Analysis ofthe Effects ofduty Cycle Constraints in Multiple-Input Converters for Photovoltaic Applications

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1 Analysis ofhe Effecs ofduy Cycle Consrains in Muliple-Inpu Converers for Phoovolaic Applicaions Junseok Song and Alexis Kwasinski Deparmen ofelecrical and Compuer Engineering The Universiy oftexas a Ausin edsong@mail.uexas.edu, akwasins@mail.uexas.edu Absrac - A muliple-inpu converer for phoovolaicpowered communicaion sies is analyzed. The sudy shows ha he duy cycles of a muliple-inpu converer, which conrol he power delivered by each source, may have pracical limiaions when each phoovolaic module aemps o achieve is own maximum power poin a he same ime. This paper discusses an alernaive circui ha has wo coupled inducors in a common oupu sage, which may enable he muliple-inpu converer o overcome he limied duy cycles. The relaionship beween he duy cycles and he power supplied by each inpu is derived mahemaically in order o heoreically analyze he effecs given by he limiaions. Power budgeing wih respec o each source is also demonsraed wih simulaions. I. INTRODUCTION Phoovolaic (PV) modules are a commonly suggesed soluion o power elecommunicaion sysems, especially in isolaed locaions [1]. One advanage of PV modules is increased sysem availabiliy by providing a diverse power supply. Hence, PV modules may conribue o meeing one of he mos imporan requiremens in elecom power sysems. However, PV sysems have a relaively high cos which impacs heir widespread applicaion. One alernaive in order o reduce he financial impac of PV sysems cos is o have a modular and scalable design. In his approach, i is no necessary o iniially insall all he PV modules required o mee he expeced load for a sie's life, bu PV modules could be added as he load grows. Alhough his goal seems o be rivial as more PV modules are added, pracical issues make adding PV modules non-rivial. The main issue is ha any given PV module usually says on he marke for only a few years before being disconinued. Thus, i is likely ha a differen model ofsolar panels will need o be chosen ifa plan's solar generaion capaciy is o be increased. Moreover, even ifhe goal is no o increase he plan capaciy, he shor commercialized life of PV modules affec mainenance and repair planning as mos PV manufacurers reserve he righ o replace disconinued PV modules wih differen models ofha originally purchased [2], [3]. If he sysem is designed wih a cenralized archiecure wih a single converer for he enire PV array wih differen PV module models, hen he overall performance of he sysem is ha of he PV module model wih he wors characerisics. One alernaive is o have individual converers for each group ofpv modules ofhe same model. In his way, he varying maximum power poin (MPP) of he differen PV module models could be racked and overall performance would no be affeced. However, his ends o be a cos ineffecive soluion. Anoher approach o inegrae several differen PV modules is o use muliple-inpu converers (MICs) [1]. This mehod has he advanage of being more cos effecive han having separae converer modules for each solar panel, wihou affecing planning flexibiliy. However, MICs have limiaions on he duy cycles [4]-[6]; wih forwardconducing-bidirecional-blocking (FCBB) swiches, he duy cycle ofeach leg is affeced by he duy cycles ofall oher legs wih a higher inpu volage [7]. Hence, he power budgeing of MICs should be considered carefully in order o achieve MPP for each PV module. Pas research on power budgeing was performed on MICs which did no have duy cycle limiaions [1] or which had oher inpus oher han PV modules [8]-[10]. To deermine he effecs of duy cycle limiaions in MICs for PV applicaions, his paper analyzes a MIC ha only has PV modules as power sources. Analysis is conduced wih he derivaion of equaions in order o verify he effecs of duy cycle consrains when each module aemps o reach is MPP. MATLAB simulaions examine he heoreical analysis and show he effecs given by hese limiaions. The feasibiliy of MPP racking wih he limied duy cycles is also explored using a proposed circui which has wo coupled inducors in he oupu sage. The paper is organized as follows. Secion II analyzes he limiaions of a muliple-inpu single ended primary inducor converer (SEPIC) ha can boh sep-up and sep-down he inpu volage. Secion III discusses an alernaive circui o overcome he limied duy cycles and finally, in Secion IV, he simulaion resuls and conclusions of he sudy are discussed. II. ANALYSIS OF A MULTIPLE-INPUT CONVERTER As shown in Fig. 1, he analysis is conduced wih a muliple-inpu SEPIC because is buck and boos characerisic allows racking he enire volage-curren (V-I) or volagepower (V-P) curve ofa PV module.

2 il1 L1 ic1 C1 qo ic2 C2 ic~ iou~ -. L C Vou -. icn C~ Fig. I. A muliple-inpu SEPle h, Assuming ideal componens and a coninuous conducion mode of operaion, he relaionship beween he inpu and oupu can be represened from [8], [II] as where DWe is each inpu's effecive duy cycle, i.e., is own duy cycle less han ha of he inpu leg wih he nex immediae higher inpu volage. An example of he characerisics ofhe swiching funcions and he exisence of an effecive duy cycle D We is shown in Fig. 2 for a woinpu converer wih FCBB swiches. In his figure, ql and q2 represen he conrol signals ofhe FCBB swiches. In his sudy, a wo-inpu SEPIC was considered for he analysis. The dynamic equaions resuling from applying Kirchhoffs curren law a one node of each capacior in Fig. I are (1) (2) { ILl(1 - DUff) - Duff(hz ld = 0 hz(1 - Dze) - Dzeff(ILl h) = 0 -Vc R DD(ILl hz h) = 0 Wih he assumpion Vj > Vb he effecive duy cycles in (3) can be subsiued by When (4) is subsiued in (3), i yields ILl(1- Dl) - D1(hz h) = 0 l:~~1 - (Dz - D1)) - (Dz - D1)(ILl h) = 0 { R (1 - Dz)(hl lcz h) = 0 The firs wo equaions of(5) can be combined ino Vc l: =- = lou R (3) (4) (5) (6) When he hird equaion of (5) is compared wih (6), a relaionship of li. and lou can be verified as follows. As shown below, (6) and (7) can be used o represen lou wih he currens from each inpu. (7) (8) To simplify he sudy, i was assumed ha all componens are ideal and he converer is operaing in coninuous conducion mode. In addiion, i was also assumed ha V 1 is higher han V 2 and hence D(i)e is defined as shown in Fig. 2. Since average seady-sae curren hrough any capacior is zero, he currens ofhe averaged model of(2) resuls in 0 1T I I I 0 2T I I I I I I I I 0 2effT I I I I I 01effT I Fig. 2. An example operaion ofdwell (I is assumed ha V 1 > V 2 ) When V ou is muliplied o boh sides of (8), he oupu power can be derived as To analyze he relaionship beween he oupu power and he duy cycles of each inpu, (1) uses he definiions shown in (4) o derive Vou in he double-inpu case as (10) The oupu power Pou can be represened wih he inpu volages and currens when (10) subsiues Vou in (9). (11)

3 Wih he definiions of he inpu and oupu power, (12) source, he circui in Fig. 4 can be divided ino wo circuis which are shown in Fig. 5. Equaion (15) and (16) show he calculaed equivalen resisance in erms ofduy cycles in boh inpus. (15) he power supplied by each inpu can be derived as D (Vou/ _ V1VouDz ) 1 R R(1- D z ) P 1 = -----~---=- (Dz - D 1 ) (~~ - 1) Pz = Pou - P 1 Equaion (13) and (14) indicae ha he power supplied by each inpu is no only affeced by is own duy cycle, bu also by hose of he oher inpus. Fig. 3, compuer represenaions conduced wih MATLAB, suppors he previous heoreical analysis; as shown, he values of P 1 and Pz vary wih he values of boh D 1 and Dz. For insance, in order o aain 200 was for P 1 and 160 was for Pz, he values of D 1 and Dz can be se as 0.30 and 0.73 respecively. Moreover, (13) and (14) esablish he fac ha he MPP canno be racked if he required effecive duy cycles in boh inpu legs are more han 0.5. For example, when boh inpus need 0.6 of effecive duy cycle o achieve MPP, required Dz becomes according o (4) - which is an impossible value. This example depics he heoreical limiaion ha seems o occur when boh inpus of a wo-inpu converer may seem o demand a sum of effecive duy cycles over 1.0. When he simulaion is exended o a hree-inpu case, he maximum effecive duy cycle of each inpu leg should be limied below 0.33 in order o avoid limiaions, or even less if one of hree inpus requires an effecive duy cycle above his value. Since he conrol ofhe power supplied by each inpu leg is affeced by he duy cycles of he oher inpu legs, he equivalen resisance seen by each source is also affeced by oher inpus. As indicaed in (12) and represened in Fig. 4, he oal oupu power is he sum of he power provided by each inpu. To calculae he equivalen resisance seen from each P1 vs Duy Raio (D1,D2) (13) (14) P2 vs Duy Raio (D1,D2} Fig.3. Simulaionresuls showingpi and P 2 as he funcions of Di and D 2 (I is assumedha Vi = 30.3 V, V 2 = 16.8 V, and R = 10 Q) loul Pi :- Vi.- MIC Voul Pz :- V V Vz Fig. 4. Simplified schemaicof wo-inpuseple R eq1 ~ : Pi R eq2 ~ : P2 Fig. 5. Equivalenresisance seen by each source : P ou _. (16) In order o achieve he MPP, he equivalen resisance should be designed o mee he poin where he PV module produces maximum power. Fig. 6 shows an example of he equivalen resisance ha implies maximum power, i.e., a he curren (4.9 A) and volage (26 V), he oupu power (127.4 W) becomes he greaes. In his example, he equivalen resisance is 5.31 ohms. I is also possible o find differen maching resisances for differen PV modules conneced a differen inpus o aain maximum power wih a MIC. This poin is inroduced and discussed in he following secion. 6 5 ~ 4 ~ e ~ u 2 o o a l :-\~ : ".".;' 5..:.:..,..:::.:. - V-I Curve V-PCurve. <.~-...'.' o..~ '..' <;\e \...,<-<~~....~ \..,:.::::.::q.e Volage [V] '\ 35 \.. Fig. 6. An examplev-i and V-P curveof a PV module \ !. II> 60 s

4 III. DISCUSSION OF A DESIGN ALTERNATIVE If all inpus of a MIC in Fig. 1 achieve MPP for a given load resisance, he oupu volage will be univocally deermined by he sum of all inpu powers a each respecive MPP. All inpu currens and volages will be fixed because hey only depend on each PV module physical characerisics and operaing condiions. Hence, boh I L and I LZ are fixed, and equal D Vou h = (1- D z ) R D ze V ou hz = (1- D z ) R Hence, D and Dz can be obained by solving he sysem of algebraic linear equaions given by (10) and (17) or (18). As a resul, Since lou will end o be zero as R approximae infmiy, Dz will approximae 1 for very large values ofr. Thus, in heory, i would always be possible o achieve he desired duy cycles o reach MPP in all inpus. In pracice, however, he duy cycle Dz could be higher han he maximum recommended duy cycle which is abou 0.85 for converers for curren source inerface as a SEPIC or boos, as high duy cycles could lead o conrol sensiiviy and sabiliy issues. Therefore, alhough heoreically i would always be possible o achieve he required duy cycles o reach MPP in all inpus, in pracice, here is a limiaion. One simple alernaive circui o overcome duy cycle limiaions in muliple-inpu SEPICs is shown in Fig. 7. The proposed MIC uses wo coupled inducors o isolae he inpu and oupu sages. This soluion may achieve he required oupu level as a sum ofhe maximum power from each source wihou exceeding he limied duy cycle in any of he inpu legs. As same as he analysis conduced in Secion II, a woinpu SEPIC was considered for he analysis of he proposed circui. Wih he coupled inducors, he oupu volage Vou,n becomes I L D = I L I LZ lou I L I LZ Dz = I L I LZ lou (17) (18) (19) (20) (21) Since he equivalen resisances and inpu powers ofboh nonisolaed and isolaed muliple-inpu SEPIC are he same when he MPP is achieved, he oupu powers are he same as well. Moreover, he oupu volages - (10) and (21) - become equal when he same load resisance is used for boh cases. Hence, Equaion (23) also sands since he inpu curren of boh cases are equal when he equivalen resisances are he same. For insance, he inpu curren for leg #1 and #2 in boh cases are (23) (24) When (23) and (24) are used o solve for he duy cycles, D,n and Dz,n for he isolaed MIC in Fig. 7 are equal o D,n = Dz,n = nc, 1 (n - l)dz ndz 1 (n - l)dz (25) (26) where n is N / Nz. As shown in (25) and (26), he proposed MIC gives anoher facor - he raio ofn o Nz - o he value of duy cycles. This is an advanage because he level of equivalen resisance can be conrolled by no only he duy cycles ofeach inpu, bu also by he raio given by he number of urns in he windings of wo coupled inducors; Le., by conrolling he urns raio, he new duy cycles may be smaller providing a soluion o overcome conrol limiaions. A numerical example is used in he following o demonsrae he validiy of he heoreical analysis. This example is summarized in Table I. When wo differen PV modules are conneced o a MIC, he required equivalen resisances o achieve MPP are differen for each inpu [12], [13]. For he MIC ha has no coupled inducors, he required duy cycles o reach MPP can be calculaed using (19) and (20). On he oher hand, he required duy cycles of MIC wih coupled inducors can be deermined by applying (25) and (26). For he calculaion, i was assumed ha N: Nz equals 1:2 or 1:3, V = 41 V, Vz = 17.1 V, and R = 120 Q. The duy cycles ofboh MICs were se o mee he equivalen resisance o achieve MPP for each conneced PV module. As shown in Table I, Dz in he case wihou coupled inducors is oo large TABLE I COMPARISON IN DUTY CYCLES BETWEEN Two MICs Inpu Maximum Model V wih wih Power MPP I MPp Req,MPP R LOAD wihou Number coupled inducors coupled inducors coupled inducors (N 1 : N 2 = 1: 2) (N 1:N2 = 1: 3) 1 SPR-220-BLK [12] 220W 41 V 5.37 A : : : NE-80EJE [13] 80W 17.1 V 4.67 A 3.660

5 - - il1 L1 ic1 C1 qo - ic2 C2 ic! iou! -icn C~ N1 N2 C Vou ili Fig. 7. Proposed muliple-inpu SEPIC for pracical purposes. However, he duy cycles of a MIC become smaller when coupled inducors are placed in common oupu sage. This resul can be exended o oher cases which have duy cycle consrains and he possibiliy of dividing an inducor ino coupled inducors. By changing he urns raio, MICs may have smaller duy cycles corresponding o he MPP and hence can provide a soluion for he pracical duy cycle limiaions. IV. CONCLUSION A sudy of duy cycle consrains in MICs for PV applicaions was presened. The analysis used a muliple-inpu SEPIC o show ha he limied duy cycles in many MIC opologies may pracically preven MPP racking in all inpus. This is because he power supplied from each inpu is no only affeced by is own duy cycle bu also he duy cycles of all oher inpus. The resuls of mahemaical derivaion and compuer simulaion suppor he heoreical analysis. This paper also discussed a soluion ha may overcome he pracical duy cycle limiaions of MICs by replacing he inducor in he common oupu sage wih wo coupled inducors which enable he MIC o have one more design facor o faciliae reaching he equivalen resisance seen by each source. The coupled inducors also provide he flexibiliy ofchoosing he smaller values for duy cycles o achieve MPP, which may be a way o avoid exceeding pracical duy cycle limis. An example was used o demonsrae he heoreical analysis. The calculaion resuls showed ha he proposed MIC may achieve MPP wih smaller values of duy cycles, which can be an answer o he pracical limiaions driven by inpu legs' duy cycle inerdependencies. This example shows ha he proposed MIC is suiable for use as a highperformance DC-DC converer for various kinds of PV modules ha require differen maximum power operaing poins in various condiions. [5] A. Kwasinski and P.T. Krein, "Sabilizaion of consan power loads in De-De converers using passiviy-based conrol," in Rec. INTELEC 2007, pp [6] A. Kwasinski, Feasibiliy muliple-inpu converers See March issue of he power elecronics ransacions [7] A. Kwasinski and P.T. Krein, "A Microgrid-based Telecom Power Sysem using Modular Muliple-Inpu DC-DC Converers," in Rec. INTELEC 2005, pp [8] N.D. Benavides and P.L. Chapman, "Power budgeing of a mulipleinpu buck-boos converer," IEEE Transacions on Power Elecronics, vo1.20, no.6, pp , Nov [9] Y.-M. Chen, Y.-C. Liu, F.-Y. Wu, and Y.-E. Wu,, "Muli-inpu converer wih power facor correcion and maximum power poin racking feaures," Applied Power Elecronics Conference and Exposiion, APEC Seveneenh Annual IEEE, vol.l, no., pp vol. I, [10] M. Veerachary, "Muli-inpu inegraed buck-boos converer for phoovolaic applicaions," Susainable Energy Technologies, ICSET IEEE Inernaional Conference on, vol., no., pp i, Nov [II] S.H. Choung and A. Kwasinski, "Muliple-inpu DC-DC converer opologies comparison," Indusrial Elecronics, IECON h Annual Conference ofieee, vol., no., pp , Nov [12] hp:// 1,2009]. [13] hp:// 1, REFERENCES [I] B. Le and A. Kwasinski, "Analysis of a flexible and rugged phoovolaic-based power sysem," in Rec. INTELEC 2008, pp [2] hp://solarechpower.com/webl/doc/warrany.pdf[jul. I, 2009l [3] hp:// DWarrany.pdf[Jul. I, 2009l [4] A. Kwasinski and P.T. Krein, "Muliple-inpu de-de converers o enhance local availabiliy in grids using disribued generaion resources," in Proc. APEC 2007, pp.l657-i663.