Backstepping PWM Control for Maximum Power Tracking in Photovoltaic Array Systems

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1 2010 American Control Conference Marriott Waterfront, Baltimore, MD, USA June 0-July 02, 2010 ThB12.6 Backstepping PWM Control for Maximum Power Tracking in Photovoltaic Array Systems E. Iyasere, E. Tatlicioglu an D. M. Dawson Abstract A power system consisting of a photovoltaic (P) array panel, c-to-c switching converter an a battery is consiere in this paper. A backstepping PWM controller is evelope to maximie the power of the solar generating system. The controller tracks a esire array voltage, esigne online using an incremental conuctance extremum-seeking algorithm, by varying the uty cycle of the switching converter. The stability of the control algorithm is emonstrate by means of Lyapunov analysis. I. INTRODUCTION olar energy is one of the more attractive sources of S energy toay owing to the rising costs of traitional energy sources, an increase in environmentalism an the inexhaustibility of the source of energy. The primary evice for harnessing solar energy is the solar cell, which uses the photovoltaic effect to transform sunlight into electricity via a semiconuctor evice. Conitions such as cell parameters an atmospheric conitions (temperature an solar irraiation) affect the instantaneous energy generate by a P array as emonstrate by the current-voltage ( i v ) characteristic shown in Fig. 1 which can be mathematically escribe as follows [1]: qv ns AKT i = ni p ph ni p rs e 1 (1) v t where ( ) i t is the P array output current; ( ) is the P array output voltage; n s is the number of cells connecte in series; n p represents the number of parallel moules; q is the charge of an electron; K is the Boltmann s constant; A is the p-n junction ieality factor; an T is the cell temperature in Kelvin (K). The reverse saturation current, I rs, an the photocurrent, I ph, can be expresse as: qego 1 1 T KT Tr T rs = or Tr I I e E. Iyasere is with the College of Engineering an Science, Clemson University, Clemson, SC 2961 USA (corresponing author, ; clemson.eu). E. Tatlicioglu, is with the Department of Electrical an Electronics Engineering, Imir Institute of Technology, Imir, Turkey ( D. M. Dawson is with the College of Engineering an Science, Clemson University, Clemson, SC 2961 USA ( clemson.eu). (2) λ I = I K T T () ( ( )) 100 ph sc l r where I or is the reverse saturation current at the reference temperature, T r ; E go is the ban gap energy of the semiconuctor; I sc is the short-circuit cell current at the reference temperature an raiation; K l is the short-circuit current temperature coefficient; an λ is the solar raiation 2 in mw / cm. Thus, the P array output power, P ( t ), can be calculate as: qv ns AKT P = iv = ni p phv niv p rs e 1 (4) It can be conclue that there exists a maximum power point (MPP) that varies with solar raiation an cell temperature as shown in Fig. 2. To this en, several control approaches have been evelope to optimie the power output when atmospheric conitions are varying. An area of particular importance is the evelopment of online extremum-seeking algorithms which are generally classifie into incremental conuctance (IncCon) [2]-[4] an perturb an observe (P&O) methos [5], [6]. Hussein et al. [2] evelope a maximum power tracking (MPT) technique that is efficient in cases of rapily changing atmospheric conitions. They showe that the maximum power operating point can be tracke accurately by measuring the solar array current an voltage, comparing the incremental an instantaneous conuctances of the P an changing the array voltage accoringly. Leyva et al. [5] emonstrate the global stability of an MPPT algorithm using Lyapunov analysis an applie it to a P system base on the perturb an observe metho. Control techniques use to irectly control photovoltaic characteristics inclue classical control [7]-[9], fuy logic control [1], robust control [6], [10], variable structure [11], [12], an artificial neural networks [1]-[15]. Lian et al. [1] regulate the output power of a solar power generating system using the Takegi-Sugeno fuy metho which inclues using virtual esire variables (Ds). Kasa et al. [6] presents a robust control metho for maximum power point (MPP) tracking in a photovoltaic system where the circuit parameters are uncertain. The MPP is tracke by varying the uty ratio of the switching evice in orer to /10/$ AACC 561

2 Fig. 1. Current-voltage characteristics of a P array Fig. 2. Power-voltage characteristics of a P array Fig.. The system structure of the photovoltaic array system control the array voltage. alenciaga et al. [11] esigne a variable structure controller to regulate the output power of a stanalone hybri generation system consisting of a P array, win turbine, a storage battery bank an a variable monophasic loa. Asie from maximiing the output power, another common application for photovoltaic arrays is loa matching [6], [16], [17]. Saie et al. [16] maximie the output mechanical energy of a DC motor, riving a mechanical loa, connecte to a P array via a c-c converter with varying atmospheric conitions. Yaaiah et al. [6] evelope a controller algorithm to match a solar cell array to a mechanical loa using artificial neural networks. In this paper, a control strategy is evelope to maximie the power of a solar generating system while incluing the ynamics of the DC-DC converter that is assume absent in some papers. The control objective is to etermine the maximum power operating point (MPOP) by tracking a esire array voltage which can be achieve by moulating the pulse with of the switch control signal (increasing or ecreasing the uty ratio of the switching converter). The esire array voltage is esigne online using a filtere incremental conuctance MPP tracking algorithm. The propose strategy ensures that the MPOP is etermine an the tracking errors are globally asymptotically regulate. The stability of the control algorithm is verifie by Lyapunov analysis. The rest of the paper is organie as follows. The ynamic moel of the solar generating system is escribe in Section II. In Section III, a backstepping array voltage tracking controller is esigne along with the corresponing close-loop error system. The stability analysis of the close-loop error system is iscusse in Section I. In Section, the esire array voltage trajectory is generate. Concluing remarks are presente in Section I. II. PHOTOOLTAIC ARRAY SYSTEM DYNAMICS The solar generation moel consists of a P array moule, c-to-c boost converter an a battery as shown in Fig.. The converter transfers power from the P array terminals to the battery bank, inirectly controlling the voltage of the P array panel, v ( t ) an thus the array power generation. The ynamic moel of the solar generation system can by expresse by an instantaneous switche moel as follows: Cv = i il (5) Li = v (1 u) (6) L b where L an il ( t ) represents the c-to-c converter storage inuctance an the current across it; b is the voltage of the storage battery an u is the switche control signal that can only take the iscrete values 0 (switch open) an 1 (switch close). Using the state averaging metho [18], the switche moel can be 562

3 reefine by the average PWM moel as follows: C = I IL (7) LI = D (8) L b where an I are the average states of the output voltage an current of the solar cell; IL is the average state of the inuctor current; D is the limite uty ratio function of the off-state of the switche control signal, u( t ). To facilitate control evelopment, the following moel characteristics are assume: Assumption 1: ( t ), ( ), measurable. I t I an ( ) Assumption 2: C an L are known constants. L t are Assumption : b is moele as a constant value ue to its slow charge ynamics [5]. Assumption 4: I ( t ) is boune provie that ( ) boune. Assumption 5: I constant such that b t is can be upper boune by a positive I < µ where µ. III. CONTROLLER DESIGN The control objective is to maximie the power P t by extracte from a solar generating system, ( ) tracking a evelope esire array voltage,, such that as t. This is achieve by varying D, the uty ratio of the off-state of the switche control signal. Remark.1: The esire array voltage, ( t ), is esigne online using a numerical-base extremumseeking algorithm, as shown in Section I, to maximie the extracte power P ( t ) such that, where is the unknown optimal array voltage, implies that P tens to P max, the maximum power point (MPP). Aitionally, is esigne to be sufficiently ifferentiable, that is ( ), ( ), ( ). t t t A. Error System Development To quantify the state control objective, tracking errors enote by enote by et ( ) an are efine as follows e= (9) = IL ID (10) where ID enotes the subsequently esigne esire storage inuctor current. From the efinition of the tracking errors in (9) an (10), an the system ynamics in (7) an (8), an open loop system is evelope as follows: Ce = C I ID (11) L = D LI (12) b D B. Control Input Design The control inputs will be esigne base on the subsequent stability analysis as well as the structure of the open loop error systems in (11) an (12). The esigne esire storage inuctor current, ID ( t ) is esigne as I = C I k e (1) D e The uty ratio, D is esigne as follows 1 I I L D = LC Lke b C C e k k1 sgn ( ). (14) where ke, k1, k are control gains, an sgn ( ) is the stanar signum function. Substituting (1) an (14) into the open loop error ynamics of (11) an (12), results in the following close loop error system Ce = kee (15) L = k e k LI (16) 1 sgn ( ) I. STABILITY ANALYSIS Theorem 1: Given the close loop error system in (15) an (16), the tracking error signals efine in (9) an (10) are globally asymptotically regulate in the sense that et, t 0 as t (17) ( ) ( ) Proof: A non-negative scalar function, enote by, is efine as = Ce L (18) After taking the time erivative of (18) an making the appropriate substitutions from (15) an (16), the following expression is obtaine = e[ k ] sgn ee k e k1 ( ) LI (19) = k e k k LI (20) From (20), e 1 can be upper boun as follows ke k k LI (21) e 1 If the control gain k 1 is esigne such that k 1 > Lµ 56

4 then from Assumption 5, can be upper boun as follows ke e k (22) From (18) an (22), it is straightforwar to see that et, (9) can be use along et ( ), t ( ). Since ( ) with Remark.1 to show that. Base on the above bouneness statements, (1) can be use along with Remark.1 an Assumption 4 to show that I. After utiliing the fact that ID,, from (10), it is clear that IL. The expression in (14), Remark.1 an Assumptions an 4, can be use along with the above bouneness statements to show that D t. The above bouneness statements can be ( ) utilie along with (7), (8) an Assumption to show that, I L. Above bouneness statements can be use along with Remark.1, an the time erivative of (9) to show that et ( ). The time erivative of (1) can be use along with the above bouneness statements, Remark.1 an Assumption 5 to show that I D. After taking the time erivative of (10), it can be conclue that. After employing a corollary to Barbalat s lemma [19], it is easy to show that et ( ), t ( ) 0 as t.. GENERATING ( ) t ONLINE In Remark.1, it is assume the esire array voltage, t, t t ( t ), can be esigne such that ( ) ( ) an ( ) are boune an, where ( ) t is the unknown optimal array voltage that maximies the solar P t. The extremum-seeking power extracte, ( ) algorithm use in this paper is the incremental conuctance MPP tracking algorithm [2]. Unlike many other MPT algorithms, there is no significant loss of efficiency in cases with rapily changing atmospheric conitions. This algorithm utilies ero slope regulation to track the maximum power point by comparing the incremental an instantaneous conuctances of the P array an varying the esire voltage, ( t ) accoringly. Aitionally, the algorithm accounts for changes in the atmospheric conitions when the array is operating at maximum power by checking if incremental current is nonero. To ensure that, an ( ) t are boune, a filter-base form of the incremental conuctance algorithm is use, wherein at each iteration, the iscrete guess, [ n ], is passe through a set of thir orer stable an proper low pass filters to generate continuous boune signals for, an ( t ). The following filters were use in this stuy = [ n] (2) s 1s 2s s = [ n] s 1s 2s (24) 2 s = [ n] s 1s 2s (25) where s is the Laplace variable, 1, 2, are filter constants an n. The algorithm waits until certain error threshols are met before making the next t n e t t e guess (i.e., if ( ) [ ] 1 an ( ) ( ) 2 then n = n 1 ; where e1, e2 are threshol constants). I. CONCLUSIONS A backstepping PWM control strategy has been evelope for a solar generating system to maximie the power extracte from a photovoltaic array in varying weather conitions. A esire array voltage is esigne online using an extremum-seeking algorithm to seek the unknown optimal array voltage while remaining boune an sufficiently ifferentiable. To track the esigne trajectory, a tracking controller is evelope to moulate the uty cycle of the boost converter. The propose controller is proven to yiel global asymptotic stability with respect to the tracking errors via Lyapunov analysis. REFERENCES [1] K. Lian, Y. Ouyang an W. Wu, Realiation of maximum power tracking approach for photovomtaic array sytems base on T-S fuy metho, IEEE International Conf. on Fuy Systems, vol. 1, no. 1, pp , Jun [2] K. H. Hussien, I. Muta, T. Hoshino an M. Osakaa, Maximum photovoltaic power tracking: an alogrithm for rapily changing atmospheric conitions, Proc. IEE-Gener., Transm., Distrib., vol. 142, no. 1, pp , Jan [] Y. Kuo, T. Liang an J. Chen, Novel maximum-power-pointtracking controller for photovoltaic energy conversion system, IEEE Trans. In. Electron,, vol. 48, no., pp , Jun [4] T. Kim, H. Ahn, S. Park an Y. Lee, A novel maximum power point tracking control for photovoltaic power systems uner rapily changing solar raiation, Proc. IEEE Int. Symp. In. Electron., vol. 2, no. 1, pp , [5] R. Leyva, C. Alonso, I. Quennec, A. Ci-Pastor, D. Lagrange an L. Martine-Salamero, MPPT of photovoltaic systrms using extremum-seeking control, IEEE Trans. Aero Electron. Syst., vol. 42, no. 1, pp , Jan [6] N. Kasa, T. Iia an G. Majumar, Robust control for maximum power point tracking in photovoltaic power systems, Proc. Power Conv. Conf., vol. 2, no. 1, pp , Apr [7] S. J. Chiang, K. T. Chang an C. Y. Yen, Resiential photvoltaic energy storage system, IEEE Trans. In. Electron., vol. 45, no. 1, pp , Jun [8] C. Hua an J. Lin, DSP-Base controller application in battery storage of photovoltaic systems, Proc. IEEE IECON Int. Conf. In. Electron., Contr. Instrum, vol., pp , Aug [9] S. Kim, E. Kim an J. Ahn, Moeling an control of a griconnecte win/p hybri generation system, Proc. IEEE PES Transmiss. Distrib. Conf. Exhib., pp , May

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