Partial Shading Analysis of Multi-String PV Arrays and Derivation of Simplified MPP Expressions

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1 Partal Shadng Analyss of Mult-Strng P Arrays and Dervaton of Smplfed MPP Expressons Georgos. Psarros, Student Member, IEEE, Efstratos I. Batzels, Student Member, IEEE, and Stavros A. Papathanassou, Senor Member, IEEE Abstract--In ths paper, the electrcal response of a partally shaded photovoltac (P) array, comprsng several strngs connected n parallel, s nvestgated. The P array s smulated by employng an enhanced verson of the wdely used sngle-dode model, reformulated n an explct manner employng the Lambert W functon. The multple maxmum power ponts (MPPs) that appear on the P- characterstc of the array n partal shadng condtons are analyzed, n terms of ther number and propertes. Smplfed emprcal expressons are then derved to calculate the voltage, current and power for each local MPP, at any rradance level and temperature, usng only datasheet nformaton, n a most smple and straghtforward manner, wthout resortng to detaled modelng and smulatons. The derved formulae are valdated usng both smulaton and expermental results. Index Terms-Drect expressons, energy model, explct, local maxma, maxmum power pont (MPP), partal shadng, photovoltac (P) array, power peaks, smplfed expressons. OMECLATURE a Modfed dode dealty factor of the P cell. a bp Modfed dode dealty factor of the bypass dode. α Imp,α Isc Temperature coeffcents of I mp and I sc. b Breakdown correcton factor of the P cell. G Irradance ncdent on cell strng group of P strng, n per unt (p.u.) of the STC value ( W/m ). I mp P strng current at MPP. I mpα P array current at MPPΑ. I mpb P array current at MPPΒ. I ph P cell photocurrent. I SC, P cell strng short crcut current. I s P cell dode saturaton current. I sbp Bypass dode saturaton current. MPP Local maxmum power pont of P strng. MPPΑ Local maxmum power pont A of the P array correlated to MPP of P strng. MPPΒ Local maxmum power pont B of the P array. m Breakdown coeffcent of the P cell. umber of cell strngs n group of strng. umber of seres-connected cell strngs wthn each P module. Manuscrpt receved May, 4, revsed ov., 4, accepted Dec. 9, 4. The work of Mr. G.. Psarros was supported by the State Scholarshps Foundaton (IKY). Mr. E. Batzels s supported n hs PhD studes by "ΙΚΥ Fellowshps of Excellence for Postgraduate Studes n Greece - Semens Program". The authors are wth the School of Electrcal and Computer Engneerng, atonal Techncal Unversty of Athens, Athens 78, Greece (e-mal: gpsarros@mal.ntua.gr; batzels@mal.ntua.gr; st@power.ece.ntua. gr). m umber of P modules n the P strng. p umber of parallel connected strngs n the P array. s umber of seres-connected cells n a cell strng. n a umber of potental local MPPs n a P array experencng several rradance levels. n umber of rradance levels on the P strng. P mp P strng power at MPP. R s Seres resstance of the P cell. R sh Shunt resstance of the P cell. S Sum of rradance levels of all P strngs, except, related to the operatng currents at voltage mp. T c Operatng temperature of a P cell strng. br Breakdown voltage of the P cell. cell P cell voltage. P cell strng voltage. mp,i mp P module MPP voltage and current. mp P strng voltage at MPP. mpα P array voltage at MPPΑ. mpb P array voltage at MPPΒ. oc P module open crcut voltage. str P strng voltage. β mp Temperature coeffcent of mp. β oc Temperature coeffcent of oc. Δ D oltage drop on a conductng bypass dode. κ λ oltage correlaton coeffcent of mpα to mp. Emprcal coeffcent for I mp. Ν I. ITRODUCTIO O unform llumnaton sgnfcantly affects the operaton of photovoltac (P) generators, gvng rse to multple local maxmum power ponts (MPPs), thus reducng the MPP trackng algorthm effectveness []-[3] and system performance [4]. In [], t s shown that two local maxmum power ponts (MPPs) appear on the P- curve of P modules, operatng under two rradance levels. Other studes examne the electrcal response of P strngs under partal shadng condtons, reachng the concluson that multple MPPs are presented n the general case []- []. The mportance of multple MPPs on energy yeld calculaton s hghlghted n [9]. In [], the correlaton between the number of local MPPs and the parameters of the P modules n a partally shaded P strng s nvestgated, concludng that one or two MPPs are presented when two rradance levels are consdered. A complete shadng analyss for the P strng s presented n [8], characterzng and dentfyng any number of MPPs n the general case of multple rradance levels. Moreover, n [] arrays consstng of parallel connected short-strngs and seres connected strngs are examned n terms of energy

2 yeld, concludng that strngs connected n seres operate less effcently under partal shadng condtons. In order to study the operaton of a P array under partal shadng condtons and nvestgate the local MPPs, a sutable smulaton model needs to be adopted, such as the commonly used sngle-dode P cell electrcal equvalent []-[3], whch s sutable for unform operatng condtons. Enhanced and more sophstcated methods are requred to smulate the electrcal response of P arrays under partal shadng, [7], [4]-[6], whch are usually teratve algorthms, ther man drawbacks beng the computatonal complexty and convergence ssues. These drawbacks may be crcumvented by the explct model proposed n [8], n whch the P strng voltage s expressed as an explct functon of ts current usng the Lambert W functon, dspensng wth the need for an teratve soluton. To avod constructng the entre I- curve n order to dentfy the global MPP, smplfed formulae have been proposed n the past, whch drectly evaluate the local and global MPPs. In [], emprcal expressons for the MPP voltage and power are developed for a partally shaded P module, whle n [8] ths approach s extended to ntroduce sem-emprcal formulae for the voltage, current and power of all local MPPs of a partally shaded P strng. Module datasheet nformaton and emprcal coeffcents are utlzed to derve a smple and suffcently accurate MPP estmaton for seres connected P structures (modules and strngs). Concernng mult-strng arrays, n [7] and [8], smple, non-electrcal models are proposed, whch are emprcal n nature and are manly based on a smplfed consderaton of the shadng phenomenon, demonstratng only moderate accuracy. In [9] more accurate expressons are developed, yet lmted to one level of shade. Inter-row shadng of P arrays s examned n [], leadng to a smplfed expresson, whle the same topc s further analyzed n []. However, these approaches do not determne the local MPPs n a quanttatve manner. Ths aspect s studed n [], where closed-form expressons of voltage, current and power are developed for local MPPs of a partally shaded P array, whch present acceptable accuracy, but may not keep up wth the complexty of the developed phenomena. The expressons proposed n [] are based on assumptons for the calculaton of current and voltage at MPP, whch ntroduce naccuraces (overestmaton of current and underestmaton of voltage), as dscussed n Secton. Further, these formulae employ emprcal coeffcents, derved expermentally for the study-case P modules of the paper, whch may not be sutable for other modules. It s therefore apparent that the dentfcaton and quantfcaton of the MPPs developed n mult-strng arrays under partal shadng s an ssue stll open to nvestgaton. Consstent and accurate MPP expressons, wth a general applcablty, are stll mssng from lterature. In ths paper, the response of a P array operatng under non-unform rradance condtons s examned and a thorough analyss of the local MPPs s performed, to shed more lght on ths mode of operaton. The modelng bass for ths analyss s the approach descrbed n [8], modfed for mult-strng P arrays and valdated by measurements. Subsequently, smplfed expressons are developed, whch provde the voltage, power and current of all MPPs of a I ph + I( R s D ) D R sh D negatve voltage term Fg.. P cell electrcal equvalent crcut. - I + cell partally shaded P array, operatng under any temperature and rradance levels, n a smple and drect manner, avodng tme-consumng modelng and smulatons. The expressons ntroduced n the paper apply for the general case of mult-strng arrays, rely only on module datasheet nformaton and are sutable for any commercal crystallne P module. Ther accuracy s valdated by smulaton of numerous shadng scenaros, as well as by expermental results obtaned by outdoor measurements. Ther utlty ncludes P energy yeld estmaton and array optmzaton applcatons, whle they may also prove a valuable tool n the development of shade resstant MPPT algorthms. Ths paper s structured as follows: In Secton II the smulaton model s descrbed and expermentally valdated. The model s utlzed n Secton III for the analyss of P array shadng. The results obtaned are then employed n Secton I to derve smplfed MPP expressons, whch are valdated n Secton by smulaton and outdoor measurements. II. P ARRAY SIMULATIO MODEL A. P Cell Electrcal Equvalent A varety of P cell electrcal equvalent crcuts are reported n the lterature [4], [7]-[], [4]-[6], [3]-[7], wth the sngle-dode model beng the most commonly used [7]-[], [4]-[], [3]-[4]. Although the double-dode model [4], [] s more accurate under certan operatng condtons, the sngle-dode equvalent combnes smplcty wth suffcent accuracy [4] and allows for the development of explct models [8]. In ths study, the electrcal equvalent depcted n Fg. s used. It s based on the sngle-dode model and properly expanded to represent accurately the negatve dode breakdown operaton [4], [6], [6]-[7]. The followng equaton apples: cell I Rs cell I R a s I I ph Is e Rsh m cell I R () s bcell I Rs br term for negatve breakdown voltage where I ph, I s, a, R s and R sh are the parameters of the model as descrbed n [] and b, br, m are coeffcents related to negatve voltage operaton. B. P Array Modelng wth the Lambert W Functon In order to reduce computatonal complexty and convergence ssues related to the transcendental form of (), eq. () s reformulated n [8] and expressed n the explct form =f(i) usng the Lambert W functon: -

3 P arr (kw) P arr (kw) I arr (A) I arr (A) 3 Rsh I ph Is Rs Rsh I Rsh I ph Is I RshI s a cell a W e, I I () ph a br I Rs zr mn, I I ph where W{x} refers to the Lambert W functon and z Rmn to the mnmum real root of the followng equaton, [8]: 4 br z I ph I z bbr z bbr (3) Rsh Rsh A group of seres connected P cells havng a bypass dode connected n parallel s denoted as a cell strng. The exstence of bypass dodes s necessary to prevent hot-spot phenomena due to reversed operaton of shaded cells under non-unform llumnaton condtons. At negatve voltages, the bypass dode conducts and the cell strng voltage s thus lmted to a typcal value of -, dependng on the dode characterst. Eq. (4) gves the cell strng voltage n an explct form, assumng a number of s cells connected n seres. s cell ( I), I ISC, (4) I I SC, abp ln, I I SC, I sbp Assumng that each P module s composed of cell strngs and each P strng comprses m modules, the P strng voltage s expressed as: m ( I) () str The model can be easly extended for the case of a P array consstng of p strngs, as reported n [8]. In order to determne the entre I- curve of the array, () s used for each ndvdual strng and then curve superposton s appled usng lnear nterpolaton. C. Expermental aldaton The modelng method descrbed above s expermentally valdated through outdoor measurements on a P array comprsng strngs connected n parallel, each consstng of P modules (datasheet characterst shown n Table I). The physcal layout of the examned P array s depcted n Fg.. The fve parameters of the model are calculated accordng to [], whereas typcal values for the negatve voltage coeffcents b, br, m are consdered, [4], [6], [6], and bypass dode coeffcents based on [] and [6] are used. In order to measure the P array electrcal response n partal shadng condtons, varous shade patterns were generated usng a semtransparent fabrc materal wth a transmsson rate (TR) of 49%. The I- characterst of the array were measured usng a varable resstor connected at the array termnals. For each pattern, the resstance was vared between mnmum and maxmum, changng the load appled to the P array and thus the operatng pont on the I- characterstc. Durng ths procedure, the voltage and current was recorded usng a portable DAQ measurng system and LabIEW. The short crcut and open crcut ponts, n partcular, were measured by shortng and opencrcutng the array termnals. The same procedure was appled for the expermental valdaton of Secton. bypass dode P-strng P-cell strng P-cell P-array P-module Fg.. Layout of the two-strng P array used n the measurements and ts buldng blocks. TABLE I DATASHEET CHARACTERISTICS OF THE P MODULES USED I THE MEASUREMETS (YIGLI YL-6) Model Type s I sc(a) I mp(a) oc () mp() YL-6 mc-s Fg. 3. Shadng scenaro wth 4 and shaded cell strngs n each of the Measured two P strngs (G=.3 p.u., TR=49%, T c= C). Smulated (a) 3 Measured arr () Smulated (a) Measured Smulated arr () (b) Fg. 4. Measured and smulated I- and P- curves of the shadng 3 scenaro depcted n Fg.3. arr () (b) Measured Smulated 3 arr () 3 An ndcatve shadng scenaro s depcted n Fg. 3 and the respectve expermental and smulated I- and P- curves are llustrated n Fg. 4. The smulaton results prove to be suffcently accurate, a fact verfed by addtonal measurements, as well as n [8]. III. P ARRAY OPERATIO UDER PARTIAL SHADIG In ths secton, the operaton of a partally shaded P array s analyzed, leadng to a systematc characterzaton of the developed local MPPs. What matters n ths process s the extent of shade, n terms of the number of affected cell strngs per P strng, whereas the exact locaton of the shaded modules wthn the array does not affect ts electrcal response [8]. A. P Strng Operaton under Partal Shadng A P strng llumnated at n rradance levels G, = n, sorted n decreasng order (G >G + ), develops up to n local MPPs [8]. At the same tme, n groups of unformly llumnated cell strngs exst, each comprsng cell strngs that operate at the same rradance level G (= n).

4 P (W) I (A) P (kw) I (A) I (A) 4 MPP MPP MPP3 strng MPP4 MPPA MPP MPPA MPPB.. 3 strng () MPP MPP3 MPP MPP4 3 3 () Fg.. I- and P- curves of a P strng llumnated at 4 rradance levels (G =., G =.96, G 3=.6, G 4=.4 p.u. / T c = C). Referrng to a specfc MPP, cell strng groups to operate at the current correspondng to the rradance level G, whle the remanng groups (+ to n) are bypassed by ther conductng bypass dodes. An ndcatve scenaro of a P strng llumnated at 4 rradance levels s shown n Fg., gvng rse to 4 dstnct MPPs. B. Characterzaton of MPPs Appearng n a Partally Shaded P Array Explanng the response of a partally shaded P array may be qute complcated, partcularly when attemptng to estmate the number of local MPPs that may appear on the P- curve and dentfy the orgn of each one of them. The defntons gven n Secton III.A for the sngle P strng are extended to the P array level. Each of the p parallel connected strngs s llumnated at a number of n (= p ) rradance levels G, = n, sorted n decreasng order (G > G +), that gve rse to a maxmum of n local MPPs on the P- curve of the partcular strng. The notaton MPP s used for these MPPs, where the superscrpt denotes the examned strng and the subscrpt the rradance level G. The number of cell strngs n each group, whch experence the same rradance level G, s denoted as. In the followng, superscrpt and subscrpt always refer to the strng and rradance level, respectvely. To facltate understandng, the smple shadng scenaro of Fg. 3 s examned. The I- curves of the two component strngs and the characterstc of the entre array are depcted n Fg. 6, for operaton at rradance levels of p.u. and. p.u., at C. In Fg. 6, strng (red lne) experences two rradance levels, G =. p.u. and G =. p.u., formng two groups of =4 and = cell strngs respectvely, leadng to the appearance of two local maxma MPP and MPP. Smlarly for strng (green lne), =3 cell strngs are llumnated at G =. p.u. and =4 shaded cell strngs at G =. p.u., gvng rse to MPP and MPP. The resultng I- curve of the entre array (blue lne) exhbts 3 local MPPs n total, whle a closer nspecton reveals a correlaton between those and the MPPs of the ndvdual strngs. Specfcally, the array MPP at the hghest current and the lowest voltage (MPPA ) s closely related to the MPP of strng, because the shape of the array I- characterstc n ths regon s dctated by the I- curve of strng, snce the characterstc of strng s practcally flat. Hence, the voltage at MPPA s very close to the voltage mp of MPP (slghtly hgher zoom box n Fg. 6), whle the array current s approxmately twce the current I mp of MPP (the sum of I mp and the current of strng at voltage mp). Smlarly, MPPA of the array s closely related to MPP of strng, appearng at a voltage only slghtly hgher than mp. array strng MPP MPP strng MPP 3 3 () Fg. 6. I- curves of a partally shaded P array and ts two component strngs, for the shadng scenaro of Fg. 3 (G =. p.u., G =. p.u.) array MPPA MPPA MPPA MPPB strng strng MPPA MPP MPP MPPA MPPA MPP MPP MPP MPP 3 MPPB MPP MPP MPP MPP () Fg. 7. I- and P- curves of a partally shaded P array presentng 3 MPPAs and one MPPB. Although the frst two MPPs of the array are related to a specfc MPP of one strng, the last MPP of the array (MPPB) does not follow the same trend, but t comes from the nteracton between the two last MPPs of the ndvdual strngs (MPP and MPP ). When a P strng operates at the rghtmost MPP n, all cells n strng operate at the lowest current mposed by the most shaded cell strng group, snce all bypass dodes are reverse based. In ths case for the array, the voltages of the dfferent MPP n of ndvdual strngs are very close to each other, regardless of the specfc shadng pattern of each strng [8], leadng to a sngle MPP for the entre array, as n Fg. 6. Therefore, two types of array MPPs may be dentfed: One that s related to a specfc strng MPP (belongng n the frst n - MPPs of strng ) and denoted as MPPA, where corresponds to strng and to the rradance level G on strng. The other type comprses a sngle MPP, denoted as MPPB, resultng from the nteracton of the rghtmost MPPs (MPP n) of all strngs. Based on ths characterzaton, the number of local MPPs for a partally shaded array s: a p p p (6) n n n comprsng n - MPPAs for each of the p strngs of the array and a sngle MPPB. otably, ths s the maxmum number of MPPs that may be observed, whle the actual number may be smaller dependng on the shadng scenaro. In Fg. 7, another scenaro of a -strng array s llustrated to further clarfy the prevous analyss. Strng s llumnated at three dfferent rradance levels (G =. p.u, G =.7 p.u. and G 3=.3 p.u., wth =6, = and 3=8 cell strngs respectvely), whle strng experences two rradance levels (G =. p.u and G =. p.u., wth = and =4 cell strngs). Accordng to (6), up to four local MPPs may appear n the P- curve of the array. MPPA, MPPA and MPPA are respectvely related to MPP of

5 I (A) strng, MPP of strng and MPP of strng. MPPB results from the nteracton of MPP 3 and MPP,.e. the last MPPs of each strng. The analyss presented n ths secton expands the concept ntroduced n [], leadng to clear characterzaton of the array MPPs. It s shown that the last MPP of the array (MPPB) dffers from the others, whle the number of potental local MPPs s not ust the sum of the strng MPPs, but t s determned accordng to (6). MPPA array strng strng I. DERIATIO OF EMPIRICAL EXPRESSIOS FOR THE MPPS OF A PARTIALLY SHADED P ARRAY In ths secton, smple explct expressons are derved to calculate the voltage, current and power at each MPP of a partally shaded P array, usng only nformaton provded n the datasheet of the P modules. Smlar expressons presented n [] and [8] deal only wth a P module or strng and they are only applcable at STC temperature ( C). evertheless, the bass for the analyss at P array level s provded by the expressons for MPP of a partally shaded P-strng, derved n [8]: n G mp G oc mp g g D ( ) a g Gg G g g g g Imp G I mp λ b m P I mp mp mp c Expressons smlar to (7) are used n [], however the frst term of (7a) s smplfed and the effect of the extent of shadng s gnored n (7b). A. Formulae for MPPA In order to derve the approprate expressons for MPPA, ts relaton to MPP s taken nto consderaton. ) Calculaton of voltage mpa: As noted n Fg. 6 and 7, MPPA appears at a slghtly hgher voltage than MPP of strng. To dentfy ths offset, the smplfed scenaro of Fg. 8 s examned. Strng s unformly llumnated at G =. p.u., whle strng s shaded at a varable extent, from fully shaded to completely unshaded. The rradance on the unshaded and shaded parts s G =. p.u. and G =. p.u. As the extent of shade changes, MPP, and hence MPPA, shft horzontally n a smlar manner. The rato mpa/ mp s llustrated n Fg. 9 as a functon of the extent of the shade on strng (.e. the fracton of the shaded cell strngs to the number of total cell strngs n the P strng). The calculaton s repeated for dfferent P modules. mpa s always hgher than mp, whle ther rato remans constant at any shade extent hgher than -%. The dfference between mpa and mp s close to 4% for all modules, leadng to the adopton of the coeffcent κ=.4 as typcal. In Fg., the mean rato mpa to mp s plotted when the rradance level G on the unshaded strng vares from. p.u. to. p.u.. Apparently, the devaton between mpa and mp s affected by the operatng current of strng, whch n turn s lnearly related to the rradance G. Hence the followng approxmaton s ntroduced: (7) MPP () Fg. 8. I- curves and MPPs of a partally shaded P array and ts component strngs. Strng remans unshaded, whereas strng s shaded at a varable extent. mpa / mp Shade extent - /( * m ) Fg. 9. Rato mpa to mp as a functon of the shade extent, for the scenaros of Fg.8. Rato shown for commercal P modules and ther mean value (sold lne). mpa / mp G =. pu G =.7 pu G =. pu Shade extent - /( * m ) Fg.. Mean value of rato mpa to mp as a functon of the shade extent, for the scenaros of Fg. 8, assumng varable rradance level on the unshaded part of the array. Same commercal P modules consdered as n Fg. 9. mpa κ G mp (8) where κ=.4 and mp s gven by (7a) for Δ D =. If strng s not unshaded as consdered above, but t s partally shaded at a smaller extent than strng, as shown n Fg. 3, then (8) stll apples, because the operatng current of strng remans proportonal to G, rrespectvely of the mult-step shape of the I- curve. To extend (8) for mult-strng P arrays, the ndcatve case of Fg. s consdered, where the array conssts of three P strngs, llumnated at dfferent rradance levels. Takng MPP as an example, a smlar analyss shows that the voltage offset between mpa and mp s lnearly dependent on the operatng currents of strng and strng 3, and thus on the rradance levels G and G 3. The effect of each strng s cumulatve, leadng to the expanson of (8): mpa 3 κ G G (9) mp Therefore, n the general case of multple rradance levels and a mult-strng P array, the voltage offset between MPPA of the array and MPP of the respectve

6 I (A) 6 3 MPP MPPA array strng strng strng () Fg.. I- curves of a partally shaded P array comprsng three strngs. strng s determned by the emprcal coeffcent κ and the sum S of the rradance levels assocated wth the operatng currents of all other strngs except strng (.e. strngs,, -, +,, p ) at the operatng voltage mpa: where S mpa mp κ S () s g s g g g : s, s r r r r r r G () Concernng the emprcal coeffcent κ, a constant value of.4 s a good approxmaton when the shade extent s greater than %. Below ths threshold, κ vares from to.4 n a non-lnear way (Fg. 9). If a better accuracy s sought, at the expense of smplcty, eq. () below can be used, whch s accurate over the entre range of shade extent: κ.4.4 mn sg, s s s s g g r r r r g g () To obtan the voltage mpa of MPPA, (7a) s substtuted n () and the followng expresson s derved when Δ D =: G mp G oc mpa κ S g (3) g Gg G g In the prevous analyss, the voltage drop on the bypass dodes was neglected for smplcty (Δ D =). If a non-zero voltage drop Δ D s consdered, the complete equaton for mpa s obtaned: G mp G oc mpa κ S g g Gg G g (4) n g g where S s gven by (), κ may be ether consdered constant at.4 or calculated from (), and Δ D can be assumed nearly equal to (or a more accurate value, f avalable). ) Calculaton of current I mpa: Assumng agan the shadng scenaros of Fg. 8, the rato I mpa /I mp s plotted n Fg. aganst the extent of shade and n Fg. 3 aganst the level of rradance on the unshaded strng. It s observed that ts average value remans practcally constant as the shaded area vares (very close to for G =. p.u.), whle t changes proportonally wth the rradance G on the unshaded strng. Ths rato obvously depends also on the rradance G on the shaded strng, D I mpa /I mp Shade Extent - /( * m ) Fg.. Rato I mpa to I mp as a functon of the shade extent, for the scenaros of Fg.8. Rato shown for commercal P modules and ther mean value (sold lne). I mpa /I mp..8.6 G =. p.u. G=.7 p.u. G =. p.u Shade Extent - /( * ) m Fg. 3. Mean rato I mpa to I mp for commercal P modules at the shadng scenaros of Fg.8, wth varable rradance level on the unshaded strng. snce t determnes the current I mp n a lnear way. In fact, the rato I mpa to the nomnal current I mp may be smply approxmated n all cases by the sum of the rradance levels G and G (snce I mp s lnearly dependent on the rradance level G ): Fg.3 Imp G Imp mpa mp mp mp mp I = I +G I G I G I I mpa G G Imp () Eq. () can be readly extended for the case of the multstrng P array of Fg., by observng that the current I mpa s lnearly dependent on the operatng currents of strng, strng and strng 3, and thus the respectve rradance levels G, G and G 3 : ImpA 3 G G G (6) I mp In the general case of multple rradance levels and a mult-strng P array, the followng expresson holds for I mpa: I I ( G S ) (7) mpa mp where S s gven by (), and the term ncluded n the parenthess corresponds to the sum of the approprate rradance levels of all strngs (ncludng the G of strng ). B. Formulae for MPPB ) Calculaton of oltage mpb : By observng the characterst n Fg. 6-8 and Fg., t s evdent that MPPB always les n the same nterval as the voltages mpn of the rghtmost MPPs of the strngs, whose proxmty leads to the formaton of a sngle MPPB for the entre array. Ths proxmty arses from the fact that, regardless of the shadng pattern and ntensty, at MPP n none of the cell strngs s bypassed and ther operatng voltages devate only slghtly dependng on ther rradance level [8]. Ths aspect s graphcally demonstrated n Fg. 4, where the mpn varaton wth the extent and ntensty of shade s shown to be lmted. Therefore, gven the close proxmty of voltages mpn and the unavalablty of a reasonably complex analytcal way to

7 mpn () 7 3 G =. G =. G =.9 G =.3 G = /( * m ) Fg. 4. araton of the voltage mpn wth the extent of the shade on a partally shaded P strng, at dfferent ntenstes. quantfy the dervaton of MPPB from the assocated MPPs of the ndvdual strngs, ts voltage mpb s smply approxmated by the average value of the strng MPP voltages mpn: mpb G G = (7 a) p p mpn p n mp oc g p g G g G g (8) ) Calculaton of Current I mpb : Based on the prevous analyss, I mpb s gven by the sum of the currents I mpn of the component strngs, derved from (7b): n p g g ImpB I mp G n (9) m where λ s an emprcal constant equal to.6 [8]. C. Temperature Effect So far, expressons (4), (7), (8) and (9) provde the voltage and current of all MPPs of a partally shaded P array, consderng the multple rradance levels G on dfferent parts of the array. To take nto account the temperature effect, the followng expressons can be used for the terms I mp, mp and oc : I I α T T (a) mp mp Imp C β T T (b) mp mp mp C β T T (c) oc oc oc C () where α Imp, β mp and β oc are the temperature coeffcents of the MPP current, the MPP voltage and the open crcut voltage, as gven n the P module datasheet, and T = C s the temperature at STC. In the absence of specfc values for α Imp and β mp, α Isc (short crcut current temperature coeffcent) and β oc can be used nstead, as a reasonable approxmaton. In (), a common temperature T c can be assumed for all cells, whle more refned approaches are possble f a sutable thermal model s avalable.. ALIDATIO OF THE MPP EXPRESSIOS IA SIMULATIO AD EXPERIMETAL RESULTS The accuracy of the smplfed formulae ntroduced n the prevous secton s valdated frst by smulaton, employng the explct P array model descrbed n Secton II, and then by outdoor measurements on a real P array. A. Smulaton Results In the smulaton process, a three-strng P array s consdered, comprsng modules per strng, whose characterst are gven n Table I (Yngl Yl-6). The TABLE II MPP ESTIMATIO ERROR OF THE PROPOSED FORMULAE FOR A PARTIALLY SHADED THREE-STRIG P ARRAY Error % MPPs Current oltage Power Rms Max Rms Max Rms Max MPPA MPPA MPPA MPPA MPPA MPPB TABLE III MPP ESTIMATIO ERROR USIG THE FORMULAE OF [], FOR THE SAME PARTIALLY SHADED THREE-STRIG P ARRAY AS I TABLE II Error % MPPs Current oltage Power Rms Max Rms Max Rms Max MPPA MPPA MPPA MPPA MPPA MPPB array experences sx dfferent rradance levels, varyng n the range of W/m to W/m, and three dfferent operatng temperatures, C, 4 C and 6 C, leadng to a total number of 9, scenaros. Estmaton errors of (4), (7), (8) and (9) for all MPPs on the array characterstc are shown n Table II, evaluated aganst the results of the detaled model of Secton II, whch s used to smulate the entre I- curve for each shadng scenaro and dentfy all local MPPs. The rms error of the smplfed equatons s very low, not exceedng 4%, whle maxmum errors up to % are observed. The latter correspond to mnor local MPPs appearng under extreme shadng scenaros, whch are not mportant, as ther actual power s very small and they never consttute the global MPP of the array. Another mportant aspect of the proposed smplfed expressons s ther computatonal effcency. Smulaton of the 9, shadng scenaros of Table II requred only mn, as compared to almost days usng the detaled model of Secton II. For comparson purposes, the formulae proposed n [] are also evaluated for the same scenaros and MPP estmaton errors are presented n Table III. The model of [] employs emprcal expressons to extrapolate mp, I mp, oc and I sc from STC to the actual rradance and temperature; such expressons are derved by measurements on the specfc P module used and cannot be generalzed. Snce such relatons are not avalable for the study-case module (Yngl Yl-6), a proportonal dependence of currents on rradance and voltages on temperature s assumed, as expressed n (). Further, the model of [] calculates multple maxma n the vcnty of the rghtmost MPP, rather than a unque MPPB. Here, the MPP wth the hghest power s selected as MPPB among them and used n the comparson. Comparng Tables II and III, the formulae ntroduced n ths paper appear to be consderably more accurate. Ths s manly due to the underestmaton of voltage ( mpa s assumed equal to mp) and overestmaton of current (I mpa s assumed to be % hgher than I mp) n the model of []. evertheless, t s noted that the largest errors n Table III occur at mnor (low power) MPPs, developng under unrealstc operatng condtons.

8 8 As a further valdaton of the proposed formulae, dfferent commercally avalable P modules are consdered and the shadng scenaros analyss s repeated for the same three-strng array confguraton. Usng the detaled model of Secton II, the entre P- characterstc s determned for each of the 9, shadng patterns. The operatng pont wth the hghest power s taken as the reference GMPP for each scenaro, to evaluate the accuracy of the maxmum local MPP estmate of the proposed expressons. The resultng devatons are gven n Table I, presentng rms and maxmum errors below 3% and % respectvely. B. Expermental Results For the expermental valdaton of the proposed equatons, measurements on an operatng P array are conducted usng the procedure descrbed n Secton II.C. The examned P array comprses strngs connected n parallel, each consstng of Yngl Yl-6 P modules connected n seres. Several shadng patterns are generated usng a semtransparent fabrc wth a transmsson rate of 49%, whle the I- curves are measured usng a varable resstor connected to the array termnals, as explaned n Secton II.C. The ncdent rradance level (G, G on the unshaded part and G, G on the shaded part), P module operatng temperature and shade extent are recorded for each scenaro and llustrated n Table. The local MPPs are dentfed from the measured I- and P- curves for each case. Thereafter, (4), (7), (8) and (9) are appled to evaluate the voltage, current and power at each MPPs, whch are then compared to the measured values. The respectve errors are summarzed n Table I. It s clear that the performance of the proposed smplfed expressons s qute satsfactory n TABLE I GLOBAL MPP ESTIMATIO ERRORS OF THE PROPOSED FORMULAE, APPLIED TO A PARTIALLY SHADED THREE-STRIG P ARRAY, ASSUMIG DIFFERET COMMERCIAL P MODULES Error % at Global MPP P module Current oltage Power Rms Max Rms Max Rms Max Slken SLK6P6L Bosch M Upsolar UP-M4P SCHOTT Perform Suntech STP TABLE EXPERIMETAL ALIDATIO SCEARIOS (P ARRAY SHOW I FIG. ) Scenaro G,G G,G T c ( o C) TABLE I MPP EALUATIO ERRORS USIG THE SIMPLIFIED EXPRESSIOS OF SECTIO I, FOR THE SCEARIOS OF TABLE Error % Scenaro MPPA MPPA MPPB P I P I P I real world condtons, presentng errors lower than % n all scenaros. I. COCLUSIO In ths paper, the electrcal response of partally shaded P arrays, comprsng several parallel-connected strngs, s nvestgated. Under non-unform llumnaton condtons, a P array presents several local MPPs, whch are shown to be closely related to the MPPs of the ndvdual strngs. Based on ths observaton, the maxmum number of array MPPs s determned and the MPPs are characterzed n terms of ther voltage and current. Subsequently, smplfed equatons are derved to evaluate the voltage, current and power of all MPPs of a P array operatng under non-unform llumnaton condtons, n a drect and straghtforward manner, usng only basc datasheet nformaton. The proposed formulae are valdated by smulaton and va outdoor measurements. Due to ther accuracy, smplcty, computatonal effcency and generc formulaton, the smplfed equatons are sutable for fast energy yeld calculatons n shaded P arrays. II. REFERECES [] H. Patel and. Agarwal, Maxmum power pont trackng scheme for P systems operatng under partally shaded condtons, IEEE Trans. Ind. Electron., vol., no.4, pp , Apr. 8. []. Fema, G. Ls, G. Petrone, G. Spagnuolo and M. tell, Dstrbuted maxmum power pont trackng of photovoltac arrays: ovel approach and system analyss, IEEE Trans. Ind. Electron., vol., no.7, pp. 6-6, Jul. 8. [3] K. Ishaque and Z. Salam, A determnstc partcle swarm optmzaton maxmum power pont tracker for photovoltac system under partal shadng condton, IEEE Trans. Ind. Electron., vol. 6, no.8, pp , Aug. 3. [4] S. Slvestre and A. Chouder, Effects of shadowng on photovoltac module performance, Prog. Photovolt.: Res. Appl., vol. 6, no., pp. 4-49, Mar. 8. [] E. Paraskevadak and S. Papathanassou, Evaluaton of MPP voltage and power of mc-s P modules n partal shadng condtons, IEEE Trans. Energy Convers., vol. 6, no. 3, pp , Sep.. [6] M.F.. Tauddn, S.M. Ayob, and Z. Salam, Trackng of maxmum power pont n partal shadng condton usng dfferental evoluton (DE), n Proc. PECon, Kota Knabalu, Malaysa, Dec., pp [7] H. Patel and. Agarwal, MATLAB-based modelng to study the effects of partal shadng on P array characterst, IEEE Trans. Energy Convers., vol. 3, no., pp. 3 3, Mar. 8. [8] E. Batzels, I. Routsolas, and S. Papathanassou, An explct P strng model based on the Lambert W functon and smplfed MPP expressons for operaton under partal shadng, IEEE Trans. Sustan. Energy, vol., no., pp. 3 3, Jan. 4. [9] A. Mak, S. alkealaht, and J. Leppaaho, Operaton of seresconnected slcon-based photovoltac modules under partal shadng condtons, Prog. Photovolt.: Res. Appl., vol., no.3, pp , May. [] A. Mak and S. alkealaht, Effect of photovoltac generator components on the number of MPPs under partal shadng condtons, IEEE Trans. Energy Convers., vol. 8, no. 4, pp. 8 7, Dec. 3. [] A. Mak and S. alkealaht, Power losses n long strng and parallelconnected short strngs of seres-connected slcon-based photovoltac modules due to partal shadng condtons, IEEE Trans. Energy Convers., vol. 7, no., pp , Mar.. [] W. De Soto, S.A. Klen, and W.A. Beckman, Improvement and valdaton of a model for photovoltac array performance, Sol. Energy, vol. 8, no., pp , Jan. 6. [3] H. Tan, F. Manclla-Davd, K. Ells, E. Mulad, and P. Jenkns, A cell to-module-to-array detaled model for photovoltac panels, Sol. Energy, vol. 88, no.9, pp , Sep.. [4]. Quaschnng and R. Hantsch, umercal smulaton of current voltage characterst of photovoltac systems wth shaded solar cells, Sol. Energy, vol. 6, no. 6, pp.3-, Jun. 996.

9 9 [] E. Karatepe, M. Boztepe, and M. Colak, Development of a sutable model for characterzng photovoltac arrays wth shaded solar cells, Sol. Energy, vol. 8, no.8, pp , 7. [6] G. Lu, S. K. guang, and A. Partrdge, A general modelng method for I- characterst of geometrcally and electrcally confgured photovoltac arrays, Energ. Convers. Manage., vol., no., pp , ov.. [7] F. Martnez-Moreno, J. Munoz, and E. Lorenzo, Expermental model to estmate shadng losses on P arrays, Sol. Energy Mater. Sol. Cells, vol. 94, no., pp , Dec.. [8]. Thakkar, D. Cormode,. Lon, S. Pulver, and A. Cronn, A smple non-lnear model for the effect of partal shade on P systems, n Proc. 3 th PSC, Honolulu, HI, Unted States, Jun., pp [9] P. Rodrgo, E. F. Fernández, F. Almonacd, and P.J. Pérez-Hgueras, A smple accurate model for the calculaton of shadng power losses n photovoltac generators, Sol. Energy, vol. 93, pp , Jul. 3. [] K. Brecl and M. Topc, Self-shadng losses of fxed free-standng P arrays, Renew. Energ., vol. 36, no., pp. 3-36, ov.. [] C. Delne, A. Dobos, S. Janzou, J. Meydbray, and M. Donovan, A smplfed model of unform shadng n large photovoltac arrays, Sol. Energy, vol. 96, pp. 74-8, Oct. 3. [] S. Moballegh ang J. Jang, Modelng, predcton, and expermental valdatons of power peaks of P arrays under partal shadng condtons, IEEE Trans. Energy Convers., vol., no., pp. 93-3, Jan. 4. [3] D. D. guyen and B. Lehman, Modelng and smulaton of solar P arrays under changng llumnaton condtons, n Proc. 6 IEEE COMPEL Workshop, Rensselaer Polytechnc Insttute, Troy, Y, USA, Jul. 6, pp [4] M. G. llalva, J. R. Gazol, and E. R. Flho, Comprehensve approach to modelng and smulaton of photovoltac arrays, IEEE Trans. Power Electron. vol. 4, no., pp. 98-8, May 9. [] Z. Salam, K. Ishaque, and H. Taher, An mproved two-dode photovoltac (P) model for P system, n Proc. PEDES & Power Inda, ew Delh, Inda, pp., Dec.. [6] H. Kawamura, K. aka,. Yonekura, S. Yamanaka, H. Kawamura, H. Ohno, and K. ato, Smulaton of characterst of a P module wth shaded P cells, Sol. Energy Mater. Sol. Cells, vol. 7, no.3-4, pp. 63 6, Feb. 3. [7] J. W. Bshop, Computer smulaton of the effects of electrcal msmatches n photovoltac cell nterconnecton crcuts, Sol. Cells, vol., no., pp , 988. Stavros A. Papathanassou (S 93 M 98 SM ) receved the Dploma degree n Electrcal Engneerng and the Ph.D. degree from the atonal Techncal Unversty of Athens (TUA), Athens, Greece, n 99 and 997, respectvely. He was wth the Dstrbuton Dvson of the Publc Power Corporaton of Greece, engaged n power qualty and DG studes. In, he oned the Electrc Power Dvson of TUA, where he s currently an Assocate Professor. Hs research nterests are n the feld of RES and DG, ncludng wnd turbne and P technology, storage applcatons and ntegraton of DG to the grd. In 9- he was a Member of the Board of the Hellenc Transmsson System Operator. III. BIOGRAPHIES Georgos. Psarros (S 4) receved the Dploma n Electrcal & Computer Engneerng, n, and hs M.Sc. degree on Energy Producton & Management, n 4, both from the atonal Techncal Unversty of Athens (TUA), Greece, where he s currently workng towards the Ph.D. degree. Hs research nterests le n the feld of renewable energy sources, especally on photovoltac technology and energy management of non-nterconnected sland grds. msmatched operaton. Efstratos I. Batzels (S 4) receved the Dploma n Electronc & Computer Engneerng from the Techncal Unversty of Crete (TUC), Chana, Greece, n 9, and hs M.Sc. degree on Energy Producton & Management from the atonal Techncal Unversty of Athens (TUA), Athens, Greece, n, where he s currently workng towards the Ph.D. degree. Hs current research nterests nclude renewable energy technologes, especally photovoltac system desgn and smulaton under