ortion inversely related to the resistane o the able (this being roortional to the amount o oer and insulation used in the able). Thus, wvar ( r ) wix + Eq. 19 r For examle, lotting ost (aording to one Canadian sulier) versus er unit length resistane or RW90 stranded oer able yields Figure 1. The data over 14 AWG u to irular mils a vast range, as suggested by the log sale on the igure. The line it to these data with wix 0.09 $/m and wvar $ ohm/m is nearly eretly oinident with the data; errors, also shown on the igure, are generally less than 6%. It would be ossible to adjust these osts to inlude the ost o onduit and installation. 5

6 Cost er unit Length ($/m) and % Error in Estimate %Error Estimated and Atual Wire Costs er unit Length Cable Resistane (Ohms/m) (Log sale) Figure 1: Cost versus Resistane: Estimated and Atual or RW90 Stranded Coer Cable Otimal Cable Resistane: Combining Equations 18 and 19 leads to an equation or the otimal resistane o the, rˆ : ˆ wvar v 0 rˆ Eq. 0 wvar rˆ ˆ v For examle, at the installation mentioned earlier, roduing 1000 Wh/W and with abling ator o 0.11, wiring osts as in Figure 1, and a aaity ost o $10/W, the otimal er unit resistane is given by the ollowing equation, whih is no more diiult to use than, or examle, a 5% voltage dro rule. r ˆ 49ˆ Eq. 1 It should be noted that there is no reerene to the length o the ondutor in this relation. This demonstrates that able length is not a onsideration in the hoie o otimum size o ondutor. Furthermore, the elimination o l rom Equation 18 is not ontingent uon the validity o the assumed distribution or outut: even i were inorretly estimated, it would still be ossible to eliminate l rom the equation. To some the absene o the length o the able run may seem ounterintuitive or surrising. It is helul to think about this in terms o a unit length o able: this will have a artiular urhase ost assoiated with it and it will ause a ertain energy, with an assoiated ost. The otimal wiring resistane balanes these two osts. A longer length o able multilies eah o these osts by the same ator, so that total osts are higher, but the balane between the two is unhanged. COMARISON WITH SIMULATION The validity o the above aroah in artiular, the use o the exonential requeny distribution and the assumtion that the voltage is onstant was veriied through simulations with the Toolbox simulation akage (Sheri et al., 003). Hourly weather data were used to determine the outut o a single hotovoltai module (see Table 1 or seiiations), oerating at its imum ower

7 oint. For eah simulated year, the integral o the squared urrent was divided by the daylight hours to ind the average squared urrent or the year. This was, in turn, averaged over all the years o simulation or the and orientation. Note that the energy dissiated as heat in the s is equal to the rodut o the resistane (assumed onstant) and the squared urrent. The average imum ower oint voltage, weighted by the ower outut o the rather than time, was also determined in the simulation. Then, rearranging Equation 9, the abling ator,, imlied by the simulation was alulated rom: isquared Eq. ˆ The average squared urrent is written in this euliar way to emhasize that it is not merely the square o the average urrent. arameter Rated voltage Rated urrent Oen-iruit voltage Short-iruit urrent alue A A Nominal Oerating Cell Temerature (NOCT) 45 ºC Oen iruit voltage temerature oeiient Short-iruit urrent temerature oeiient %/ºC %/ºC Eiieny (or ell temerature estimate only) 1 % Table 1: Module Charateristis The CWEEDS data set (Environment Canada, 003) rovided hourly measurements o ambient air temerature and the insolation on the horizontal or s within Canada. Only s having measured, rather than modeled, values or the horizontal insolation were hosen, and only years with a large majority o their data being measured values were used. Thus, or eah o ive s (anouver, Inuvik, Edmonton, Toronto, and St. John s), the simulation ran through between 0 and 46 years o weather data. For loations outside o Canada, hourly data were not available. Rather, average monthly values or temerature and learness index (Duie and Bekman, 1991) served as inuts to the Watgen syntheti weather generator (Watsun, 199). As a hek o the synthesized weather data, the value o the abling ator alulated using the syntheti data was omared to that alulated using CWEEDS data or ive orientations in Toronto and one orientation in eah o Edmonton and Inuvik; the agreement was within several erent. Table omares the values o rom simulation and the above-desribed analytial method. The average relative error is or these s is 7.1%; errors tend to be largest or east and west aing s. This level o auray should be aetable or the uroses o wiring otimization. It should be noted that in alying the analytial method, the values o average imum ower oint voltage, average outut, and imum outut were taken rom the simulation. ALICATIONS Finding Otimal Wire Size or an Installation: Consider an examle taken rom the web o a harge ontroller manuaturer. A nominally 360 W, 1 is loated 60 m rom the oint o interonnetion (i.e., a total wiring distane o 10 m). I the size is seleted to ahieve a 5% voltage 7

8 dro, then the resistane should be Ω/m. The losest size is 4/0, with a resistane o Ω/m. This osts $10.86/m aording to one sulier, and thus wiring osts are $1303. Site Data Azimuth Tilt m Outut Error * º º Wh/W Sim. Analyti % anouver C % anouver C % anouver C % anouver C 90 (W) % anouver C -90 (E) % Inuvik C % Inuvik W % Inuvik C % Inuvik C % Inuvik C % Inuvik C 90 (W) % Inuvik C -90 (E) % Edmonton C % Edmonton W % Edmonton C % Edmonton C % Edmonton C 90 (W) % Edmonton C -90 (E) % Toronto C % Toronto W % Toronto C % Toronto W % Toronto C % Toronto W % Toronto C 90 (W) % Toronto W 90 (W) % Toronto C -90 (E) % Toronto W -90 (E) % Toronto C 180 (N) % St. John s C % St. John s C % St. John s C % St. John s C 90 (W) % St. John s C -90 (E) % Albuquerque, NM W % Albuquerque, NM W % Albuquerque, NM W % Albuquerque, NM W 90 (W) % Albuquerque, NM W -90 (E) % Athens, Greee W % El Fasher, Sudan W % * C indiates CWEEDS data, W indiates tyial year synthesized by Watgen Table : as Calulated by Toolbox Simulation and roosed Method 8

9 Let us now aly the analytial method, and orget or a moment that the Canadian Eletrial Code limits the voltage dro to 5%. We assume that this will be installed near anouver, aing due south, at a tilt angle o 50º, and will ost $10/W. To get a quik estimate o the average outut o the, we use the RETSreen model (RETSreen, 005), with no es in inverter or : it suggests 181 Wh/W. We assume that there are 4000 daylight hours in the year, that the will at imum rodue 360 W, and that its average imum ower oint voltage will be On this basis, we alulate the normalized average ower outut o the,, to be 0.31 and the abling ator,, to be (omare with a simulated value o 0.179, in Table ). Then, realling Equation 0 and the ost inormation o Figure 1, we ind the otimal er unit length resistane: rˆ ˆ wvar v Ω / m The losest size is AWG #4, whih has a resistane o Ω/m. This is hal as large in diameter as the 4/0 suggested by the 5% voltage dro rule, and osts $.36/m, rather than $10.86/m. Thus wiring osts all rom $1303 to $83 a dierene o roughly 0% o the total initial ost o this system! The total wiring osts, that is, with the ost o es inluded, are ound rom Equation 17. With the 4/0, they are $1351, and with the #4, they are $56 again, a very substantial dierene. This is illustrated in Figure, whih shows the wiring urhase ost, the ost o es and the total wiring ost. With the 4/0, wiring urhase osts onstitute virtually the entirety o the total wiring osts, with the ost o es only $48 the equivalent o 5 W o additional aaity. At the otimal size o AWG #4, the ower es are worth $43, nearly equal to the urhase ost /0 Wire Size Ditated by 5% oltage Dro Costs ($) Total Wiring Cost (urhase + Losses) Otimal: AWG #4 500 Wiring urhase Cost Cost o Losses AWG Wire Size Figure : Finding the Otimal Wiring Size or a Nominally 360 W, 1 Array with 10 m o Wiring This examle is robably atyial in that the inanial enalty o ollowing the 5% voltage dro rule is so large. This largely stems rom it being a 1 system, with high urrents in omarison to the size o the system. But the 5% voltage dro rule an lead the system designer astray, even at higher voltages, and not always by seiying a subotimally large. For examle, i the above system was

10 atually a nominally 1000 W, 48 with a 15 m total wiring distane, then the 5% voltage dro rule would ermit an AWG #14 to be used. Reusing the alulations rom the revious examle, or this and orientation: rˆ 53.5 ˆ The otimal resistane is Ω/m, whih is most losely aroximated by AWG #6. On the basis o this simle alulation, the designer would ay $18 more in the urhase ost o the abling but save $15 in redued es. Note also that the otimal wiring size lies in a broad valley. The small errors in the analytial method resented here do not, thereore, muh aet the results. Using the simulated value or rom Table in the above two examles, AWG #4 and #6 would have still been hosen. Beneit o Swithing to a Higher oltage: The irst examle in the revious setion ame rom the web o a ontroller manuaturer. They used this examle to show that a ontroller that ermitted the to oerate at a voltage muh higher than the system voltage would ay or itsel in the redued wiring osts. They argued that with the same 360 W d in a 48 oniguration, wiring savings o around $1000 would result, and their ontroller would still ermit a 1 system voltage. What is the real value o suh a ontroller? The exression or the otimal resistane given above indiates that with a nominally 48, the otimal resistane is Ω/m. AWG #10 would be used (by hane, it would also have been the hoie using the 5% rule). The wiring urhase rie would all by around $00 omared with the #4 that is otimal or the 1. Total wiring osts would be around $140, or a redution o a little under $400. This demonstrates how the aroah develoed here an ailitate the investigation o this tye o roblem. Imliations or the Canadian Eletrial Code: Unortunately, the Canadian Eletrial Code (Canadian Standards Assoiation, 00a) renders moot muh o the above disussion. Rule 8-10 ditates that the voltage dro shall not exeed 5% rom the suly side to the oint o utilization, and not exeed 3% in a eeder or branh iruit. The rationale or this rule is that the voltage dro may result in a lower than aetable aliation voltage [that] will derease the oerating eiieny o eletrial equiment suh as motors, heating systems, and lighting systems. The establishment o riteria or imum allowable voltage dro in a iruit ensures aetable utilization voltages, to obtain otimum erormane rom eletrial equiment (Canadian Standards Assoiation, 00b). This rationale does not aly to hotovoltai systems with imum ower oint traking: over the ourse o a day, natural variation in the irradiane will ause the imum ower oint voltage to vary over a large range, bounded on the low side by zero. A higher voltage dro will lead to more raid hanges in the imum ower oint voltage when irradiane onditions hange, but it seems erverse to limit the natural variation o the system when there is no good tehnial reason that the system should not be able to aommodate it. This is eseially true when the rule an ditate signiiant additional osts that do not add to system saety. A undamental law in the rule is that it relates the hoie o the to its length. This aer has shown that the otimal size is indeendent o the length. The use o the rule will, thereore, neessarily lead to subotimal size hoies in ertain systems. Based on these arguments, the Canadian Eletrial Code should exet hotovoltai systems with imum ower oint trakers rom Rule CONCLUSIONS A simle exonential untion an be used to aroximate the requeny distribution o the outut o a hotovoltai onneted to a imum ower oint traker. Beause it is easily integrated, it an be used to exress, in losed orm, the annual average es in the wiring onneting the and the oint o interonnetion. With urhase osts exressed, with good auray, as a simle 10

11 untion o the wiring resistane, the total wiring osts, omrising es and wiring urhase osts, an be minimized by setting the derivative to zero and solving or the resistane. A unit length o between the and the oint o interonnetion inurs ertain es due to its resistane. The resent value o this stream o es an be ound by determining the urhase ost o the aaity that would generate energy equivalent to these es. The otimal wiring size an be ound by onsidering a unit length o : that is, the length o the wiring run is irrelevant. resently, the Canadian Eletrial Code stiulates that the voltage dro in the wiring between the and the oint o interonnetion shall not exeed 5%, or, deending on the installation, 3% o the system voltage. Sine this relates the hoie o size diretly to the length o the wiring run, it will neessarily lead to subotimal hoie o wiring sizes in many installations. In ertain ases, the dierene between the wiring osts o the otimally sized ondutor and the ondutor hosen on the basis o the 5% rule an be in the neighborhood o 0% o the total osts o the system. Sine the rationale or the Eletrial Code rule does not make sense in the ontext o hotovoltai systems, this artile argues that hotovoltai systems with imum ower oint trakers should be exeted rom Rule 8-10 o the Canadian Eletrial Code. REFERENCES Bendt,., M. Collares-ereira, and A. Rabl. The Frequeny Distribution o Daily Insolation alues. Solar Energy. ol. 7, No. 1.,. 1-5, Duie, J. A. and W. A. Bekman. Solar Engineering o Thermal roesses. New York, N.Y.: Wiley- Intersiene, Environment Canada Atmosheri Environment Servie. Canadian Weather Energy and Engineering Data Sets (CWEEDS Files). Toronto, Ontario: Environment Canada, 003. Canadian Standards Assoiation. Canadian Eletrial Code, art I. Rexdale, Ontario: Canadian Standards Assoiation, 00a. Canadian Standards Assoiation. CE Code Handbook, art I. Rexdale, Ontario: Canadian Standards Assoiation, 00b. Freilih, Je and J. M. Gordon. Case Study o a Central-Station Grid-Intertie hotovoltai System with -Trough Conentration. Solar Energy. ol. 46, No. 5, , Jantsh, Martin, Heribert Shmidt, and Jürgen Shmid. Results o the Conerted Ation on ower Conditioning and Control. roeedings o the 11th Euroean hotovoltai Solar Energy Conerene, Montreux, Switzerland, , 199. eio, K. and. D. Lund. Otimal Sizing o Solar Array and Inverter in Grid-Conneted hotovoltai Systems. Solar Energy Materials and Solar Cells. ol. 3, , RETSreen International. RETSreen International hotovoltai rojet Model. Natural Resoures Canada, 005. Sheri, Farah, Dave Turotte, and Mihael Ross. TOOLBOX: A Comrehensive Set o System Comonents or the Matlab/Simulink Environment. roeedings o the 003 Conerene o the Solar Energy Soiety o Canada In., Kingston, Ontario, Canada, August 18 to 0, 003. WATSUN Simulation Laboratory. WATGEN 1.0 User s Manual and rogram Doumentation. Waterloo, Ontario, Canada: WATSUN Simulation Laboratory, 199. ACKNOWLEDGEMENTS The method desribed here was originally develoed by the author in the ontext o a study unded by the EU-Joule rogram. Dr. Eero artiainen, resently o Gaia Grou Oy, managed the study, whih was onduted at the Advaned Energy Systems Grou, headed by ro. eter Lund, o the Deartment o Tehnial hysis at the Helsinki University o Tehnology. The author also thanks Dave Turotte o the Natural Resoures Canada CANMET Energy Tehnology Centre-arennes or his lariiation o the relevant setions o the Canadian Eletrial Code. 11