Nonlnear dynamcs and bfurcaton analyss of a boost converter for battery chargng n photovoltac applcatons Mohammed M. Al-Hndaw, Abdullah Abusorrah, Yusuf Al-Turk Department of Electrcal and Computer Engneerng, College of Engneerng and Renewable Energy Research Group, Kng Abdulazz Unversty, Jeddah, Saud Araba Daman Gaours Chemcal Process Engneerng Research Insttute, Centre for Research and Technology, Hellas, Therm-Thessalonk, Greece Kuntal Mandal *, Soumtro Banerjee Department of Physcal Scences, Indan Insttute of Scence Educaton and Research - Kolkata, Mohanpur Campus - 741252, West Bengal, Inda and Kng Abdulazz Unversty, Jeddah, Saud Araba Receved (to be nserted by publsher) Photovoltac (PV) systems wth a battery back-up form an ntegral part of dstrbuted generaton systems and therefore have recently attracted a lot of nterest. In ths paper, we consder a system of chargng a battery from a PV panel through a current mode controlled boost dc-dc converter. We analyze ts complete nonlnear/nonsmooth dynamcs, usng a pecewse model of the converter and realstc nonlnear v characterstcs of the PV panel. Through ths study, t s revealed that system desgn wthout takng nto account the nonsmooth dynamcs of the converter combned wth the nonlnear v characterstcs of the PV panel can lead to unpredctable responses of the overall system wth hgh current rpple and other undesrable phenomena. Ths analyss can lead to better desgned converters that can operate under a wde varaton of the solar rradaton and the battery s state of charge. We show that the v characterstcs of the PV panel combned wth the battery s output voltage varaton can ncrease or decrease the converter s robustness, both under peak current mode control and average current mode control. We justfy the observaton n terms of the change n the dscretetme map caused by the nonlnear v characterstcs of the PV panel. The theoretcal results are valdated expermentally. Keywords: Boost converter, Photovoltac source, Battery chargng, Bfurcaton analyss 1. Introducton Dstrbuted power generaton systems that employ hybrd forms of energy producton play an mportant role n today s power grds [Pepermans et al., 25]. Ther man characterstcs are hgh effcency, good power qualty and low carbon emssons. In countres wth a hgh number of sun peak hours, photovoltac (PV) panels can be extensvely used n such dstrbuted energy systems. The produced energy can be drectly njected nto the man grd or t can be stored as another form of energy, lke chemcal energy n batteres Correspondng author s emal address: kmandal198@gmal.com 1
2 Mohammed M. Al-Hndaw et al. and hydrogen tanks, or as mechancal energy n flywheels [Nehrr et al., 211]. In solated areas where the connecton to the power grd s not feasble or s fnancally prohbtng, the most common approach s to store the produced energy n a battery, to be used later by a resdental or ndustral consumer. Electrc vehcle chargng statons can also utlze ths energy. Therefore a very common and mportant applcaton of PV power s to charge a battery. Snce, for a gven amount of ncdent solar energy, PV panels gve out maxmum power only at a specfc voltage and current, t s desrable to use a dc-dc converter as nterface between the PV panel and the battery [Barreto et al., 214]. Ths nterface wll perform the maxmum power pont trackng (MPPT). Unfortunately these converters can exhbt varous nonlnear phenomena that cannot be studed or predcted usng conventonal analyss tools lke small sgnal averaged models [El Aroud et al., 25]. The nonlnear dynamcs of dc-dc converters were frst nvestgated by Verghese et al. and Deane and Hamll, n the late 198s and the early 199s [Verghese et al., 1986; Verghese, 1989; Deane & Hamll, 199], usng a sampled model of the converter, and snce then a lot of work has been done n ths area [Alfayyoum et al., 2; Banerjee & Verghese, 21; Tse, 23]. The nstabltes or bfurcatons that can occur n such converters are closely related to ther performance [Tse & L, 211] and therefore the analyss and thorough understandng of such nstabltes s of great mportance [Gaours et al., 28]. For example, when a perod doublng bfurcaton occurs the current rpple ncreases by a large extent, and ths greatly decreases the overall effcency and lfetme of the system. Even though t s well known that varaton n the nput voltage and output load can cause varous nonlnear phenomena [Chen et al., 27], [Xong et al., 213], lttle work (to the best of the authors knowledge) has been done on the stuaton when the converter s fed by a PV source and the load s a battery. The man ssue that needs to be addressed here, s how the overall system wll behave when, for example, the solar rradaton vares over a wde range through the day, or when the battery voltage changes from a very low value when s dscharged to ts nomnal value when s charged. Ths s the man focus of ths paper,.e., to study the overall dynamcs of the PV-converter-battery system, to demonstrate usng experments, and numercal and analytcal tools the man nonlnear phenomena that can occur, and therefore to provde a desgn methodology so that more effcent power converters can be desgned for such dstrbuted energy applcatons whch reman stable n spte of varatons n the PV nput or the battery load. As a case study n ths paper a boost converter s used, that s fed from a PV panel and charges a battery. The v characterstcs of the PV panel as well as the dfferent battery voltage levels are taken nto account n the overall analyss. As the man chargng phase of a battery s under constant current, a current mode control of the converter s consdered. In ths method of control, the swtch s turned on by a free runnng clock, and s turned off when the nductor current reaches a reference value I ref. In the usual current mode control, ths I ref s controlled by an outer voltage loop so that the output voltage s kept constant. In battery chargng applcaton, there s no need for the outer voltage loop. Instead, the settng of I ref may be used to perform the MPPT of the PV panel. Furthermore, many algorthms of MPPT have been proposed to date, each wth ts own advantages and dsadvantages [Hohm & Ropp, 23; Salas et al., 26; Esram & Chapman, 27]. It has also been shown n [Elgendy et al., 212] that the MPPT algorthm tself can gve rse to oscllatons and other dynamcal phenomena. In ths paper we do not consder the dynamcal phenomena nduced by the MPPT system. Instead we concentrate our attenton on the nherent dynamcs of the converter when the source has the nonlnear characterstcs of a PV panel. 2. System Characterstcs 2.1. The photovoltac panel The PV panel s a current source, whose value depends on the ncdent solar radaton whch vares from tme to tme wthn the day. The charge separaton s carred out by a P-N juncton, whch acts as a forward based dode connected across the current source. The dode current depends on v D, the voltage across the dode, as D = I ( e Av D 1 )
Nonlnear dynamcs and bfurcaton analyss of a boost converter for battery chargng n photovoltac applcatons 3 where A = q γkt e, and I o s the saturaton current of the dode, q s the charge of an electron = 1.6 1 19 coulombs, k s the Boltzmann s constant =1.38 1 23 J/K, T e s the absolute temperature, and γ s the dode dealty factor. The whole system s represented by an equvalent crcut [Chouder et al., 212] as shown n Fg. 1, whch ncludes a shunt resstance R sh (representng the charge recombnaton nsde the solar cells) and a seres resstance (representng the resstance n the current path through the PV panel). v D R s L I ph D R sh v L To the Load Fg. 1. Equvalent crcut of a PV panel. The v characterstcs of the PV panel s gven by the transcendental equaton I ph I ( e A (v L + L R s ) 1 ) v L + L R s R sh = L (1) whch can be solved by the Newton-Raphson method to gve the v characterstcs as shown n Fg. 2. It may be noted that, for every value of the photocurrent, the power output maxmzes at a defnte value of voltage and current the maxmum power pont. PV, A 1.8.6.4.2 2 4 6 v, V PV P PV, W 5 4 3 2 1 2 4 6 v, V PV Fg. 2. Typcal curves for the PV panel current-voltage (v P V P V ) and power-voltage (v P V P P V ) for dfferent values of the photocurrent I ph : 1 A,.8 A,.6 A, and.4 A, showng the maxmum power ponts. The parameters of the panel equvalent crcut are R s =.1 Ω, R sh =1 Ω, T e =3 K, I o =1 9 A, A=3.8647. When the PV panel s connected to a dc-dc converter, the nput current and voltage are constraned to reman on one of the curves shown n Fg. 2 dependng on the ncdent solar radaton at that tme. Ths nonlnear characterstcs of the source, and ts nteracton wth the nonlnear characterstcs of the dc-dc converter are the focal ponts of the nvestgaton here [Gaours et al., 212; De et al., 211; Zhusubalyev et al., 211]. 2.2. The battery Snce the dynamcs of the voltage fluctuaton of the battery s much slower than the clock perod of the converter, n ths work we represent the battery smply as a voltage source, whose voltage vares slowly dependng on the state of charge, but does not vary durng the short duraton of a clock cycle. The dynamcs of the system dependng on dfferent battery voltages are studed, but the battery voltage s not consdered to be a dynamcal varable n the model. 2.3. The dc-dc boost converter wth peak current mode control In a normal boost converter, the nput sde has an nductor whch stores energy when the swtch s on and delvers energy to the load when t s off and the output sde has a capactor whch helps to reduce the
4 Mohammed M. Al-Hndaw et al. output voltage rpple. In case the load s a battery whch s beng charged, there s no need for a capactor at the output stage, as the output s clamped to the battery voltage. In some of the earler lterature [Hamll & Deane, 1992; Deane, 1992; Banerjee & Verghese, 21; Tse, 23] a one-dmensonal dscrete-tme map for the current mode controlled boost converter has been derved under the constant output voltage assumpton. In the battery chargng system, the battery voltage changes at a much slower rate than the system clock, and so ths assumpton s qute vald when we study the cycle-to-cycle dynamcs. Iref Iref Iref n n+1 n n+1 n n+1 t (c) Fg. 3. Three basc types of trajectory between consecutve clock nstants. We frst brefly present the one-dmensonal map of the system when fed from a constant voltage source for the sake of completeness. In that stuaton, durng on perod the current rses wth the slope m 1 =V n /L and durng the off perod the current falls wth a slope m 2 =(V out V n )/L. There can be three basc types of transtons between one clock nstant and the next, as shown n Fg. 3. Let the nductor current at the nth clock nstant be n. If n >I ref m 1 T, then the swtch remans on for a perod T on = (I ref n )/m 1, and remans off for the rest of the clock perod T T on. Therefore at the end of the clock perod the nductor current s n+1 =I ref m 2 (T T on ) =I ref m 2 {T (I ref n )/m 1 } ( = 1 + m ) 2 I ref m 2 T m 2 n. (2) m 1 m 1 However, f n < I ref m 1 T then the swtch remans on for the whole of the clock perod, and the nductor current at the end of the perod becomes n+1 = n + m 1 T (3) If n =I ref (m 1 /m 2 + 1) m 1 T the current reaches zero exactly at the next clock nstant. Therefore f n s greater than ths value, the system goes nto the dscontnuous conducton mode (DCM) and n that case n+1 =. I ref slope=1 I n slope= m 2 /m 1 n+1 a I b I ref Fg. 4. The graph of the map gven by (4).
Nonlnear dynamcs and bfurcaton analyss of a boost converter for battery chargng n photovoltac applcatons 5 The three functonal forms together defne how the nductor current at one clock nstant maps to that at the next clock nstant. The resultng map s gven by ( n + m 1 T, ) for n I a n+1 = 1 + m 2 m 1 I ref m 2 T m 2 m 1 n, for I a < n < I b (4) for n I b where n s the sampled value of the nductor current at a clock nstant, and n+1 s that at the next clock nstant. The borderlne values of the current are I a =I ref m 1 T, and I b =I ref (m 1 /m 2 + 1) m 1 T. It s known that such a converter loses stablty at a duty rato of.5, whch can also be nferred from the above map functon shown n Fg. 4. A pont n the orbt fallng n the horzontal chunk represents operaton n DCM. Ths segment dsappears f I b >I ref. The dervatve at the fxed pont s m 2 /m 1, whch should be less than 1 for stablty. When such a converter s fed from a PV source, the nput voltage s no longer constant, and vares wth the nput current accordng to (1). Ths alters the dynamcs whch wll be analyzed n the next secton. 11 L v L=v n PV Panel L S D Battery Fg. 5. The combned system: the photovoltac source, the boost converter, and the battery. The parameters are chosen as: L=3.125 mh, and clock perod T =.1 ms. v out 3. Dynamcs of the combned system When the three elements the PV panel source, the boost converter, and the battery are combned to form a system (Fg. 5), a few natural ssues arse. Frst, when the battery starts chargng, ts voltage s expected to be low, whch slowly rses as the chargng progresses. Hence the output voltage s a varable quantty that vares very slowly much slower than the clock of the converter. Second, the power fed by the PV panel s also a varable quantty that depends on the ncdent solar radaton (whch determnes I ph ), and the poston of the operatng pont on the v curve. The poston of the operatng pont moves on the v curve durng the on-off cycles of the swtch. The response of the PV panel to changes n current or voltage s nstantaneous, gven by (1). In ths paper we assume that a maxmum power pont tracker s n operaton that sets the reference value I ref of the current mode controller at the maxmum power pont I MPP. Suppose a 12 V battery s beng charged from the converter, and ntally the battery voltage s 1 V. Suppose the ncdent solar radaton s low, say I ph =.15 A. At that value of the photocurrent, the maxmum power pont occurs at V n = 4.546 V and I n =.1 A. If t were a constant voltage supply, wth 4.546 V as the nput voltage, the converter would be unstable and n perod-2 subharmonc (n DCM), as shown n Fg. 6. But f t s connected to a PV panel, the behavor s n CCM, and s stable as shown n Fg. 6. Thus the nonlnear character of the source helps n stablzng the system. Whle n operaton the voltage and the current are constraned to be on the v curve (Fg. 7), and t wll be shown later that ths nonlnearty changes the slope of the dscrete tme map governng ts dynamcs. We now nvestgate the change n dynamcs as the ncdent solar radaton ncreases. For a battery voltage of 1 V, the bfurcaton dagram of the PV-fed system s shown n Fg. 8, whch shows that the
6 Mohammed M. Al-Hndaw et al. Fg. 6. The waveforms of the nductor current: when fed from a constant voltage source wth V n =4.546 V, and when fed from a PV panel, wth the same voltage at the maxmum power pont. Battery voltage: 1 V. Fg. 7. The varaton of the panel output voltage and current on the v curve durng the operaton of the boost converter, when I ph =.15 A and battery voltage s 1 V..4.3 PV, A.2.1.1.1.2.3.4 I ph, A Fg. 8. The bfurcaton dagram of the PV-fed converter wth the varaton of I ph, when V out =1 V and V out =12 V. system does not undergo any nstablty. However, f the battery voltage ncreases to 12 V (Fg. 8), we fnd stable behavor only for low values of I ph when the perod-1 orbt s n DCM. Ths behavor becomes unstable at around I ph =.15 A, through a nonsmooth perod-doublng bfurcaton n whch the perod-2 orbt has one cycle n CCM and the other n DCM. Here the subsystem sequence does change, and so t s a border collson bfurcaton (BCB). As the ncdent solar radaton ncreases, at I ph =.22 A another BCB occurs whch s caused by the saturaton of the duty cycle, where the perodcty does not change. The next perod doublng s also of the BCB type, as t goes from a CCM-DCM perod-2 orbt to a CCM-CCM-CCM-DCM perod-4 orbt. But at around I ph =.27 A, the operaton changes to contnuous conducton mode (CCM). Ths results n a change n subsystem sequence, and hence s a border collson bfurcaton. At that pont we see a sudden change n the nature of the orbt; t changes nto chaos. The chaotc behavor contnues for a large range of I ph. Fnally at hgh values of I ph the behavor saturates and the converter fals to functon. In Fg. 9 we present the bfurcaton dagrams when the converter s fed from a voltage source, and the voltage s vared n the same range correspondng to the maxmum power ponts of the PV source consdered n Fg. 8. It shows that, for low values of the nput voltage the behavor s chaotc, whch becomes perodc when the nput voltage exceeds half the output voltage. The basc reason for the drastc change n the bfurcaton behavor must be contaned n the structure of the map functon when PV nput s used. Fg. 1 shows the graphs of the map functons for the PV-fed system and the constant voltage fed system (4). It shows that the nonlnear characterstcs of the PV panel
Nonlnear dynamcs and bfurcaton analyss of a boost converter for battery chargng n photovoltac applcatons 7 Fg. 9. The bfurcaton dagrams of the converter when fed from a voltage source, when V out =1 V and V out =12 V. The nput voltages correspond to the maxmum power ponts for each I ph n Fg. 8. reduces the slope of the map functon at the fxed pont, and tends to ncrease the stablty. The dfference s more drastc for low values of I ph (and consequently the low values of I MPP ). As the ncdent solar radaton s ncreased, the dfference reduces, and so the system fed from the two types of sources tend to show smlar response. x n+1.25 Constant Source.2.15 PV.1.5.5.1.15 x n.2 x n+1.6.4.2 PV Constant Source -.2.5.1 x n Fg. 1. The graphs of the map functon when PV nput s used, and when constant voltage nput s used, for I ph =.3 A (for whch I MPP =.24 A and V MPP = 4.787 V) and for I ph =.15 A (for whch I MPP =.1 A and V MPP = 4.546 V). Here V out =1 V. Now we consder the stuaton where the battery voltage ncreases gradually due to chargng. To avod complcaton we hold I ph constant at 1 A, and plot the bfurcaton dagram whle V out ncreases as the bfurcaton parameter (Fg. 11, ). It shows that ntally the converter exhbts a stable perod-1 behavor, whch becomes unstable at a perod doublng bfurcaton, but the orbt dverges fast, and hts the border. Subsequently the orbts nclude a skpped cycle. The perod-2 behavor also becomes unstable at around 11 V, and the resultng perod-4 orbt undergoes a border collson. For larger values of V out, the behavor s chaotc. Fg. 11. The bfurcaton dagrams. for the varaton of the battery voltage; for varaton of I ref, I ph =1 A. Even though a smple control strategy could be to fx the I ref at the MPPT, we note that ths strategy allows varaton of the panel voltage and current on one sde of the MPP, and hence the operaton s away from the MPP most of the tme. Therefore t may be logcal to set I ref at a pont above the MPP, but below I ph. What s the rght settng? How does the system behavor vary as I ref s vared? In order to explore
8 Mohammed M. Al-Hndaw et al. these questons, we keep I ph = 1 A and V out = 11 V constant, and vary I ref. The resultng bfurcaton dagram s presented n Fg. 11. It shows that the system s stable for I ref <.8 A, and loses stablty through a perod doublng bfurcaton subsequently gong nto chaos through a border collson bfurcaton. Note that for I ph = 1 A, the system s at the maxmum power pont for I ref =.9 A, and f ths parameter s set at that value, the system wll be unstable. Hence n general t can be concluded that a choce of I ref too close to I ph may make the system unstable, and a suboptmal choce of I ref may be necessary to ensure stablty. 4. Average current mode control L L D v out v n S I ref PI + v con v ramp Fg. 12. The schematc dagram of the boost converter under average current mode control feedng a battery load. In the average current mode control (Fg. 12), the dfference between the nductor current and a reference current s used to produce a control voltage sgnal: t v con =K p (I ref L ) + K I (I ref L ) dt Here the unt of K p s V/A, and that of K I s V/As. The control voltage v con s compared wth a ramp voltage v ramp, where wthn a clock cycle v ramp = V L + V U V L t T When v con v ramp, the swtch s on, otherwse t s off. In ths case also, we frst ask: How wll the stablty of the system change as the ncdent solar radaton changes? For ths, we obtan the bfurcaton dagram for the varaton of I ph (Fg. 13). We assume that the I ref s set equal to the maxmum power pont. Fg. 13. The bfurcaton dagrams of the converter wth average current mode control for varaton of I ph. when fed from a PV source. when fed from a voltage source, wth the values of V n set equal to the maxmum power pont for each value of I ph. Other parameters are: K p =8 V/A, I ref =I MPP, V out =12 V, V L = V and V U =2 V. It s clear that the PV-fed system remans stable for the full range of the photocurrent, but the constant-voltage fed system s stable only for very low values of the nput voltage (whch corresponds to low values of the photocurrent).
Nonlnear dynamcs and bfurcaton analyss of a boost converter for battery chargng n photovoltac applcatons 9 Fg. 14. The bfurcaton dagrams of the converter wth average current mode control for varaton of the battery voltage. when fed from a PV source, when fed from a voltage source, wth V n = V MPP for I ph =.3 A. Other parameters are: K p =12 V/A, I ref =I MPP, V L = V and V U =2 V. Next we ask, how wll the stablty status change when the battery charges up,.e., V out ncreases. We obtan ths by plottng the bfurcaton dagram wth V out as parameter (Fg. 14). We fnd that the PV-fed system remans stable for a larger range of the battery voltage. Now we proceed to obtan the explanaton of the observed bfurcaton behavor n terms of the dscretetme maps of the system wth constant voltage supply and wth PV supply. Snce the expresson for the map for constant voltage supply s not avalable n lterature, we derve t below. The ntegrator gan s chosen wth a vew to nullfy the steady state error wthn a stpulated span of tme followng a dsturbance. We notced that the choce of the ntegrator gan has neglgble effect on the pont of occurrence of the perod doublng bfurcaton. To demonstrate ths, we plot n Fg.15 the bfurcaton curve n the K p versus v out parameter space for varous values of K I. The curves obtaned for varous values of K I practcally overlap, demonstratng the correctness of the observaton. The underlyng reason for the above observaton s that the perod doublng bfurcaton s a fast-scale phenomenon, and the dynamcs of the proportonal controller s faster than that of the ntegral controller. That s why the value of K p does nfluence ths bfurcaton sgnfcantly, but the value of K I does not. Ths mples that, t s possble to smplfy the model by neglectng the ntegrator when developng a Poncaré map to predct the perod doublng bfurcaton. Fg. 15. The regon of stablty n the K p v out parameter space for the average current mode controlled converter wth constant voltage source, for varous values of K I. T on T off L n n+1 n L n L n+1 v ramp v ramp v con v con Case 1 Case 2 v con v ramp Case 3 Fg. 16. The three types of transton wth a ramp nterval.
1 Mohammed M. Al-Hndaw et al. Let the nductor current at the begnnng of a ramp perod be n and that at the end of that ramp perod be n+1. There can be three types of transton wthn a ramp cycle as shown n Fg. 16. In Case 1, f the transton from on to off state happens after a perod T on, then Ths gves K p (I ref n m 1 T on )=V L + V U V L T on (5) T T on = K pi ref K p n V L V U V L T + K p m 1 (6) The value of the current after the on perod s n + m 1 T on. So the value of the current at the end of the ramp nterval s n+1 = n + m 1 T on m 2 (T T on ) = (m ( 1 + m 2 )(K p I ref V L )T m 2 T + 1 (m ) 1 + m 2 )K p T n (7) V U V L + K p m 1 T V U V L + K p m 1 T In Case 2,.e., f K p (I ref n ) V L, the functon s gven by and n Case 3,.e., f K p (I ref n m 1 T ) V U, then n+1 = n m 2 T (8) n+1 = n + m 1 T (9) Now, n Case 1 after the swtch s turned off, f the slope of v con s greater than that of v ramp, then the swtch wll be turned on and off n quck successon whch s an undesrable condton. Hence, to avod that condton, a latch has to be used whch ensures that, f the swtch s turned off, t can turn on only at the end of that clock perod. Wth that functonalty, the v con waveform can reman wholly above v ramp. Notce that the slope of the map functon at the fxed pont s gven by the coeffcent of the n term n (7), and the perod doublng bfurcaton occurs when ts value s 1. Ths gves a closed form expresson for the bfurcaton curve whch s exactly the same as that obtaned numercally n Fg. 15. Fg. 17. The graphs of the map functon for average current mode control when PV nput s used, and when constant voltage nput s used. correspondng to the parameters n Fg.13 and I ph =.5 A; correspondng to the parameters n Fg.14, V out =11.5 V. When a PV source s connected, the structure of ths map wll change due to the v characterstcs of the PV panel whch s obtaned numercally. The two graphs correspondng to the parameter values of Fg. 13 and Fg. 14 are presented n Fg. 17. It shows that the characterstcs of the PV source changes the slope of the graph at the fxed pont, and hence t makes t stable over a larger range of the parameters. 5. Expermental valdaton For the purpose of the expermental nvestgaton, we have used a PV panel that gves an open crcut voltage of 18.4 V and short crcut current of.5 A at ambent sunlght condton. It was connected through a peak current mode controlled boost converter to a 25.2 V battery. The parameters of the panel were
Nonlnear dynamcs and bfurcaton analyss of a boost converter for battery chargng n photovoltac applcatons 11 Fg. 18. The nductor current and the reference current waveforms for constant voltage source of 15.5 V and PV source for the same nput voltage. In both cases I ref =.2 A and I ph =.35 A. Fg. 19. The nductor current and the reference current waveforms for the PV-fed system: for I ref =.22 A and I ref =.3 A. In both cases I ph =.35 A. calculated from the measured v characterstcs of the panel as R s =.8 Ω, R sh = 19 Ω, I o = 2.5 1 7 A, γ = 59.5, and the temperature was T e = 295 K. The boost converter had the parameters as used n the smulaton. We frst nvestgate what happens when the converter s fed from a PV panel that gves a voltage whch s below the crtcal value. The result s shown n Fg. 18. Fg. 18 shows that the converter s unstable when fed from a voltage source of V n =15.5 V (note that the waveform s a perod-2 subharmonc, as the same state repeats after two clock cycles). When the same converter s fed from a PV panel gvng the same value of V n (Fg. 18), t becomes stable. Ths shows that the nonlnear characterstcs of the PV panel has a stablzng effect on the converter s fast-scale dynamcs. Next, we nvestgate the effect of the varaton n reference current, because n a peak current mode controlled converter the maxmum power pont tracker sets the reference current. Comparng the expermental waveforms n Fg. 19 wth Fg. 18 we conclude that an ncrease n the reference current may destablze the converter. We thus see that the results obtaned n the earler sectons are valdated n the experments. 6. Conclusons It s known that dc-dc converters are nherently nonlnear systems that exhbt nonlnear nstabltes for large parameter fluctuatons. For a boost converter fed from a voltage source, f the nput voltage falls below a certan value, the converter undergoes a fast-scale nstablty resultng n subharmonc oscllatons. If such a converter s fed by a PV panel, such large parameter fluctuatons are expected to occur as the solar ntensty vares wdely through the day. The battery voltage, whch s the output voltage of the converter, may also fluctuate dependng on the battery s state of charge. In ths paper we have derved the one-dmensonal maps for both peak as well as the average current mode controlled boost converter and have demonstrated how the converter reacts to the fluctuaton of the
12 REFERENCES external parameters lke the nput voltage and the battery voltage. These smple dscrete-tme nonlnear models may be used to nfer many conclusons of engneerng mportance. In ths work we show that the converter fed from a PV panel may reman stable for a larger range of nput voltage than a converter fed from a voltage source. Usng the one-dmensonal map of the combned system we explan ths observaton by showng that the nonlnear v characterstcs of the PV cell works toward ncreasng the stablty of the converter by alterng the slope of the map functon at the fxed pont. It s a common practce to put a capactor between the PV panel and the converter. Ths s necessary n case of a buck converter because of ts dscontnuous nput current characterstcs. But n a boost converter the nput current s contnuous, and the nput capactor s not really necessary. Stll t has become a common practce n the belef that t wll reduce the chances of nstablty. The above result shows that the fact s n the contrary. If the nput voltage s allowed to fluctuate accordng to the nherent v characterstcs of the PV panel, t n fact ncreases the stablty of the converter. For a peak current mode controlled converter, a reference current settng above the I MPP s desrable for extractng maxmum power from the panel. We show that whle such a settng s desrable from a energy pont of vew, t may destablze the converter. Thus the actual settng has to take both these aspects nto consderaton, the engneerng decson may be taken on the bass of the derved model. We also show that for peak- as well as average current mode control, the converter tends to lose stablty as the battery voltage ncreases. Therefore f t s necessary to charge the battery up to a certan voltage, the other parameters have to be chosen such that the converter does not lose stablty at that output voltage. Acknowledgements Ths project was funded by the Deanshp of Scentfc Research (DSR), Kng Abdulazz Unversty, Jeddah, under grant No. (3-135-1433/HC). The authors, therefore, acknowledge wth thanks DSR techncal and fnancal support. References Alfayyoum, M., Nayfeh, A. H. & Borojevc, D. [2] Modelng and analyss of swtchng-mode dcdc regulators, Internatonal Journal of Bfurcaton and Chaos 1, 373 39. Banerjee, S. & Verghese, G. C. (eds.) [21] Nonlnear Phenomena n Power Electroncs: Attractors, Bfurcatons, Chaos, and Nonlnear Control (IEEE Press, New York). Barreto, L., Praca, P. P., Jr., D. S. O. & Slva, R. [214] Hgh-voltage gan boost converter based on three-state commutaton cell for battery chargng usng PV panels n a sngle converson stage, IEEE Transactons on Power Electroncs 29, 15 158. Chen, Y., Tse, C. K., Wong, S.-C. & Qu, S.-S. [27] Interacton of fast-scale and slow-scale bfurcatons n current-mode controlled DC/DC converters, Internatonal Journal of Bfurcaton and Chaos 17, 169 1622. Chouder, A., Slvestre, S., Sadaou, N. & Rahman, L. [212] Modelng and smulaton of a grd connected PV system based on the evaluaton of man PV module parameters, Smulaton Modellng Practce and Theory 2, 46 58. De, S., Dutta, P. S., Banerjee, S. & Roy, A. R. [211] Local and global bfurcatons n three-dmensonal, contnuous, pecewse smooth maps, Internatonal Journal of Bfurcaton and Chaos 21, 1617 1636. Deane, J. H. B. [1992] Chaos n a current-mode controlled boost dc-dc converter, IEEE Transactons on Crcuts and Systems I 39, 68 683. Deane, J. H. B. & Hamll, D. C. [199] Instablty, subharmoncs, and chaos n power electroncs crcuts, IEEE Transactons on Power Electroncs 5, 26 268. El Aroud, A., Debbat, M., Gral, R., Olvar, G., Benadero, L. & Torbo, E. [25] Bfurcatons n dc-dc swtchng converters: revew of methods and applcatons, Internatonal Journal of Bfurcaton and Chaos 15, 1549 1578. Elgendy, M., Zahaw, B. & Atknson, D. J. [212] Assessment of perturb and observe MPPT algorthm
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