Passivity-Based Control for Charging Batteries in Photovoltaic Systems

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1 Passivity-Based Control for Charging Batteries in Photovoltaic Systems Valentim E. Neto, Filipe Perez,André G. Tôrres, Allan F. Cupertino, and Heverton A. Pereira, Gerência de Especialistas em Sistemas Elétricos de Potência Universidade Federal de Viçosa Av. P. H. Rolfs s/nº, Viçosa, MG, Brazil Abstract This paper proposes the use of passivity-based control in a battery charger for isolated photovoltaic systems. It uses a boost converter acting as maximum power point tracker (MPPT). This technique is compared with the traditional strategy based on proportional-integral () linear controllers. The dynamic of the controlled system during variations in the solar radiation and the influence of the inductor parameters uncertainty in the response of the battery charger are analyzed. Index Terms Battery, Boost Converter, Passivity Based Control, Maximum Power Point Tracker, Photovoltaic Panel. I. INTRODUCTION The generation of electricity through alternative sources has been widely studied, due to countries wanting to diversify its energy matrix focusing on power generation without pollution and with as low environmental degradation as possible. In this context, the solar photovoltaic energy has been growth considerably in the last years []. Photovoltaic systems can be classified into three main categories: hybrid, grid-connected and isolated. The use of each is a function of the application and availability of energy resources. The isolated system can be applied in rural areas of difficult access. This system is easily adapted to the buildings and roofs of houses []. Until had been installed in the world GW in photovoltaic systems, as shown in Fig., and more than 95% of these are connected to the grid. This represents an increase of 3 GW in the last year. With this power level is possible to save 53 million tons of []. Installed Power (MWp) 5,3 6,9 9,5 6, 3,6 4,6 7,, Year Fig. Installed power on solar photovoltaic systems until []. The authors would like to thank CNPq, FAPEMIG and CAPES by their financial support. Graduate Program in Electrical Engineering Federal University of Minas Gerais Av. Antônio Carlos 667, 37-9 Belo Horizonte, MG, Brazil allan.cupertino@yahoo.com.br, heverton.pereira@ufv.br The battery bank is an important element of the isolated system and can represent until 5% of initial costs for installation, reaching to 46% if considered the maintenance costs [3]. This is explained by the fact that battery s life cycle is smaller than the others components of the photovoltaic system. Furthermore, the batteries are subjected to the most diverse operating conditions, due to nonlinear behavior of photovoltaic panels, [4], [5]. Nickel-metal hydride batteries have as advantages the following points [6], [7]: It can be loaded several times a day; reduced memory effect; long life cycle; low maintenance requirements ; it is less toxic than others. However, this type of battery has a high cost, high risk of damage to overload and high self-discharge [6], [7]. In order to increase the lifetime of the batteries, the charging method needs to consider the charge state of the battery and its rated values. When the battery is discharged the system can works at maximum power point if the rated current of the battery is not exceeded. If the rated current is exceeded or the battery reaches full charge, the MPPT algorithm is maintained in standby mode. Therefore, the battery is protected against overload. Many works in literature study the control of battery chargers. A traditional methodology is the proportionalintegral () control technique [8]. Several authors have been proposed nonlinear techniques like the passivity-based control (), due to the existence of large disturbances in the generated power [9], []. The is based on energy functions and finds a state of operation in which the plant stores the minimum energy. This method of control has some advantages over traditional techniques, because it is not necessary to linearize the system around the operation point. This characteristic can improve the response of non-minimum phase systems, like switching converters. Using it is possible to obtain a robust battery charger using a boost converter. This work proposes the use of passivity-based control in a battery charger for a photovoltaic system. It is used a boost converter acting as maximum power point tracker (MPPT). Anais do V Simpósio Brasileiro de Sistemas Elétricos, Foz do Iguaçu PR, Brasil. -5/4/4 ISSN

2 This technique is compared with the traditional linear strategy. The topology of isolated photovoltaic system is presented in Fig.. Fig. 4 Photovoltaic panel model. II. Fig. Battery charger topology. MODELING OF THE SYSTEM A. Maximum Power Point Tracker (MPPT) The MPPT is an algorithm that maintains the photovoltaic panel delivering maximum power to the system at various levels of solar radiation. It was used the incremental conductance algorithm. The incremental conductance algorithm is considered better than other techniques based on the principle of perturbation and observation, because it has a faster response to changes in solar radiation. The operating principle of the algorithm is the incremental calculation of the derivative of the curve of power once this value is zero at the maximum power point []. Fig. 3 shows the algorithm. TABLE I. MAIN PARAMETERS OF A SOLAR PANEL. Parameter Symbol Value Maximum Power ( ) 48 Maximum Power Voltage ( ) 8.6 Maximum Power Current ( ).59 Open Vircuit Voltage ( ). Short Circuit Current ( ).89 Temperature Coefficient of ( ) -.7 Temperature Coefficient of ( ).66 C. Boost converter modeling and control The modeling of the boost converter assumes that the battery voltage is constant. This is a good approximation because the battery voltage variation during the charging process is very small. Besides, the curve of the solar panel is linearized around the maximum power point, like is shown in Fig. 5 [8]. In this situation, the solar panel can be modeled as a voltage source in series with a resistance. Thus, the battery charger equivalent circuit is shown in Fig I x V curve Linearization Fig. 3 Incremental conductance algorithm [8]. B. Solar panel modeling The electrical model of the solar panel used in this work is shown in Fig. 4 and is presented in details by []. The resistances represent the voltage drop and losses for both the current flowing through the load ( ) and the reverse leakage current of the diode ( ), respectively. is a controlled DC current source. In Table I the parameters of a solar panel SM48KSM, manufactured by Kyocera are presented. Fig. 5 I x V curve of the solar panel SM48KSM and its linearization around the maximum power point. Fig. 6 Battery charger equivalent circuit. Anais do V Simpósio Brasileiro de Sistemas Elétricos, Foz do Iguaçu PR, Brasil. -5/4/4 ISSN

3 cycle. The average model of the converter is expressed by: Where: ( ( () [ [ ( ) ( ) ( ) ( ) ( ) ( ) ]; ] [ ( ) ( ) [ ] [ ] [ ]. denotes the average value of and is the duty The purpose of this modeling is to find the converter transfer function that relates the small-signal input voltage of the converter ( ) with the control variable. One important detail is that an increase in the duty cycle reduces the input voltage of the converter. According to reference [3], a small-signal model is obtained considering that each variable of () can be represented by a value in steady state with a small disturbance, like shown in (). { Where, and. Developing () and applying the Laplace transform, it can be obtained: Where: In a matrix form: [ ]; () ( ( ( (3) ( [ ] [ ] ] [ ( ( ( ( ( ( ] [ ] (4) The transfer function ( relates the capacitor voltage and the duty cycle. The equation (5) relates the input voltage of the converter and the capacitor voltage: The small signal model of (5) is: ( ( The multiplication ( by (6) results in: ( ( ( It is used a compensator, given by: (5) (6) (7) ( (8) Fig. 7 shows the Bode diagram of (, and the compensated transfer function ( (. It can be observed in open loop, a resonance frequency and a small phase margin. Magnitude (db) Phase (deg) Fig. 7 Bode diagram of open-loop system G vd (s) and the compensated G(s)G vd (s). D. Boost converter modeling and control Considering the equivalent circuit presented in Fig. 6 and neglecting the effect of the capacitor resistance, the Euler- Lagrange model of the battery charger is given as [4]: Where: Bode Diagram Frequency (Hz) ( ( (9) [ ] [ ] [ ] [ ] [ ] and [ ] represents the duty cycle and and represent the inductor current and capacitor voltage, respectively. The desired values for the average inductor current and average capacitor voltage are respectively the panel current and voltage on maximum power point. The vector of averaged dynamic error is defined by [5]: ( [ ( ( ] [ ( ( ( ( ] () As ( ( ( with ( [ ( ( ] the desired state, it can be obtained that: ( ( [ ( ] Open loop Compensated () The design of the consists in modifying the system energy by adding damping through the dissipative structure [5]. This modification is accomplished through the addition, in closed loop, of a dissipative term that emulates a resistor connected in series with the inductor, denoted by. This strategy is denominated indirect control, or series control. The dissipative term added is: [ ] () Anais do V Simpósio Brasileiro de Sistemas Elétricos, Foz do Iguaçu PR, Brasil. -5/4/4 ISSN

4 And the new dissipative structure is given as: [ ] (3) Given a desired (, it is possible to verify the following change in the dynamic averaged error equation (4): ( ( [ ( ] The energy adjustment of the system is obtained doing: (4) ( [ ( ] (5) In this circumstance, the error dynamic equation is: ( (6) The desired energy in terms of the error can be modeled by : (7) is a Lyapunov function candidate for (6). The time derivative of (7) along the paths (6) results in: ( ( ( (8) Where is strictly positive and constant. The condition (8) is ensured for (6), and satisfied if: ( [ ( ] (9) Doing the matrix products of (9), the result is: { [ ( ] () The equations () contains the expressions of the control law. To avoid the influence of parasite elements, reference [6] proposed an integral action, as: ( () Equation () gives the duty cycle of the converter for the control of the input voltage. The variables and are parameters of the controller. Where and. E. Simulation It was simulated in Matlab/Simulink a photovoltaic system of 48. The solar array consists in a single panel model SM 48KSM whose parameters are shown in TABLE I. The parameters of the boost converter applied in photovoltaic system are showed in TABLE II. Finally the parameters of the batteries used in the simulation are showed in TABLE III. The algorithm of incremental conductance (MPPT) uses a sampling frequency of and a step voltage of. It is necessary calculate the maximum power point current of the panel for the technique. As this algorithm is only obtained maximum output voltage of the panel, this value is obtained using (). () Where is the maximum power point current of the panel, is the output power of the panel while is the voltage calculated by the algorithm. TABLE II. PARAMETERS OF THE CONVERTER. Boost Converter Inductor 8. Capacitor in Panel.5 Capacitor in Battery 5 Output Voltage 36 Frequency Switching Inductor Resistance. Ω Capacitor Resistance.5 Ω Diode Voltage.8 IGBT Voltage. TABLE III. PARAMETERS OF THE BATTERY. Battery (Nickel-Metal-Hydride) Nominal Voltage 36 Nominal Capacity 6 Initial Stage of Charge % Maximum Capacity 64.6 Fully Charged Voltage 4.4 Nominal Discharge Current Internal Resistance.6 Ω The Fig. 8 shows the solar radiation profile used to test and compare the performance of the two control techniques. This profile consists in ramp variations and will impact in the maximum power point voltage of the panel. In order to analyze the impact of parameters uncertain in the system response, variations of in the inductance and in the resistance of the inductor are simulated. Solar irradiance (W/m ) Fig. 8 Solar radiation profile. III. RESULTS A. Performance during solar radiation variation The Fig. 9 shows the electrical variables of the solar panel for both and technique. The panel voltage and current follows the maximum power at all levels of radiation. Anais do V Simpósio Brasileiro de Sistemas Elétricos, Foz do Iguaçu PR, Brasil. -5/4/4 ISSN

5 During solar radiation variations, the transient response of each controller is different. As can be seen in Fig., the technique follows the maximum power point voltage with a smaller current overshoot and is faster than technique. The battery current ripple reduces when the technique is used, like is shown in Fig.. This fact can be justified by the duty cycle calculated for each technique. In the same operation point there is a larger oscillation in the duty cycle calculated by the technique. In this case, the control has the advantages of improving the battery current and a reduction of the switch stress. Power (W) Fig. 9 Electrical parameters of the photovoltaic panel for and techniques Fig. Details in the voltage and current response for and techniques.. Current in the batteries (A) Duty Cycle (%) Fig. Battery current and duty cycle of converter for and techniques. B. Performance during parameter variation Variations in internal resistance and inductance of the inductor influence in the dynamic of both control techniques, like shown in Fig. and Fig. 3. However, in the technique there is an increase in the response time, which does not happen in the technique. The behavior of the current in the battery is presented in Fig. 4 and Fig. 5. Variations in the inductor did not impact in the current dynamic response. It can be observed that the inductance has an impact in the current ripple. On the other hand, the increase of the resistance impacts in the efficiency of the converter, reducing the average current in the battery L L +% L -% Fig. Voltage response of the and techniques during variations in the inductance. IV. CONCLUSION This work presented a battery charger supplied by a 48 photovoltaic panel connected to a boost converter. The incremental conductance algorithm was used to find the maximum power from the panel. It was proposed the technique and this was compared with the traditional technique. The control had a faster response in the maximum power point tracker and a smaller ripple in the duty cycle. Both components lifetime and charger process are improved using this technique. Besides, the strategy was more stable during parameters variation than technique. L L +% L -% 7. Anais do V Simpósio Brasileiro de Sistemas Elétricos, Foz do Iguaçu PR, Brasil. -5/4/4 ISSN

6 Fig. 3 Voltage response for and techniques during variations in the resistance. Fig. 4 Battery current response for and techniques during variations in the inductance RL RL +% RL -% Fig. 5 Battery current response for and techniques during variations in the resistance. RL RL +% RL -% L +% L L -%.8.6 L +% L L -% RL -% RL RL +% RL -% RL RL +% REFERENCES [] EUROPEAN PHOTOVOLTAIC INDUSTRY ASSOCIATION. Global Market Outlook for Photovoltaics 3-7. European Photovoltaic Industry Association. [S.l.], p. 6.. [] CASTRO, R. M. G. Energias Renováveis e Produção Descentralizada. Universidade Técnica de Lisboa. Lisboa.. [3] ENSLIN, J. H. R.; WOLF, M. S.; SNYMAN, D. B. Integrated photovoltaic maximum power point tracking converter. IEEE Transactions on Industrial Electronics, 44, [4] IEA. Management of Storage Batteries used in Stand-Alone Photovoltaic Power Systems - Report_IEA_PVPS_T3-:. International Energy Agency (IEA). [S.l.].. [5] SOUSA, J. M. N. D. Sistema bidirecional de carga de baterias para o FEUP VEC. Universidade do Porto. Porto, p.. 3. [6] LINDEN, D.; REDDY, T. B. Handbook of Batteries. 3. ed. [S.l.]: McGraw- Hill,. [7] AMBROSIO, R. C.; TICIANELLI, E. A. Baterias de níquelhidreto metálico, uma alternativa para as baterias de níquelcádmio. Scielo,. ISSN ISSN -44. Disponivel em: < 445&lng=en&nrm=iso>. Acesso em: 8 jan. 4. [8] VILLALVA, M. G.; SIQUEIRA, T. G. D.; FILHO, E. R. Voltage regulation of photovoltaic arrays: small-signal analysis and control design. IET Transactions on Power Electronics, v. 3, p ,. [9] BECHERIF, M.; AYAD, M. Y.; ABOUBOU, A. Hybridization of Solar Panel and Batteries for Street Lighting by Passity Based Control. IEEE International Energy Conference, Al Manamah, p ,. [] MU, K.; MA, X.; ZHU, D. A New Nonlinear Control Strategy for Three-Phase Photovoltaic Grid-Connected Inverter. International Conference on Eletronic & Mechanical Engineering and Information Technology, Harbin, p ,. [] ALMEIDA, P. M. D. Modelagem e Controle de Conversores Estáticos Fonte de Tensão utilizados em Sistemas de Geração Fotovoltaicos Conectados à Rede Elétrica de Distribuição. UFJF. Juiz de Fora, p. 9.. (Master thesis). [] VILLALVA, M. G.; GAZOLI, J. R.; FILHO, E. R. Comprehensive Approach to Modeling and Simulation of Photovoltaic Arrays. IEEE Transactions on Power Electronics, v. 4, n., p. 98-8, March 9. [3] ERICKSON, R. W.; MAKSIMOVIC, D. Fundamentals of Power Eletronics. ª. ed. New York: Klumer Academic Publishers, 4. [4] CUPERTINO, A. F. et al. A Grid Connected Photovoltaic System with a Maximum Power Point Tracker using Passivity Based Control applied in a Boost Converter. UFV. Fortaleza, p. 8.. [5] JELTSEMA, D.; SCHERPEN, J. M. A. Tuning of Passivity- Preserving Controllers for Switched Mode Power Converters. IEEE Transactions on Automatic Control, v. 49, p , August 4. [6] LEYVA, R. et al. Passivity-based integral control of a boost converter for large-signal stability. IEE Proceedings. Control Theory and Applications, v. 53, p , March 6. Anais do V Simpósio Brasileiro de Sistemas Elétricos, Foz do Iguaçu PR, Brasil. -5/4/4 ISSN

7 BIOGRAPHIES Valentim Ernandes Neto was born in Aimorés, Brazil. He is student of Electrical Engineering at Federal University of Viçosa (UFV), Viçosa, Brazil, since. Currently is integrant of GESEP, where develop works about power electronics applied in renewable energy systems. His research interests include photovoltaic energy and control applied in power converters. Filipe Perez was born in Uberaba-MG, 99. He is student of Electrical Engineering at Federal University of Viçosa (UFV), Viçosa, Brazil, since 8. Currently is integrant of GESEP, where develop works in the area of power systems and renewable sources, especially solar energy. Currently working with control converters for photovoltaic panels. His research interests include power systems, automation and control. André Gomes Tôrres received the B.S. degree, the M.S. and the Ph.D. degree in electrical engineering from the Federal University of Minas Gerais (UFMG), Belo Horizonte, Brazil, in 998, and 4 respectively. Since 5 he has been with the Department of Electric Engineering, UFV, Brazil. His research interests include power electronics, electrical drives and process automation. Allan Fagner Cupertino received the B.S. degree in electrical engineering from the Federal University of Viçosa (UFV), Viçosa, Brazil, in 3. He is integrant of GESEP, where developed works about power electronics applied in renewable energy systems. Currently he is Master student from Federal University of Minas Gerais (UFMG), Belo Horizonte, Brazil. His research interests include solar photovoltaic, wind energy, control applied in power electronics and grid integration of dispersed generation systems. Heverton Augusto Pereira received the B.S. degree in electrical engineering from the Federal University of Viçosa (UFV), Viçosa, Brazil, in 7, the M.S. degree in electrical engineering from the State University of Campinas (UNICAMP), Campinas, Brazil, in 9. Currently he is Ph.D. student from the Federal University of Minas Gerais (UFMG), Belo Horizonte, Brazil. Since 9 he has been with the Department of Electric Engineering, UFV, Brazil. His research interests are wind power, solar energy and power quality. Anais do V Simpósio Brasileiro de Sistemas Elétricos, Foz do Iguaçu PR, Brasil. -5/4/4 ISSN