MODELING AND CIRCUIT-BASED SIMULATION OF PHOTOVOLTAIC ARRAYS - Ematlab

[W/m2 ] is the irradiation on the device surface, and Gn is thenominal irradiation.The diode saturation current I0 and its dependence on thetemperature may be expressed by (4) [32, 33, 35–38]: I0 I0,nTnT 3 expqEgak 11 TnT (4)where Eg is the bandgap energy of the semiconductor (Eg 1.12 eV for the polycrystalline Si at 25 C [11, 32]), and I0,nis the nominal saturation current: I0,n expIsc,n Voc,n 1aVt,n(5)with Vt,n being the thermal voltage of Ns series-connectedcells at the nominal temperature Tn .The saturation current I0 of the photovoltaic cells that compose the device depend on the saturation current density of thesemiconductor (J0 , generally given in [A/cm2 ]) and on the effective area of the cells. The current density J0 depends on theintrinsic characteristics of the photovoltaic cell, which dependon several physical parameters such as the coefficient of diffusion of electrons in the semiconductor, the lifetime of minoritycarriers, the intrinsic carrier density, and others [7]. This kindof information is not usually available for commercial photovoltaic arrays. In this paper the nominal saturation current I0,nis indirectly obtained from the experimental data through (5),which is obtained by evaluating (2) at the nominal open-circuitcondition, with V Voc,n , I 0, and Ipv Isc,n .The value of the diode constant a may be arbitrarily chosen.Many authors discuss ways to estimate the correct value of thisconstant [8, 11]. Usually 1 a 1.5 and the choice dependson other parameters of the I-V model. Some values for aare found in [32] based on empirical analysis. As [8] says,there are different opinions about the best way to choose a.Because a expresses the degree of ideality of the diode and it istotally empirical, any initial value of a can be chosen in orderto adjust the model. The value of a can be later modified inorder to improve the model fitting if necessary. This constantaffects the curvature of the I-V characteristic and varying acan slightly improve the model accuracy.C. Improving the modelThe photovoltaic model described in the previous sectioncan be improved if equation (4) is replaced by: I0 expIsc,n KI T Voc,n KV T 1aVt(6)This modification aims to match the open-circuit voltagesof the model with the experimental data for a very large rangeof temperatures. Eq. (6) is obtained from (5) by includingin the equation the current and voltage coefficients KV andKI . The saturation current I0 is strongly dependent on thetemperature and (6) proposes a different approach to expressthe dependence of I0 on the temperature so that the net effectof the temperature is the linear variation of the open-circuit978-1-4244-3370-4/09/ 25.00 2009 IEEEvoltage according the the practical voltage/temperature coefficient. This equation simplifies the model and cancels themodel error at the vicinities of the open-circuit voltages andconsequently at other regions of the I-V curve.The validity of the model with this new equation has beentested through computer simulation and through comparisonwith experimental data. One interesting fact about the correction introduced with (6) is that the coefficient KV from themanufacturer’s datasheet appears in the equation. The voltage/temperature coefficient KV brings important informationnecessary to achieve the best possible I-V curve fitting fortemperatures different of the nominal value.If one wish to keep the traditional equation (4) [32, 33, 35–38], instead of using (6), it is possible to obtain the best valueof Eg for the model so that the open-circuit voltages of themodel are matched with the open-circuit voltages of the realarray in the range Tn T Tmax . By equaling (4) and (6)and solving for Eg at T Tmax one gets: Isc,TmaxTn 3I0,nTmax · Eg ln qVoc,Tmaxexp 1aNs kTmaxakTTmaxn·q (Tn Tmax ) (7)where Isc,Tmax Isc,n KI T and Voc,Tmax Voc,n KV T , with T Tmax Tn .D. Adjusting the modelTwo parameters remain unknown in (2), which are Rs andRp . A few authors have proposed ways to mathematically determine these resistances. Although it may be useful to havea mathematical formula to determine these unknown parameters, any expression for Rs and Rp will always rely on experimental data. Some authors propose varying Rs in an iterativeprocess, incrementing Rs until the I-V curve visually fits theexperimental data and then vary Rp in the same fashion. Thisis a quite poor and inaccurate fitting method, mainly becauseRs and Rp may not be adjusted separately if a good I-V modelis desired.This paper proposes a method for adjusting Rs and Rpbased on the fact that there is an only pair {Rs ,Rp } that warranties that Pmax,m Pmax,e Vmp Imp at the (Vmp , Imp )point of the I-V curve, i.e. the maximum power calculatedby the I-V model of (2), Pmax,m , is equal to the maximum experimental power from the datasheet, Pmax,e , at the maximumpower point (MPP). Conventional modeling methods found inthe literature take care of the I-V curve but forget that theP -V (power vs. voltage) curve must match the experimentaldata too. Works like [26, 39] gave attention to the necessityof matching the power curve but with different or simplifiedmodels. In [26], for example, the series resistance of the arraymodel is neglected.The relation between Rs and Rp , the only unknowns of (2),may be found by making Pmax,m Pmax,e and solving theresulting equation for Rs , as (8) and (9) show.1246

250200V: 26.3P: 200.1180200MPP160V: 26.3P: 200.1Pmax [W]P [W]140120100Rs V [V]25Fig. 4. P -V curves plotted for different values of Rs and Rp . Pmax,m q Vmp Rs Imp 1Ipv I0 exp (8)kT aNsVmp Rs Imp Pmax,eRpV: 26.3I: 7.617MPP65I [A] Vmp35Fig. 5. Pmax,m vs. V for several values of Rs 0.8 30V [V]432Eq. (9) means that for any value of Rs there will be a valueof Rp that makes the mathematical I-V curve cross the experimental (Vmp , Imp ) point.E. Iterative solution of Rs and RpThe goal is to find the value of Rs (and hence Rp ) thatmakes the peak of the mathematical P -V curve coincide withthe experimental peak power at the (Vmp , Imp ) point. Thisrequires several iterations until Pmax,m Pmax,e .In the iterative process Rs must be slowly incremented starting from Rs 0. Adjusting the P -V curve to match the experimental data requires finding the curve for several valuesof Rs and Rp . Actually plotting the curve is not necessary, asonly the peak power value is required. Figs. 4 and 6 illustratehow this iterative process works. In Fig. 4 as Rs increasesthe P -V curve moves to the left and the peak power (Pmax,m )goes towards the experimental MPP. Fig. 5 shows the contour drawn by the peaks of the power curves for several values of Rs (this example uses the parameters of the KyoceraKC200GT solar array [40]). For every P -V curve of Fig. 4there is a corresponding I-V curve in Fig. 6. As expectedfrom (9), all I-V curves cross the desired experimental MPPpoint at (Vmp , Imp ).Plotting the P -V and I-V curves requires solving (2) forI [0, Isc,n ] and V [0, Voc,n ]. Eq. (2) does not have adirect solution because I f (V, I) and V f (I, V ). Thistranscendental equation must be solved by a numerical methodand this imposes no difficulty. The I-V points are easily obtained by numerically solving g(V, I) I f (V, I) 0 for aset of V values and obtaining the corresponding set of I points.Obtaining the P -V points is straightforward.978-1-4244-3370-4/09/ 25.00 2009 IEEE1(9)005101520253035V [V]Fig. 6. I-V curves plotted for different values of Rs and Rp .The iterative method gives the solution Rs 0.221 Ω forthe KC200GT array. Fig. 5 shows a plot of Pmax,m as a function of V for several values of Rs . There is an only point, corresponding to a single value of Rs , that satisfies the imposedcondition Pmax,m Vmp Imp at the (Vmp , Imp ) point. Fig. 7shows a plot of Pmax,m as a function of Rs for I Imp andV Vmp . This plot shows that Rs 0.221 Ω is the desiredsolution, in accordance with the result of the iterative method.This plot may be an alternative way for graphically finding thesolution for Rs .220218216214Pmax [W]Rp Vmp (Vmp Imp Rs )/(Vmp Imp Rs ) q{ Vmp Ipv Vmp I0 exp Ns akT Vmp I0 Pmax,e }2122102082062042022000R: 0.221P: 200.10.10.20.30.40.50.60.7Rs [Ω]Fig. 7. Pmax f (Rs ) with I Imp and V Vmp .1247

8TABLE IIParameters of the adjusted model of the KC200GTsolar array at nominal operating conditions.V: 0I: 8.21V: 26.3I: 7.617I [A]654321V: 32.9I: 0005101520253035V [V]Fig. 8. I-V curve adjusted to three remarkable points.ImpVmpPmax,mIscVocI0,nIpvaRpRs7.61 A26.3 V200.143 W8.21 A32.9 V9.825 · 10 8 A8.214 A1.3415.405 Ω0.221 Ω200best model solution, so equation (10) may be introduced in themodel.V: 26.3P: 200.1180160P [W]140Ipv,n 120Rp RsIsc,nRp(10)1008060402000V: 0P: 0V: 32.9P: 05101520253035V [V]Fig. 9. P -V curve adjusted to three remarkable points.Figs. 8 and 9 show the I-V and P -V curves of theKC200GT photovoltaic array adjusted with the proposedmethod. The model curves exactly match with the experimental data at the three remarkable points provided by thedatasheet: short circuit, maximum power, and open circuit.Table I shows the experimental parameters of the array obtained from the datasheeet and Table II shows the adjusted parameters and model constants.F. Further improving the modelThe model developed in the preceding sections may be further improved by taking advantage of the iterative solution ofRs and Rp . Each iteration updates Rs and Rp towards theTABLE IParameters of the KC200GT PV array at25 C, AM1.5, 1000 W/m2 .ImpVmpPmax,eIscVocKVKINs7.61 A26.3 V200.143 W8.21 A32.9 V 0.1230 V/K0.0032 A/K54978-1-4244-3370-4/09/ 25.00 2009 IEEEEq. (10) uses the resistances Rs and Rp to determine Ipv 6 Isc . The values of Rs and Rp are initially unknown but as thesolution of the algorithm is refined along successive iterationsthe values of Rs and Rp tend to the best solution and (10)becomes valid and effectively determines the light-generatedcurrent Ipv taking in account the influence of the series andparallel resistances of the array. Initial guesses for Rs and Rpare necessary before the iterative process starts. The initialvalue of Rs may be zero. The initial value of Rp may be givenby:Rp,min Voc,n VmpVmp Isc,n ImpImp(11)Eq. (11) determines the minimum value of Rp , which isthe slope of the line segment between the short-circuit and themaximum-power remarkable points. Although Rp is still unknown, it surely is greater than Rp,min and this is a good initialguess.G. Modeling algorithmThe simplified flowchart of the iterative modeling algorithmis illustrated in Fig. 10.III. VALIDATING THE MODELAs Tables I and II and Figs. 8 and 9 have shown, the developed model and the experimental data are exactly matchedat the nominal remarkable points of the I-V curve and the experimental and mathematical maximum peak powers coincide.The objective of adjusting the mathematical I-V curve at thethree remarkable points was successfully achieved.In order to test the validity of the model a comparison withother experimental data (different of the nominal remarkablepoints) is very useful. Fig. 11 shows the mathematical I-Vcurves of the KC200GT solar panel plotted with the experimental data at three different temperature conditions. Fig.12 shows the I-V curves at different irradiations. The circular markers in the graphs represent experimental (V, I) points1248

Inputs: T , GI0 , eq. (4) or (6)Rs 0Rp Rp,min , eq. (11)KYOCERA KC200GT - 25 C91000 W/m2872800 W/m6I [A]εP max tolENDnoyes600 W/m254400 W/m23Ipv,n , eq. (10)Ipv and Isc , eq. (3)Rp , eq. (9)Solve eq. (2) for 0 V Voc,nCalculate P for 0 V Voc,nFind PmaxεP max kPmax Pmax,e kIncrement Rs210051015202530V [V]Fig. 12. I-V model curves and experimental data of the KC200GTsolar array at different irradiations, 25 C.Fig. 10. Algorithm of the method used to adjust the I-V model.KYOCERA KC200GT - 1000 W/m29SOLAREX MSX60 - 1000 W/m284.5775 C50 C425 C575 C3432.5221.510025 C3.5I [A]I [A]615101520250.53000V [V]5solar array at different temperatures, 1000 W/m .1520V [V]Fig. 11. I-V model curves and experimental data of the KC200GT210Fig. 13. I-V model curves and experimental data of the MSX60solar array at different temperatures, 1000 W/m2 .SOLAREX MSX60 - 1000 W/m26025 C5030201000IV. SIMULATION OF THE PHOTOVOLTAIC ARRAYThe photovoltaic array can be simulated with an equivalentcircuit model based on the photovoltaic model of Fig. 1. Twosimulation strategies are possible.Fig. 15 shows a circuit model using one current source978-1-4244-3370-4/09/ 25.00 2009 IEEE75 C40P [W]extracted from the datasheet. Some points are not exactlymatched because the model is not perfect, although it is exact at the remarkable points and sufficiently accurate for otherpoints. The model accuracy may be slightly improved by running more iterations with other values of the constant a, without modifications in the algorithm.Fig. 13 shows the mathematical I-V curves of the SolarexMSX60 solar panel [41] plotted with the experimental dataat two different temperature conditions. Fig. 14 shows theP -V curves obtained at the two temperatures. The circularmarkers in the graphs represent experimental (V, I) and (V, P )points extracted from the datasheet. Fig. 14 proves that themodel accurately matches with the experimental data both inthe current and power curves, as expected.5101520V [V]Fig. 14. P -V model curves and experimental data of the MSX60solar array at different temperatures, 1000 W/m2 .1249

IRsImRp VI-I -VI V Rs IIpv I0 exp 1Vt a -VVIpvNumerical solution of eq. (2)I0Fig. 16. Photovoltaic array model circuit with a controlled currentFig. 15. Photovoltaic array model circuit with a controlled currentsource and a computational block that solves the I-V equation.source, equivalent resistors and the equation of the model current(Im ).(Im ) and two resistors (Rs and Rp ). This circuit can be implemented with any circuit simulator. The value of the modelcurrent Im is calculated by the computational block that hasV , I, I0 and Ipv as inputs. I0 is obtained from (4) or (6) andIvp is obtained from (3). This computational block may beimplemented with any circuit simulator able to evaluate mathfunctions.Fig. 16 shows another circuit model composed of only onecurrent source. The value of the current is obtained by numerically solving the I-V equation. For every value of V a corresponding I that satisfies the I-V equation (2) is obtained. Thesolution of (2) can be implemented with a numerical methodin any circuit simulator that accepts embedded programming.This is the simulation strategy proposed in [42].Other authors have proposed circuits for simulating photovoltaic arrays that are based on simplified equations and/orrequire lots of computational effort [12, 26, 27, 43]. In [12]a circuit-based photovoltaic model is composed of a currentsource driven by an intricate and inaccurate equation wherethe parallel resistance is neglected. In [26] an intricate PSpicebased simulation was presented, where the I-V equation isnumerically solved within the PSpice software. Although interesting, the approach found in [26] is excessively elaboratedand concerns the simplified photovoltaic model without theseries resistance. In [27] a simple circuit-based photovoltaicmodel is proposed where the parallel resistance is neglected.In [43] a circuit-based model was proposed based on the piecewise approximation of the I-V curve. Although interestingand relatively simple, this method [43] does not provide a solution to find the parameters of the I-V equation and the circuitmodel requires many components.Figs. 17 and 18 show the photovoltaic model circuits implemented with MATLAB/SIMULINK (using the SymPowerSystems blockset) and PSIM using the simulation strategyof Fig. 15. Both circuit models work perfectly and may beused in the simulation of power electronics converters for photovoltaic systems. Figs. 19 and 20 show the I-V curves ofthe Solarex MSX60 solar panel [41] simulated with the MATLAB/SIMULINK and PSIM circuits.V. CONCLUSIONThis paper has analyzed the development of a method forthe mathematical modeling of photovoltaic arrays. The objective of the method is to fit the mathematical I-V equation to978-1-4244-3370-4/09/ 25.00 2009 IEEEFig. 17. Photovoltaic circuit model built withMATLAB/SIMULINK.the experimental remarkable points of the I-V curve of thepractical array. The method obtains the parameters of the IV equation by using the following nominal information fromthe array datasheet: open-circuit voltage, short-circuit current,maximum output power, voltage and current at the maximumpower point, current/temperature and voltage/temperature coefficients. This paper has proposed an effective and straightforward method to fit the mathematical I-V curve to the three1250

Fig. 18. Photovoltaic circuit model built with PSIM.43.532.521.525 C,1000W/m2125 C,800W/m2275 C,1000W/m0.5275 C,800W/m005101520Fig. 19. I-V curves of the model simulated withMATLAB/SIMULINK.(V, I) remarkable points without the need to guess or to estimate any other parameters except the diode constant a. Thispaper has proposed a closed solution for the problem of findingthe parameters of the single-diode model equation of a practical photovoltaic array. Other authors have tried to proposesingle-diode models and methods for estimating the model parameters, but these methods always require visually fitting themathematical curve to the I-V points and/or graphically extracting the slope of the I-V curve at a given point and/or successively solving and adjusting the model in a trial and errorprocess. Some authors have proposed indirect methods to adjust the I-V curve through artificial intelligence [15, 44–46]and interpolation techniques [25]. Although interesting, suchmethods are not very practical and are unnecessarily complicated and require more computational effort than it would beexpected for this problem. Moreover, frequently in these models Rs and Rp are neglected or treated as independent parameters, which is not true if one wish to correctly adjust the modelso that the maximum power of the model is equal to the maximum power of the practical array.An equation to express the dependence of the diode saturation current I0 on the temperature was proposed and used inthe model. The results obtained in the modeling of two practical photovoltaic arrays have demonstrated that the equationis effective and permits to exactly adjust the I-V curve at theopen-circuit voltages at temperatures different of the nominal.Moreover, the assumption Ipv Isc used in most of previous works on photovoltaic modeling was replaced in thismethod by a relation between Ipv and Isc based on the seriesand parallel resistances. The proposed iterative method forsolving the unknown parameters of the I-V equation allows todetermine the value of Ipv , which is different of Isc .This paper has presented in details the equations that constitute the single-diode photovoltaic I-V model and the algorithm necessary to obtain the parameters of the equation. In order to show the practical use of the proposed modeling methodthis paper has presented two circuit models that can be used tosimulate photovoltaic arrays with circuit simulators.This paper provides the reader with all necessary information to easily develop a single-diode photovoltaic array modelfor analyzing and simulating a photovoltaic array. Programsand ready-to-use circuit models are available for download APPENDIX - ASSOCIATION OF PV ARRAYSFig. 20. I-V curves of the model simulated with PSIM.978-1-4244-3370-4/09/ 25.00 2009 IEEEIn the previous sections this paper has dealt with the modeling and simulation of photovoltaic arrays that are single panelsor modules composed of several interconnected basic photovoltaic cells. Large arrays composed of several panels may bemodeled in the same way, provided th

Keywords - PV array, modeling, simulation. I. INTRODUCTION A photovoltaic system converts sunlight into electricity. The basic device of a photovoltaic system is the photovoltaic cell. Cells may be grouped to form panels or modules. Panels can be groupedto form large photovoltaicarrays. The term ar-