DC DC converter in order to increase or decrease the de Ix and Vx represent the short circuit current and open. sired voltage It is then connected directly to the load circuit voltage at a given temperature and solar irradiance. which is composed of a 12 V battery The duty cycle of the V is the PVM output voltage T is the PVM temperature TN. converter is controlled by a sliding mode controller is the standard conditions temperature Ei is the effective. The proposed model will guarantee the extraction of solar irradiance at the PVM Ein is the standard condition. the maximum power that can be produced by the PVM solar irradiance TCV is the open circuit voltage tempera. while regulating the load voltage to the battery s voltage ture coefficient and TCi is the short circuit current temper. That way we can have a workable load voltage that can be ature coefficient Vmax is the open circuit voltage at 25oC. connected to an inverter while matching the load resistance and more than 1200W m2 Vmin is the open circuit voltage. to the PV optimal resistance at 25oC and less than 1000W m2. Figure 1 shows the non linear relation between the cur. III Mathematical Background rent and the voltage given by the 1 Figure 2 shows the. This section presents mathematical terms used in the paper non linear relation between the power and the voltage giv. en by 2 under standard conditions We can see that the. Re Function It extracts the real part a of a complex maximum power produced by the PVM occurs at a certain. number written in the form a bi A complex number is voltage level Since the function of power depends only of. a number which can be formally defined as an ordered the voltage and it is differentiable for all values of voltage. pair of a real number then maximum power that can be extracted from the PVM. Lambert s W Function lambertw x The Lambert s will occur when the partial derivate of the power with re. W function which was named after Johann Heinrich spect to the voltage is equal to cero The partial derivate of. Lambert is defined to be the solution W x of the non the power against voltage is given in the following equa. linear equation W x exp W x x tion, Ix Ix exp Ix exp. P V b Vx b V b Vx b, IV PV Model equations 5, In the past there have been different types of models V 1 1. 1 exp b Vx b Vx exp, to estimate the non linear equations of the photovoltaic b b. module PVM Some of these models are the Anderson s By equaling 5 to zero and solving by the voltage we. Bleasser and the most common the one diode model All can obtain the optimal voltage which is given by 6. these models present a good approach into estimating the 1. solar cell voltage and currents but most of them need too Vop Re b Vx lambertw 0 36787944. much computational power or need information not availa. ble in the manufacturer s sheet A more suitable model to From 6 we can obtain a very approximate estimate of. simulate a PV module is proposed by 13 15 The PVM PV cell s output voltage at which maximum power occurs. model is know as the Ortiz PVM model In that work a The advantage of this equation is its dynamic property The. PV model was proposed where analytical equations relates only variant term is Vx Given that we can express the eq. the PV output current with the PV output voltage tempera uation as the following. ture and solar irradiance over the PV module It also shows Vop C Vx 7. experimental results validating the accuracy and effective 1. ness of the proposed model An advantage of this model is C Re b lambertw 0 36787944. that all the needed information can be found in the manu Since it depends on Vx and Vx vary with respect to. facturer s data sheet Also it shows how the PV power is temperature and solar irradiance the optimal voltage will. affected by changes in the temperature and solar irradiance vary with respect to the conditions of the temperature and. The equations are the followings irradiance giving always an estimation of the required vol. Ix V 1 tage necessary to extract the maximum power from the PV. 1 b Vx b 1 cell for all external conditions, 1 exp V Sliding Mode Controller Surface. A sliding mode controller is a variable structure con. P V V I V 1 exp 2 trol where the dynamics of a non linear system is altered. 1 b Vx b via the application of a high frequency switching control In. E b sliding mode control the trajectories of the system are. TCV T TN s V max forced to reach a sliding manifold of surface where it exhi. bit desirable features in finite time and to stay on the mani. E V Voc fold for all future time It is achieved by suitable control. s Vmax Vmin exp i ln max, strategy To apply sliding mode control we have to know if. E in V max V min, Ei the system can reach the sliding manifold Once the sys. Ix p I SC TCi T T N 4 tems reach the sliding manifold the controller has to force. the system to stay in the manifold for all future time. This law of control also guarantees us that the system. trajectories will reach the proposed manifold and will stay. there for all future time This can be explained in a practical. way At first because the PVM is not connected the PVM. output voltage will be equal to its open circuit voltage. Since the open circuit voltage is greater than the optimal. voltage the switching device will be on When the switch is. on the PVM output voltage will begin to drop because of. Figure 2 PV Output Power Vs Time the load mismatching until it reaches below the optimal. voltage Then the switch will turn off creating an open cir. cuit condition forcing the PVM output voltage to increase. up to its open circuit voltage When the output voltage. passes the optimal voltage then the switch will turn off and. the sequence will start again and will continue for all future. time This control law works for a variety of DC DC con. verters like the Buck converter SEPIC converter and Buck. Boost converter, Figure 3 PV Output Power Vs Time, VI Simulation. Sliding Mode Control is widely use for a lot of appli. The system was simulated using Matlab s Simulink, cations including control systems for DC DC converters. software with the power system toolbox With this software. 8 14 power supply electric grid connections 9 motors. we simulate and test the sliding mode controller and the. speed regulator 14 position control system among others. proposed model The simulink model is shown at figure 4. To design the sliding mode controller we have to select the. desired surface We want to obtain the maximum power. that can be extracted from the PV module at the given tem. perature and irradiance conditions From 6 we can relate. that maximum power to an optimal voltage Since we know. the output voltage we have to have in order to extract the. maximum power from the PV system we choose a surface. that will force the system to reach that voltage in a finite. time and stay there for infinite time With that in mind we. chose the following sliding manifold, V is the output voltage of the PV cell and Vop is the. Fig 4 Simulation Scheme for the Proposed Model, optimal voltage This sliding manifold will assure us to. force all the trajectories of the system to reach the optimal The solar Cell model was represented by a single block. voltage and to keep it in the optimal voltage for all future composed by a Matlab Embedded function containing the. time Since the optimal voltage is dynamic since it change equations of a solar cell 13 15 The system was simu. when changes occur in the temperature and irradiance this lated under constant ambient temperature and solar irra. sliding surface is also changing with respet to the diance and under varying ambient temperature and solar. temperature and irradiance giving us a dynamic sliding irradiance in order to validate the effectiveness of the con. surface The sliding mode will be controlling the duty cycle troller 1 and 6 were calculated and compare to the. of a switching device So the switchin device will have two manufacturer s data sheet for several PV commercial mod. operation state els The simulation results of the PV system with constant. ambient conditions connected directly to the battery and. On V Vop 0, connected to the battery by a non inverting Buck Boost. Off V Vop 0 converter in Buck mode are shown in figure 5 and 6 Si. Now the controller will behave in the following way mulations of the PV system with varying ambient condi. tions connected directly to the battery and connected to the. 1 V Vop 0 battery by a non inverting Buck Boost converter in Buck. 0 V Vop 0 mode were done The converter in Buck mode only uses. one Mosfet the left one and the second Mosfet is turn off. A control law that guarantees us that our controller will. Finally the simulations showed in figure 5 are for standard. behave in that way is given by the following equation. conditions for each PVM and figure 6 are for varying tem. u 1 1 sign V Vop 10, perature and solar irradiance. VII Results, Table 1 and 2 shows the results of the estimated op. timal voltage Vop Iop and Pmax compared to the manufac. turer s datasheet value These tables validate 6 and 1 for. the optimal voltage and optimal current estimation The. error percent for the voltage and current stays within ac. ceptable values Since the estimation are very near to the Fig 5 Temperature and irradiance variation over PVM. optimal values given by the manufacturer by forcing the. system to operate at the estimated voltage guarantee us to. be working in a near maximum power point Figure 5, shows the temperature and irradiance over the PVM Fig. ures 6 and 7 validate the sliding mode controller and ensure. that the PV operation point is in the knee point of the power. vs voltage graph were the PV operates at its maximum. power even under standard conditions STC and under va. rying ambient temperature and solar irradiation condition. while supplying a higher power to the battery Table 3. shows the percentage of increment in the power given to. the battery It can be seen that the proposed method in. crease significantly the available power delivered to the. battery Figures 8 and 9 show the simulation results of the. power that is given to the battery for two different connec. tion modes directly to the battery and through a converter. PV Vop Vop Error Iop Iop Error, Model Data Esti Data Esti Fig 6 PV maximum possible output power blue vs actual PV. sheet mated sheet mated Power when connected to proposed MPPT green at STC. SiemenSP75 17 0 17 593 3 49 4 40 4 2524 3 35, Shell SQ80 17 5 18 156 3 74 4 58 4 4305 3 27. SLK60M6 30 6 30 762 0 53 6 86 6 8193 0 59, Solare SX 5 16 5 16 673 1 05 0 27 0 2669 1 17. SolarxSX 10 16 8 17 098 1 774 0 59 0 5789 1 87, Table 1 Comparison of PV voltage and current estimated. values vs Datasheet values, PV Pmax Pmax Error, Model Datasheet Estimated. Siemens SP75 74 8 74 815 0 02, Shell SQ80 80 15 80 44 0 36. SLK60M6 209 92 209 777 0 068, Solarex SX 5 4 455 4 449 0 135. Solarex SX 10 9 912 9 8989 0 132, Table 2 Comparison of PV Power estimated value vs Data. sheet value Fig 7 PV maximum possible output power blue vs actual PV. Power when connected to proposed MPPT green at varying. PV Power at battery Power at battery Increment conditions. Model connected directly connected through in Power. W converter W Those graphs reflects the importance of the use of a MPPT. Siemens SP75 57 26 71 61 25 06 device since it can be seen that the power given to the bat. Shell SQ80 58 12 76 96 32 24 tery is greater when connected through the MPPT converter. than connected directly to the PV cell, SLK60M6 90 23 191 111 68. Solarex SX 5 3 573 4 43 32 98, Solare SX 10 7 753 9 815 26 59. Table 3 Comparison of the power supplied to the battery. Fig 7 Power at the Battery for PV Modules connected Fig 8 Power at the Battery for PV Modules. directly to battery connected to battery through DC DC Converter. VIII Conclusion References, This paper presents a simple photovoltaic solar cell dynam 1 Femia N Petrone G Spagnuolo G Vitelli M Perturb and Observe. ic sliding mode controlled maximum power point tracker MPPT technique robustness improved 2004 IEEE International Sympo. for battery charging applications capable of compute the sium on Industrial Electronics Volume 2 4 7 May 2004 Page s 845. maximum power point under constant and varying ambient. 2 Femia N Petrone G Spagnuolo G Vitelli M Optimization of. temperature and solar irradiation The proposed controller. perturb and observe maximum power point tracking method IEEE Tran. is capable of changing the duty cycle of the Mosfet switch actions on Power Electronics Volume 20 Issue 4 July 2005 Page s 963. in order to move the operation point of the PV system to 973. the optimal operation point and to maintain this operation 3 Weidong Xiao Dunford W G A modified adaptive hill climbing. point with time The proposed algorithm uses a non invert MPPT method for photovoltaic power systems Power Electronics Spe. ing Buck Boost converter in order to easily change the op cialists Conference 2004 PESC 04 2004 IEEE 35th Annual Volume 3. 20 25 June 2004 Page s 1957 1963 Vol 3, eration mode of the converter that can be necessary if the. optimal voltage of the PV module is lower than the battery 4 Atrash H Batarseh I Rustom K Statistical modeling of DSP based. Hill climbing MPPT algorithms in noisy environments Applied Power. voltage The proposed algorithm is capable of calculating Electronics Conference and Exposition 2005 APEC 2005 Twentieth. Simple Photovoltaic Solar Cell Dynamic Sliding Mode Controlled Maximum Power Point Tracker for Battery Charging Applications Emil A Jimenez Brea Eduardo I Ortiz Rivera IEEE Member Andres Salazar Llinas Jesus Gonzalez Llorente Electrical Engineering Department University of Puerto Rico Mayaguez Campus Mayaguez Puerto Rico

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