Nonlinear Controller of Single-Stage Photovoltaic Multicellular Inverter System

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1 Nonlinear Controller o Single-Stage Photovoltaic Multicellular Inverter System Taghaoui Chaimaa Abouloia Abdelmajid Elallali Aicha Mchaouar yousse Hamdoun Abdelati LTI Lab HASSAN II University o Casablanca Faculty o Science Ben M'sik Casablanca Morocco taghaouichaimaa@gmail.com abouloia@yahoo.r aichaelallali@gmail.com unsmchaouar@gmail.com alhamdoun@yahoo.r Lachkar Ibtissam LISER Lab HASSAN II University o Casablanca ENSEM Casablanca Casablanca Morocco lachkaribtissam@gmail.com Abstract In this paper we are considering the problem o controlling a single-phase -celles inverter with LC output ilter connected photovoltaic systems. The control objectives are: i) achievement o the maimum power point tracking (MPPT) or the PV array ii) ensuring a good convergence o the voltages across the lying capacitors iii) generating a perectly sinusoidal voltage to the inverter output with amplitude and requency ied by the reerence signal. The controller is designed using the backstepping approach in its adaptive versions and the Lyapunov stability argument. The whole system is described by 6 th order nonlinear mathematical model and controlled by using the backstepping approach and tools rom Lyapunov stability. To demonstrate the perormance o methods some results o the overall system simulations are presented under MATLAB/Simulink sotware. Keywords Multicellular Converter MPPT Lyapunov stability Backstepping approach.. INTODUCTION In the last years new energy sources have been proposed and have been developed due to the dependency and constant increase o costs o ossil uels. On other hand ossil uels have a huge negative impact on the environment. In this contet local generation o heat and electricity and the local use o renewable energy resources are considered as some o the most promising options to provide a more secure clean and more eicient energy supply. Among these renewable energy sources photovoltaic (PV) technology has received a great attention because o distinctive advantages such as simplicity o allocation high dependability low maintenance absence o uel cost and lack o noise. In addition to these actors there are other advantages such as the declining cost and prices o solar modules. Hence the need to master series connection has grown because o the increased market o high - power applications and the signiicant decrease in semiconductors perormance with the voltage ratin g. Thus twenty years ago Meynard and Foch (99a) (99b) have proposed the multicellular also called converter lying capacitor multicellular (FCM) as a multilevel converter. Recently this topology becomes one o the solutions used or the conversion o energy or strong powers in the industrial workplace. This kind o power converters is used in large scale PV applications or its superior advantages such as lower switching power dissipation lower harmonic distortion and lower electromagnetic intererence (EMI) outputs. Several strategies or control have been proposed in the literature in order to improve the characteriation o this converter or eample: sliding modes in (Morvan et al. 00; Amet et al. (0 0); Djemaï et al. 06) hybrid 85

2 control in (Elmetennani et al. 06; Benmiloud et al. (06a 06b)) predictive control in (Patino et al. (0080); Deay et al. ( )) and passivity based control in (Cormerais et al. 008). In general the power converter interace rom the DC source to the load or to the grid consists o two-stage converter: the DC/DC converter and the DC/AC converter. The DC-DC converter which is used to buck boost buck-boosts the PV output voltage o PV panels as well as MPPT. In some paper such as (Elmetennani et al. 06) authors suggested the multicellular inverter instead o a classical DC DC converter or boosting the output voltage with sel MPPT capability to eliminate DC DC converter to take advantage o the properties o its particular topology by sharing the constraints in voltage while working in high voltage installations. In the classical solution with two-stage converter the DC/DC converter requires several additional devices producing a large amount o conduction losses sluggish transient response and high cost while the advantages o the single stage converters are: good eiciency a lower price and easier implementation. Despite the eectiveness o this approach two -stage inverter systems tend to be eceedingly large and epensive. A single-stage topology could help to reduce the number o components thereby reducing the physical sie increasing eiciency and acilitating implementation (Wu et al. 0). An alternative solution could be the use o a single-stage converter where the DC/DC converter is avoided and in order to ensure the necessary DC/DC voltage level the PV array can be a string o PV panels or a multitude o parallel strings o PV panels. In this paper a new coniguration o single phase single-stage inverter system based on the FCM converter is proposed to increase the number o output voltage levels and as a result reduce the output voltage THD with reduced ratings and losses o lying capacitors and semiconductors. We propose a nonlinear control o single-stage photovoltaic multicellular inverter system using backstepping approach and Lyapunov stability tools. The rest o the paper is organied as ollows: the description and the modeling o the system are presented in section ; section is devoted to the controller design. Finally the controller perormances are illustrated through simulations under MATLAB/Simulink sotware in Section.. SYSTEM DESCRIPTION AND MODELING. Presentation o PV system The proposed power system presented in igure is composed o a PV source with two DC link capacitors C and C on the DC side a multicellular inverter and a low-pass output ilter on the AC side in order to minimie the harmonics distortion o current and voltage. The structure o the serial multicellular inverter comprises elementary switching cells in series each switching cell o the converter is composed o pairs o complementary switches and controlled by a binary input signals where ( 0 ) indicates that the i th upper switch is closed i and i i (open) and the i th lower switch is open (closed). Between the two cells a loating capacitor is inserted or high input voltage constraint share among the cells. The sae and proper unctioning o the power converter directly depends on the suitable distribution o the voltages across each cell. That is why it is important to realise an internal regulation o the voltages across the terminals o the two capacitors. i ipv C V pv C V V µ µ C µ V c µ C µ Vc µ I C V0 PV Generator DC Link Inverter Figure. PV system with -cells inverter Filter Load 85

3 . PV array Model Photovoltaic cells use the photovoltaic eect to convert sunlight irradiations energy to electric energy. Thus the PV model used in this paper is based on the equivalent circuit (Bonkoungou et al. 0) shown in Figure. This is the simplest equivalent circuit o a solar cell. The current source and diode represent the ideal behaviour o a solar cell and the series and shunt resistors are used to model real-world losses such as current leaks and resistance between the metallic contacts and the semiconductor. R s I I ph I d I R p + V - where Figure. PV module equivalent circuit Iph is the photocurrent (generated current under a given radiation). Rs is the series resistance. Rp is the shunt resistance. I denotes the PV array output current. V is the PV output voltage. Using Kirchho s current law the photovoltaic current equation can be represented by the equations (a -d) as ollows: I I I I (a) ph d r V I R s V I R s I I ph I0 ep (b) Vt Rp where I I 0 ph V t N KT q T qe GO I 0r ep (c) Tre K Tre T I k T T (d) SC i re 00 where: T r is the reerence temperature; T is the cell temperature; K is the Boltman s constant and q is the electron charge; K i is the short circuit current temperature coeicient at I sc ; λ is the solar radiation; γ is the ideality actor; E GO is the band gap or silicon. I o I or and I sc represent respectively the cell reverse saturation current the cell saturation current at T re and the short circuit current at 985K and kw/m. The photovoltaic generator (PVG) (also called PV array) is an assembly o many PV panels connected in Ns series and Np parallel panels in order to provide the desired values o output voltage and current required or particular applications. The output voltage and current o a PVG can be given by the ollowing equations: Vm N sv (e) Im N I () p The PV array module considered in this paper is composed by a solar panel array o solar panels. The corresponding electrical characteristics are listed in Table. 855

4 . PV system modeling Table. Electrical speciications or the solar module Parameter Symbol Value Maimum Power P m 00 W Short circuit current I sc 8. A Open circuit voltage V oc.9v Maimum power voltage V m 6. V Maimum power current I m 7.6A Number o series cells N sc 5 Applying Kirchho s laws the general model o the PV chain with cells converter under study is given by the ollowing set o dierential equations: v0 C v 0 i (a) R L i = - v - v v - v (b) c c c 0 C v ( - ) i (c) c - C v i (d) C v i (e) pv i v i pv i C i where () 0 i i v i pv 0 v v and i v ci represent respectively measured values o the conductor current the C i and current generated by PV array the output voltage the voltage across the Dc link capacitor capacitors i Flying capacitor C. R C and L represent respectively the resistive load the ilter capacitor and the ilter inductance. For control design purpose it is more convenient to consider the ollowing averaged model obtained by averaging the model (a-d) over one switching period. C i (a) R L = u - u u - u u - (b) 5 u - u u - u C (c) C (d) Where 5 v and 0 i. i and u i denote the average values over cutting periods o the signals Note that the mathematical model (a-d) is nonlinear because o the products involving the state variables and input signals. Taking into account these nonlinearities a nonlinear controller using the backstepping approach will be done in the net paragraph.. CONTROLLER DESIGN. Control objectives In order to deine the control strategy irst one has to establish the control objectives which must be ormulated as ollowing: v 0 i v c v c 856

5 i. A perect MPPT. Speciically the controller must enorce the voltage the average value o the voltage V pv provided by the solar array to track as possible a given desired value o the reerence voltage V pv. To achieve this objective we use the well-known "Perturb and Observe algorithm" to generate the reerence voltage signal. ii.regulate the capacitors voltages at its desired reerence value. Vpv = () V = pv iii.generate a sinusoidal voltage in the network with amplitude and requency ied by the reerence signal: = V sin( t) (6). MPPT Control Algorithm To ensure the correct operation o PV module at its maimum power point as the temperature insolation and load vary we use well known perturb and observe algorithm (P&O). This algorithm uses simple eedback arrangement and little measured parameters. It is based on the ollowing criterion: i the operating voltage o the PV array is perturbed in a given direction and i the power drawn rom the PV array increases this means that the operating point has moved toward the MPP and thereore the operating voltage must be urther perturbed in the same direction. Otherwise i the power drawn rom the PV array decreases the operating point has moved away rom the MPP thereore the direction o the operating voltage perturbation must be reversed and when the stable condition is arrived the algorithm oscillates around the peak power point. (Jain and Agarwal 00; Atallah et al. 0) The algorithm steps are described as shown in the ollowing lowchart (igure ) (Bhandari et al. 0).The algorithm irst calculates the power (P) by sensing the voltage and current. Then it provides a perturbation in the voltage based on the change o power where the delta Vpv is known as the perturbed voltage I delta Vpv is large the convergence is ast but that causes large luctuation in the steady state power and vice versa. The oscillation causes undesirable losses o PV power during steady state. In order to maintain the power variation s mall the perturbation sie is remain very small. The technique is advanced in such a Style that it sets a reerence voltage o the module corresponding to the peak voltage o the module. (5) Figure. P&O lowchart. 857

6 . Nonlinear controller design Backstepping is a recently developed design tool or constructing globally stabiliing control laws or a certain class o nonlinear dynamic systems.it is a recursive procedure which breaks a design problem or the ull system into sequence o design problems or lower order systems. Backstepping designs by breaking down comple nonlinear systems into smaller subsystems. Then designing control Lyapunov unctions and virtual controls or these systems and inally integrating these individual controllers into an actual controller by stepping back through the subsystem and reassembling it rom its component subsystems. (Arun Kumar et al. 05) Following the backstepping technique the controller is designed in two steps. Step : Let us deine the ollowing tracking error: C (7a) C (7b) C with (7c) and are respectively the measured capacitor voltage and the measured output voltage represent respectively the desired trajectories o Deriving and u - u - C u - u - C and with respect to time yields and accounting or (a) and (c-d) implies: = (8a) = (8b) = - - C (8c) R We need to select a Lyapunov unction or such system. As the objective is to drive its state to ero it is natural to choose the ollowing unction: V = (9a) Its time derivative is given by the ollowing equation: V = + + (9b) V = The choice where i... i. are positive constants o synthesis leads to a Lyapunov candidate unction whose dynamics is negative deinite. In view o (9b) and using (8a-c) this suggests the ollowing choices: u - u - C (0a) u - u - C R - - C (0b) (0c) i we choose as virtual control input we deduce the stabiliing unction namely : C () R As is not the control input a new error variable introduced: L and between the virtual control and its desired value is () Then using () and (8c) dynamics o error L became: () In the same way 858

7 = L () V Step : The objective now is to enorce the error variables dynamics o we obtain: u -u u -u u 5 L to vanish. To this end let us irst determine the (5) We are inally in a position to make a convenient choice o the parameter update law and eedback control to stabilie the whole system with state vector Consider the augmented Lyapunov unction candidate. V V 0.5 (6a) Its dynamics is given by: V =V (6b) = ( ) (6c) L V As our goal is to make V non-positive deinite V = 0 this suggests choosing that the bracketed term on the right side o (6c) is equal to (7) L The equations (5) and (7) lead to: = u - u u - u u5 L (8) L where is a positive constant o synthesis. Finally the control input can be solved rom the ollowing system equations. u u u u u u u u u C C C C L C 5 L : L (9) (0) (). SIMULATION RESULTS The proposed control system will have the structure shown in Figure. The controller will be synthesied by a technique using Backstepping approach and validated through simulations perormed using the Matlab/Simulink and its SimPower SystemToolbo. 859

8 Figure. Control block or multicellular inverter The characteristics o the controlled system are listed in table. Backstepping controller parameters which proved to be convenient are given values o table and the simulation results have been obtained under standard climatic conditions (000 W/m² and 5 C).and the resulting control perormances are shown by Figures 5 to 8. Table. Main parameters values o simulated circuit Components symbol Value Flying capacitor Dc link capacitors Filter capacitor Filter inductor Load impedance C CC C R 000 µf mf 9 µf mh 5 Ω Table. Backstepping controller parameters Parameter λ λ λ λ Value Figure 5. PV voltage with its reerence 860

9 Figure 6. Actual and desired voltage o cell- s lying capacitor (vc) Figure 7. Actual and desired voltage o cell- s lying capacitor (vc) Figure 8(a). Actual and desired output voltage v0 Figure 8(b). Zoom o output voltage v0 Figure 5 shows the evolution o the converter input voltage in particular it is noted that this voltage ollows the reerence voltage generated by the P&O algorithm. The capacitor voltages vc and vc shown in Figures (6-7) converge towards their desired values given by () and (5) ater 0.7 seconds they reach their reerences voltages. Figure 8 shows that the output voltage v0 is sinusoidal reaches the desired voltage 0 sin(00 t) at time 0.75 seconds. 86

10 5. Conclusion The present paper is ocusing on the problem o controlling single stage multicellular inverter. The latter is described by sith order nonlinear averaged model. The synthesis o the regulator was achieved by having recourse to advanced tools o nonlinear control such as stability in the sense o Lyapunov. It is ormally shown by simulation under Matlab/Simulink that the proposed controller guarantees the desired tracking perormance and satisies all the objectives or which it was designed namely: i) maimum power point tracking (MPPT) o (PV) module; ii) Regulating the voltages o the capacitors to their desired reerence levels and iii) generating a sinusoidal voltage at the output. Reerences Ahmed M. Atallah Almoata Y. Abdelai and Raihan S. Jumaah. Implementation o perturb and observe MPPT o PV system with direct control method using buck and buck-boost converters Emerging Trends in Electrical Electronics & Instrumentation Engineering: An international Journal (EEIEJ) Vol. no. pp Amet L. Ghanes M. and Barbot J. P. Commande directe d un convertisseur multicellulaire: résultats epérimentau CIFA 7ème Conérence Internationale Francophone d Automatique 0. Amet L. Ghanes M. and Barbot J. Direct control based on sliding mode techniques or multicell serial chopper American Control Conerence (ACC) pp Arun Kumar V V. and Laila Beebi M. Adaptive Backstepping Sliding Mode Control or Roll Channel o Launch Vehicle Int J Aeronautics Aerospace Res. Vol. no. 5 pp Benmiloud M. and Benalia A. Finite-time stabiliation o the limit cycle o two-cell DC/DC converter: hybrid approach Nonlinear Dyn vol. 8 pp Benmiloud M. Benalia Deoort M. and Djemaï M. On the limit cycle stabiliation o a DC/DC three -cell converter Control Engineering Practice Vol. 9 pp. 9- April 06. Bhandari B. Poudel S. Kyung-Tae L. and Sung-Hoon A. Mathematical Modeling o Hybrid Renewable Energy System: A Review on Small Hydro-Solar-Wind Power Generation International Journal o Precision Engineering and Manuacturing Green Technology Vol. no. pp Bonkoungou D. Koalaga Z. and Njomo D. Modelling and Simulation o Photovoltaic Module Considering Single-Diode Equivalent Circuit Model in MATLAB International Journal o Emerging Technology and Advanced Engineering vol. no. pp Deay F. Llor A. and Fadel M. An active power ilter using a sensorless muticell inverter in Proc. IEEE ISIE pp Deaÿ F. Llor A.M. and Fadel M. A. Predictive Control with Flying Capacitor Balancing o a Multicell Active Power Filter IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS Vol. 55 No 9 pp Djemaï M. Busawon K. Benmansour K. and Marou A. High-Order Sliding Mode Control o a DC Motor Drive Via a Switched Controlled Multi-Cellular Converter International Journal o Systems Science Vol. pp Hosseini S.H. Khoshkbar Sadigh A. and Sharii A. Estimation o lying capacitors voltages in multicell converters 6th Int. Con. on Electrical Engineering/Electronics Computer Telecommunications and Inormation Technology pp Jain S. and Agarwal V. A new algorithm or rapid tracking o approimate maimum power point in photovoltaic systems IEEE Power Electron.Lett. vol. no. pp. 6 9 Mar. 00. Meynard T. and Foch H. French patent N du 5 juillet 99; dpt. international PCT (Europe Japon USA Canada) N du 8 juillet Meynard T. and Foch H. Multilevel choppers or high voltage applications EPEJ (March ()) Meynard T.A. Fadel M. and Aouda N. Modeling o multilevel converter IEEE Trans. Ind. Electron. Vol. no. pp Morvan C. Richard P.Y Cormerais H. and Buisson J. Sliding mode control o switching systems with Boolean inputs in: IFAC Conerence NOLCOS Stuttgart Allemagne pp September 00. Turpin C. Depre L. Forest F. Richardeau F. and. Meynard T.A. A ZVS imbricated cell multilevel inverter with auiliary resonant commutated poles IEEE Trans. Power Electron. Vol.7 no. 6 pp Wu Chang C.H. Lin L.C. and Kuo C.L. Power Loss Comparison o Single- and Two-Stage Grid Connected Photovoltaic Systems IEEE Transactions on Energy Conversation Vol. 6 pp

11 Biography Taghaoui Chaimaa is a doctoral student in LTI laboratory Hassan II o Casablanca University. He holds a Master s degree in inormation processing (05) and a bachelor s degree in Electronics and Industrial data processing (0).Her research interests ocus on the area o simulation modeling Nonlinear Control and observation o converters. Elallali Aicha: She holds a Bachelor degree in Electronic and Industrial data processing (0) and a Master in Inormation Processing (TI) (05) rom Hassan II University o Casablanca. Currently in the second year o a Doctorate at the Hassan II University o Casablanca in LTI laboratory under the direction o Pr. Hamdoun Abdelati on a thesis entitled Multi-level inverters used in active iltering. In addition to the participation in an international symposium and an international communication on the monitoring o industrial systems 9-0 October 06 Fe Morocco and the 8th National Meeting o Young Researchers in Physics Department o Physics Faculty o Science Ben M'Sik there is a work submitted to an international conerence in Toulouse as well as the preparation o an article to be submitted to an indeed journal. 86