Chapter 2 Modeling of Solar Irradiance and Cells Symbols A C(N) CT dX E Et fcirc G Gref g gh GMT H_ HA h hastr Hd H d0 H d0 H d0 n Hdn H*dn e Hdnk e Hdn H0e Hs Diode ideality factor Distance correction Civil time Elementary solid angle Normal irradiance of a beam radiation Time equation The circumsolar fraction Global irradiance on a plane (W/m2) Reference irradiance (1000 W/m2) Asymmetry factor of the phase function P(H) Asymmetry factor of the hemispherical phase function Greenwich Mean Time Global irradiance on a horizontal surface Sun hour angle Apparent Sun elevation Astronomic (real) sun elevation Direct irradiance Directional irradiance Directional irradiance on a horizontal surface Directional Sun irradiance on a plane perpendicular to the Sun beam Direct Sun irradiance on a plane perpendicular to Sun beam Value of Hdn by sunshine time Normal direct irradiance outside the atmosphere for wave length k Normal direct irradiance outside the atmosphere Solar constant Diffuse (scattered) irradiance D. Rekioua and E. Matagne, Optimization of Photovoltaic Power Systems, Green Energy and Technology, DOI: 10.1007/978-1-4471-2403-0_2, Springer-Verlag London Limited 2012 31 32 Hsg Hsh Hsh Hsh 2 Modeling of Solar Irradiance and Cells sky sky- I0 Iapp(X) Id Is app hem Iph IRsh Isc Isc-tjref i J(l, u) J0(l,u) k K k0 kk dm k0 (m) (lat) (long) m m0 m0 z ref MST N P(H) P1, P2 and P3 p p0 psea level q Rloc Rreg Rs Rsh RST STC T Hemispherical irradiance coming from ground Hemispherical irradiance Hemispherical irradiance coming from the sky Hemispherical irradiance on a horizontal surface coming from the sky Reverse saturation current of a diode (A) Global apparent radiance of the sky coming from the direction X Current shunted through the intrinsic diode Hemispherical diffuse apparent radiance of the sky Photocurrent Current of the shunt resistance Short-circuit current Short-circuit current at rated temperature Angle between a direction X and the normal to a plane Radiance source due to multiple scattering Radiance source due to first scattering Extinction coefficient in atmospheric model Boltzman constant (K = 1.381910-23 J/K) in cell model Directional extinction coefficient Elementary relative extinction coefficient at wave length k Value of k corresponding to the standard atmosphere North latitude of the place East longitude of the place Relative (without physical dimension) Air Mass Absolute air mass Absolute air mass of a standard atmosphere in the vertical direction (between TOA and the sea level) Local mean solar time Number of the day of the year So-called phase function Constant parameters Atmospheric pressure in hPa (mbars) Pressure used in the definition of the standard atmosphere (p0 = 1013.25 hPa) Pressure at the sea level Quantum of charge (1.602910-19 C) Local ground albedo (the fraction of the light received by the ground which is reflected) Regional soil albedo Series resistance Shunt resistance Real solar time at the place Standard conditions Temperature 2 Modeling of Solar Irradiance and Cells Tj Tj ref TLinke TD TOA UT Vpv z asc boc c1 and c2 d H h0 h0 astr hp k l, l0, lp u u0 up x x0 -0 ; -1 ; -2 33 Junction temperature Reference temperature Linke turbidity factor Time difference (which is defined for each country, with in some countries a seasonal change) Top of Atmosphere Universal Time Voltage across the PV cell Altitude Temperature coefficient of short-circuit current found from the datasheet (absolute or relative) Temperature coefficient of open-circuit voltage found from the datasheet (absolute or relative) Coefficients Declination Angle between the incident and scattered light Apparent Sun zenithal angle Astronomic (real) Sun zenithal angle u0 Inclination of a plane Wave length Cosines of the corresponding zenithal angles Azimuth Sun azimuth Orientation of a plane Ratio of the scattering coefficient to the sum of the scattering and absorption coefficients The ratio of the hemispherical scattering coefficient to the sum of the hemispherical scattering and absorption coefficients Coefficients of the decomposition of Ih in Legendre polynomials In order to obtain a realistic view of the behavior of a photovoltaic system, it is necessary to achieve computer simulations. For that purpose, the most important data is the light irradiance of the photovoltaic array at a small time scale (some minutes) during a significant duration (one year or more). Unfortunately, complete experimental data (irradiance for all module inclination and orientation) are never available. For example, the only available measurement result is often hourly or daily global irradiance on a horizontal plane. Sky modeling is thus necessary in order to deduce from the available partial data, a realistic estimation of the module irradiance and some spectral characteristics of that irradiance. Of course, that first result is useful only in conjunction with a model of the photovoltaic modules, in order to deduce from it the electrical power generation for varying irradiance, spectrum, and temperature. When one is concerned by optimization of a system, it is important to use models well suited in order to achieve the performance evaluation of each tested 34 2 Modeling of Solar Irradiance and Cells configuration in a short time, and so to have the possibility of comparing a large number of possibilities. The first part of this chapter is devoted to irradiation estimation using simplified sky or atmosphere models. The second part is devoted to module modeling. 2.1 Irradiance Modeling 2.1.1 Principles and First Simplifying Assumption 2.1.1.1 Sun Light Travel In order to reach photovoltaic modules, sunlight must go through the atmosphere, where it is subject to absorption and scattering. A part of sunlight reaches the module without undergoing these phenomena: it is named the direct radiation. During its travel through atmosphere, a part of the light is scattered by air molecules, aerosols (dust), water drops or ice crystals, and also by the ground surface. That light has still a chance to arrive on the photovoltaic module after one or several scatterings. That part of module irradiation is named the diffuse fraction. By clear sky, the main part of irradiance is the direct one. By overcast sky, global irradiation is lower and the diffuse to global ratio is higher. So, the light which reaches a photovoltaic module can come from Sun by a variety of ways, as schematically shown on Fig. 2.1. In this chapter, we do not consider the infrared radiation coming from the atmosphere, since this one has wavelengths too larger for inducing photovoltaic generation. However, we consider ‘‘light’’ as also the infrared radiation coming from Sun, because the part of that radiation which is not stopped by atmosphere has wavelengths able to cause photovoltaic effect. 2.1.1.2 Angles Definition In order to correctly describe the different directions, we define a spherical coordinate system called horizontal coordinates, as shown in Fig. 2.2, where ZN is the local vertical. Then, any direction X can be specified by the two angles h and u. In particular, the normal to the receiving plane can be specified by hp and up, which are respectively the inclination of the plane and its orientation. A solid angle is defined as the surface intersected by a cone on a unit radius sphere centered on its top. In spherical coordinate, the elementary solid angle is dX ¼ sin h dh du ð2:1Þ and is expressed in steradians (sr) if h and u are expressed in radians (rad). Defining global irradiance G on a plane as the power received from light by unit area, we have then 2.1 Irradiance Modeling 35 Fig. 2.1 Ways of the solar radiation Fig. 2.2 Definition of horizontal coordinates. PP0 is the direction of the Earth rotation axis Fig. 2.3 Angle i definition G¼ ZZ Iapp ðXÞ dX ¼ i\90 ZZ Iapp ðcos h; /Þ cos i sin h dh du ð2:2Þ i\90 where Iapp(X) is the global apparent luminance of the sky coming from the direction X and i the angle between that direction and the normal to the plane (Fig. 2.3). 36 2 Modeling of Solar Irradiance and Cells Fig. 2.4 Equatorial coordinates definition The spherical geometry allows to compute the angle i by cos i ¼ cos h cos hp þ sin h sin hp cos ðu up Þ ð2:3Þ 2.1.1.3 Sun Position Computation The Sun position is defined by angles h0 and u0, respectively the Sun zenithal angle and the Sun azimuth. Instead of h0, we often use angle h¼ p h0 2 ð2:4Þ named Sun elevation. h is the angle between the sun direction and the horizontal plane. Sun position is in practice never measured, since it is easy to compute it knowing date and time, longitude and latitude of the place, and some astronomic data. The computation is made easier in a coordinates system different from Fig. 2.2, namely the equatorial coordinates. These ones are defined as shown at Fig. 2.4, where (m) is the local meridian as defined on Fig. 2.2. In that system, the coordinates are angles d and HA, called respectively the declination and the hour angle. Sun declination and hour angle can be computed with a very large accuracy by astronomical methods. Simplified computation methods without significant error for the present purpose can be found in the literature. An example of such code is given in [13]. If some degrees inaccuracy is acceptable, we can use the following formulae [14]. Sun declination is obtained as 2p 2 p ðN 2Þ sin dðNÞ ¼ 0:398 sin N 82 þ 2 sin ð2:5Þ 365 365 2.1 Irradiance Modeling 37 where N is the number of the day of the year In order to obtain the hour angle H, the following operations are leaded starting with the civil time CT in hours (a) UT (Universal Time) or GMT (Greenwich Mean Time) is obtained by subtracting from CT the time difference TD (which is defined for each country, with in some countries a seasonal change). UT ¼ CT TD ð2:6Þ (b) Using the east longitude of the place, we obtain the local mean solar time of the place by MST ¼ UT þ ðlongÞ 15 ð2:7Þ where (long) is the east longitude of the place expressed in , MST and UT being in hours. (c) Then, the real solar time at the place is obtained using the equation RST ¼ MST þ Et ð2:8aÞ where Et is the time equation, which take into account the fact that the rotation speed of the Earth around Sun is not uniform. We have approximately, in hours, Et ðNÞ ¼ 1 ½9:87 sin ð2N 0 Þ 7:53 cos ðN 0 Þ 1:5 sin ðN 0 Þ 60 ð2:8bÞ with N0 ¼ 2p ðN 81Þ 365 ð2:8cÞ (d) The hour angle is linked to the real solar time by the relation HA ¼ p ðRST 12Þ 12 ð2:9Þ Once the angles d and HA are known, we can compute the angles h0 astr or hastr and u0 by a change of coordinates: sin hastr ¼ cos HA cos d cos ðlatÞ þ sin d sin ðlatÞ ð2:10aÞ cos u0 cos hastr ¼ cos HA cos d sin ðlatÞ sin d cos ðlatÞ ð2:10bÞ sin u0 cos hastr ¼ sin HA cos ðdÞ where (lat) is the north latitude. ð2:10cÞ 38 2 Modeling of Solar Irradiance and Cells It is to be noticed that the determination of u0 without ambiguity is possible only using all the two Eqs. 2.10b and 2.10c. The apparent Sun zenithal angle h0 is approximately equal to the astronomical angle h0 astr: a small difference occurs due to the atmospheric refraction. In photovoltaic studies, we assume frequently h0 hastr or; equivalently; h hastr ð2:11aÞ None correction to Eq. 2.11a is useful for small zenithal angle (Sun elevation near to 90). For larger zenithal angles (Sun elevation near to 0), the enhanced Saemundsson formula [15] is a little bit better: p 283:15 1:02 10:3 cotg hastr þ h hastr þ ð2:11bÞ 1010 273:15 þ T 60 hastr þ 5:14 for 2 \hastr \89 where p is the atmospheric pressure in hPa (mbars) and T the temperature in C, h and hastr being expressed in degrees. Of course, the formulae are not relevant when hastr \ -2 since it is then certainly the night. Taking into account the sun radius, sunrise or sunset arrives when h 0:27 : ð2:12Þ 2.1.2 Sky and Ground Radiance Modeling The global irradiance Eq. 2.2 can be split is two parts: direct irradiance and diffuse irradiance G ¼ Hd þ Hs ð2:13Þ where the index ‘‘d’’ stands for ‘‘direct’’ and the index ‘‘s’’ for ‘‘scattered’’ (diffuse). 2.1.2.1 Direct Radiation Direct Sun radiation (also named beam radiation) is assumed coming from a point at the Sun disk center. The direct radiance is thus a Dirac delta on the point of coordinate h0 (or h) and u0. Then, the corresponding part of Eq. 2.2 reduces to Hd ¼ 0 ð2:14aÞ if cos i\0 Hd ¼ Hdn cos i if cos i [ 0 ði\90 Þ ð2:14bÞ 2.1 Irradiance Modeling 39 where the index ‘‘n’’ is for ‘‘normal’’ and, thus, Hdn is the direct Sun irradiance on a plane perpendicular to Sun beam. i is the incidence angle of direct radiation on the plane. It is easy to compute that incidence angle i, using the particular case of Eq. 2.3 where h = h0, cos i ¼ cos h0 cos hp þ sin h0 sin hp cosðu0 up Þ ¼ sin h cos hp þ cos h sin hp cos ðu0 up Þ ð2:15Þ So, using twice (2.14), it is sufficient to have only one measurement of direct radiation on one irradiated plane to compute Hdn and thus the value of Hd on any plane. When the sky is uniform (clear or uniformly cloudy), Hdn is a smooth function of time. However, in case of incomplete cover by thick clouds, Hdn experiences quick changes between a value H*dn for sunshine time and nearly 0 when there is a cloud in front of Sun. In that case, we speak about ‘‘bimodal state’’. A related notion is the sunshine duration. The sunshine duration is the time for which Hdn [ 120 W. Thus, if H*dn [ 120 W, the ratio between the sunshine duration and the time of recording is equal to the probability to have Hdn = H*dn at a particular time. That probability is also approximately equal to the complement to unity of the cloud cover (in per unit) 2.1.2.2 Circumsolar Diffuse Radiation Restraining the Eq. 2.2 to the diffuse part, we have ZZ ZZ Hs ¼ Is app ðXÞdX ¼ Is app ðcos h; uÞ cos i sin h dh du i\90 ð2:16Þ i\90 Searching for a simplified expression of Is app (cos h, u), a common approximation [16] is to split that function of two coordinates in a sum of two function of only one coordinate, namely a circumsolar part and an hemispherical part: Is app ¼ Is app circ ðHÞ þ Is app hem ðhÞ ð2:17Þ where the circumsolar part is function only of the angle between the Sun direction and the observation direction, and the hemispheric part is only function of h. That splitting comes from the common observation that the radiance of the part of the sky which is near to the Sun is higher than the other parts of the sky. Angle H is simply cos H ¼ cos h cos h0 þ sin h sin h0 cosðu u0 Þ ð2:18Þ For the irradiance computation, usually, we do not use Eq. 2.18, but we consider that all the circumsolar part comes from the Sun disk center. Then, direct irradiance and circumsolar irradiance can be treated as a whole, which is named ‘‘directional’’ ‘‘irradiance’’. We use the index ‘‘d’’’ for ‘‘directional’’. We obtain 40 2 Modeling of Solar Irradiance and Cells Fig. 2.5 Total irradiance on a surface in function of the inclination for different orientations. * are experimental values [17], curves are computed by Eq. 2.20 with Hd0 n and Hsh fitted for the best approximation for the directional component an expression very similar to that obtained for direct component (2.14): Hd0 ¼ 0 if cos i\0 Hd0 ¼ Hd0n cos i if cos i [ 0 ði\90 Þ ð2:19aÞ ð2:19bÞ As the radiance Is app hem depends only of h, it is clear that the corresponding irradiance depends only of the plane inclination hp. We have then G ¼ Hd0n cos i þ Hsh ðhp Þ ð2:20Þ That partition can give a good approximation of real irradiance, as it is shown at Fig. 2.5, which uses global irradiance experimental data from the literature [17] versus inclination for different orientations at a fixed time (clear sky, Davis California, h0 = 90–34 = 56). From Eq. 2.20, it is obvious that, in order to compute the directional part of irradiation, it is sufficient to have two experimental values of global irradiance on two planes of same inclination hp but different orientations up provided they are dissymmetric with regard to Sun azimuth u0. However, it remains useful to split the directional irradiance into direct and circumsolar irradiances since only the direct component is submitted to bimodal state. An important particular case of Eq. 2.20 is the irradiance on the horizontal plane, which is often measured in meteorological stations. In that case, following Eq. 2.15 for hp = 0, we have cos i = sin h and thus Eq. 2.20 becomes: G ¼ Hd0 n sin h þ Hsh ð2:21Þ 2.1.2.3 Ground Diffuse Radiation The next step in our analysis is to split the hemispherical irradiance into a part coming from the sky and a part coming from ground. Hsh ¼ Hsh sky þ Hsg ð2:22Þ 2.1 Irradiance Modeling 41 This ground radiance is often assumed to be isotropic, and thus purely hemispherical. For that reason, we have omitted the index ‘‘h ’’ in the last term of Eq. 2.22. Then, reducing the integral Eq. 2.2 to the ground diffuse radiation, we can show that the corresponding irradiance is Hsg ¼ Rloc G 1 cos hp 2 ð2:23Þ where Rloc is the local ground albedo (the fraction of the light received by the ground which is reflected). In the expression Eq. 2.23, the value of G- is given by (2.21). It shall be noticed that the ground diffuse irradiance does not contribute to Hs h-. G ¼ Hd0 þ Hsh sky ð2:24Þ 2.1.2.4 Sky Hemispherical Diffuse Irradiance It remains for achieving the analysis to give an expression for the sky hemispherical diffuse irradiance Hsh sky(hp). The simplest assumption is that the sky hemispherical irradiance results from an isotropic radiance. Then, the corresponding part of Eq. 2.2 leads to the expression Hsh sky ¼ Hsi ¼ Hsi 1 þ cosðhp Þ 2 ð2:25Þ where the index ‘‘i’’ stands for ‘‘isotropic sky’’ and thus implicitly hemispherical. In that case, the full description of irradiance as a function of the inclination and orientation of a plane is obtained using only three parameters: Hd0 n, Hsi- and Rloc. If Rloc is known, the two irradiance measurement requested at Sect. 2.1.2.2. are sufficient in order to identify that function. Unfortunately, the sky isotropy assumption is contrary to the common observation that the part of the sky near to the horizon is often clearer than the other parts of the sky. So, different authors have introduced empirical additional terms, the most known being the horizon circle diffuse irradiance. We shall not consider such terms in that book because we prefer obtain the expression of Hsh sky from a physical analysis of atmosphere, which is achieved below. 2.1.3 Use of an Atmospheric Model In many cases, the available experimental irradiance data are insufficient, because • they do not allow computing the irradiance on the considered plane using the methods described in 2.1.2, 42 2 Modeling of Solar Irradiance and Cells • a best approximation of Hsh sky than Eq. 2.25 is required and the number of irradiance measurements is insufficient for that, • they are given as mean values on too large time intervals, • information on spectral characteristics is needed but not available. In particular, we stress the fact that average values of irradiance are not suited for direct use in photovoltaic studies since • the power generation of photovoltaic modules is not proportional to irradiance, • for the systems simulations, the energy production timing is of importance. Then, atmospheric modeling can be used to extend in a realistic way the available experimental data. 2.1.3.1 Air Mass Notion As the air density increases from 0 at the TOA (top of atmosphere) to its value at the ground level, the air influence on a light ray is more important at low altitude. For that reason, it is useful to use as coordinate not the distance l covered by the light ray but the Air Mass Z 0 m ¼ q dl ð2:26Þ def The use of Air Mass in the equation is related to the assumptions that • the air molecules and particles act on light individually, • their effect is independent of temperature. Besides being not totally exact, these assumptions are acceptable for our purpose. A particular case of Eq. 2.26 is the case of a vertical ray. We define m0 z ref as the Air Mass of a standard atmosphere in the vertical direction between TOA and the sea level. Then, we can use instead of absolute Air Mass Eq. 2.26 a relative (without physical dimension) Air Mass m ¼ def m0 m0z ref ð2:27Þ Vertical Air Mass mz is strongly related to the atmospheric pressure. Assuming that the gravitational field g is constant in the entire atmosphere (g & 9.81 m/s2), vertical Air Mass is given by mz ¼ p p0 ð2:28Þ 2.1 Irradiance Modeling 43 Fig. 2.6 Definition of the extinction coefficient where p0 is the pressure used in the definition of the standard atmosphere (p0 = 1013.25 hPa). If one knows the pressure at the sea level, we can find the pressure at an altitude z (in km) using the formula [14] p ¼ psea level ð0:89Þz for z \3 km ð2:29Þ Light can be considered as the sum of electromagnetic radiations with different wave length k. Visible light has a spectrum going from k = 400 nm (violet) until k = 800 nm (red) (1 nm = 10-9 m). Sun radiation includes UV (ultraviolet) of shorter wave lengths and IR (infrared) of longer wave length. Some solar modules are able to get energy even from invisible Sun radiation. When a beam of light crosses an air volume, its energy is lowered by absorption and diffusion phenomena. The elementary relative lowering is given by the product kk dm, where dm is the elementary relative Air Mass in the beam direction and where kk is a coefficient named ‘‘extinction coefficient’’. kk is function only of k and of the atmosphere composition. Figure 2.6 gives an illustration of that phenomenon. 2.1.3.2 Direct Radiation Direct radiation, also called ‘‘beam radiation’’, is considered as coming from the direction computed at 2.2.1.3. It experiences an Air Mass m which is given, assuming the atmosphere plane and without refraction, by: m mz mz mz ¼ cosðh0 Þ sin h sin hastr ð2:30Þ If we consider the atmospheric refraction, the way followed by the beam radiation is lightly curved. We can take that fact into account replacing Eq. 2.30 by a best approximation, such as the Kasten formula [14] m mz 1 sin h þ 9:40 104 ðsin h þ 0:0678Þ1:253 ð2:31Þ The direct radiation is characterized by its normal irradiance Hdn, as in Sect. 2.1.2.1. The extinction mechanism of Fig. 2.6 leads thus to the equation: 44 2 Modeling of Solar Irradiance and Cells dHdnk ¼ kk Hdnk dm ð2:32Þ Assuming the atmosphere uniform, the solution of Eq. 2.32 is simply e ekk m Hdnk ¼ Hdnk ð2:33Þ e where Hdnk is the normal irradiance outside the atmosphere (at the TOA) Integrating on all the wave length, we obtain the total normal irradiance outside the atmosphere Z e e Hdn ¼ Hdnk dk ð2:34Þ When the Sun–Earth distance is equal to its mean value (1 astronomical unit), the e is called the ‘‘solar constant’’ H0e The value of that last one is value of Hdn e approximately H0e ¼ 1361 W=m2 [18]. Similarly, the mean values of the Hdnk can be found in the literature [19]. However, the Sun irradiation received at TOA varies during a year due to the e e variation of Sun–Earth distance. So, Hdn and Hdnk have to be deduced from their mean values using a distance correction 2p ðN 2Þ ð2:35Þ CðNÞ 1 þ 0:034 cos 365 where N is the number of the day of the year. For each wave length, kk can be decomposed in a sum. Each term of the sum take into account the effect of one atmospheric component. In fact, we can consider as only one component the set of all constant gas (N2, O2, .. CO2). Variable gases are the water vapor H2O, ozone O3 and sometime NO2. We have also to take into consideration the aerosol content (dust) and the water condensed in drops or in ice crystals (clouds). For simplified computation, we prefer to replace the infinite sum Eq. 2.34 by a finite one: X X e Hdn ¼ Hdn ai eki m with ai \1 ð2:36Þ i i For terrestrial applications, the sum of the coefficients ai can be lower than unity since some wave length are so strongly absorbed in the atmosphere that they do not contribute to the irradiance at the ground level. The decomposition Eq. 2.36 can maintain information on the light spectrum. For that, the spectrum can be divided in bands of wave length, each band corresponding to one or several exponential terms. On the other hand, for a very simplified computation, we can use only one exponential for the entire spectrum. In that case, it is convenient to admit that the extinction factor k is a function of m. A classical method is to use 2.1 Irradiance Modeling 45 kðmÞ ¼ k0 ðmÞTLinke ð2:37Þ where k0 (m) is the value corresponding to the standard atmosphere. TLinke is the Linke turbidity factor, which is a characteristic of the state of the atmosphere. Once k0(m) has been determined by computation, one measurement of the direct irradiation is sufficient in order to compute TLinke. Historically, several approximations of k0(m) have been proposed and lead to different value of TLinke. Thus, old experimental values of TLinke can need revision. Kasten [48] has given in 1996 an accurate expression: 1 6:6296 þ 1:7513 m 0:1202 m2 þ 0:0065 m3 0:00013 m4 1\m\20 k0 ðmÞ ¼ for ð2:38aÞ For higher values of m, one can use [49] k0 ðmÞ ¼ 1 10:4 þ 0:718 m for m [ 20 ð2:38bÞ Contrary to the irradiance, the value of TLinke depends only on little of the Sun elevation. Then, assuming that the meteorological conditions are constant during a time period, if one knows the experimental mean value of Hdn during that period, it is possible to search the value of TLinke which is constant on that period and which, by computation, leads to the same mean value of Hdn. Using that value of TLinke, it is then possible to compute at each time the value of Hdn. Of course, if one is in bimodal mode, that computation has to be adapted taking into account the sunshine duration. For photovoltaic computation, it is better to use H*dn with a sunshine probability in place of a smoothed value of Hdn for the above mentioned reasons. For the systems simulations where the energy production timing is of importance, it is possible to generate realistic variations of the direct irradiation at small time scale using the method of Markov chains. 2.1.3.3 Generalities About the Diffuse Radiation In the following, we use the variables: l ¼ cos h; l0 ¼ cos h0 ; lp ¼ cos hp . . . ð2:39Þ Behind the considered functions are wave length dependent, we shall make abstraction of the subscript ‘‘k’’, hoping that the formulae could be use for broadband calculations. The diffuse light is described at each point of the atmosphere by the radiance I(h, u) going to the direction (h, u). The apparent sky and ground radiances are then related to I(h, u) by: Is app ðl; uÞ ¼ Iðl; u þ pÞ ð2:40Þ 46 2 Modeling of Solar Irradiance and Cells When an unpolarized parallel beam of normal irradiance E crosses an elementary volume of thickness dm (given in relative Air Mass), the increase of scattered radiance is given by the equation dIðHÞ ¼ kx PðHÞE dm 4p ð2:41Þ where x is the ratio of the scattering coefficient to the sum of the scattering and absorption coefficients, H is the angle between the incident and scattered light and P(H) is the so-called phase function. The phase function is normalized in such a way that [20] 1 4p Z PðHÞdX ¼ 1 2 4p Zp PðHÞ sin HdH ¼ 1 ð2:42Þ 0 where dX is the elementary solid angle In presence of aerosols or water drops, the phase function is very complicated and, even when long numerical computations are achieved, we use for radiative transfer computations approximations of this function which depend only of a few parameters, the most important of them being the asymmetry factor defined as 1 g¼ 4p Z 1 PðHÞ cosðHÞdX ¼ 2 4p Zp PðHÞ cosðHÞ sinðHÞdðHÞ ð2:43Þ 0 From the above definitions, it can be proved that, in a plane atmosphere, the diffuse radiance I(l, u) obeys to the equation l dIðl; uÞ ¼ kIðl; uÞ Jðl; uÞ J0 ðl; uÞ dmz ð2:44Þ where J(l, u) is the source due to multiple scattering and J0(l, u) is the source due to first scattering. The source due to multiple scattering is given by the equation kx Jðl; uÞ ¼ 4p Z2p Z1 Iðl0 ; u0 ÞPðcos HÞdl0 du0 ð2:45Þ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃpﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 l2 1 l02 cosðu u0 Þ ð2:46Þ 0 1 with cos H ¼ ll0 þ On the other hand, the primary source of diffuse radiation is J0 ðl; uÞ ¼ kx H0 Pð cos H0 Þekmz =l0 4p ð2:47Þ 2.1 Irradiance Modeling 47 with cosðH0 Þ ¼ ll0 þ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃqﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 l2 1 l20 cosðu u0 Þ ð2:48Þ where h0 and u0 are the coordinates of the Sun and l0 is defined by Eq. 2.39. The equation system Eqs. 2.44–2.48 is very difficult to solve. Currently, an accurate solution of these equations can be obtained only by using time-consuming numerical calculations. Numerous empirical models have been developed. Reviews of these models can be found in the literature [21, 22]. Such models use functions whose numerous parameters are to be extracted from experimental data. They can thus be geography or climate dependent and present unpredictable limitations. In the following, we describe a realistic simplified model using only as data a few atmospheric parameters with well defined physical signification. 2.1.3.4 d-Approximation and Circumsolar Component The d-approximation [35] consist to assume that a fraction fcirc of the scattered light keeps the direction of the incident light, let’s be PðHÞ ¼ 2 fcirc dð1 cos HÞ þ ð1 fcirc ÞPh ðHÞ ð2:49Þ where the first term is a Dirac delta We consider then that the circumsolar component is due to the first term and has then exactly the same direction as the direct rays. On the other hand, the residual phase function Ph(H) shall be related to hemispherical diffuse radiation. Using the subscript d0 for the directional irradiance (direct ? circumsolar), we obtain an equation similar to Eq. 2.33: 0 e k m Hd0 n ¼ Hdn e ð2:50Þ where k0 is the directional extinction coefficient and k0 ¼ kð1 x fcirc Þ ð2:51Þ If necessary, the circumsolar component can be obtained as the difference between Eqs. 2.50 and 2.33. 2.1.3.5 Equations for Hemispherical Component We obtain equations for the residual diffuse radiance Ih (l, u) by replacing in Eqs. 2.44, 2.45 and 2.47 k by k0 and x by: x0 ¼ ð1 fcirc Þx 1 x fcirc ð2:52Þ 48 2 Modeling of Solar Irradiance and Cells Defining Ih (l) as the mean value of Ih (l, u) on –p \ u \ p, the equation for Ih (l) is obtained by replacing in the corresponding part of Eq. 2.44 I, J and J0 by their mean values on u. We get [24]: l dIh ðmz ; lÞ k0 x0 ¼ k0 Ih ðmz ; lÞ dmz 2 Z1 Ih ðmz ; l0 ÞPh ðl; l0 Þdl0 1 k 0 x0 0 Ph ðl; l0 ÞH0 ek mz =l0 4p ð2:53Þ where Z2p 1 Ph ðl; l Þ ¼ 2p 0 Ph ðHÞ du ð2:54Þ 0 The approximation is to consider that Ih (mz, l, u) can be identified with Ih (mz, l), and thus considered as a hemispherical radiance. There are some approximate solutions of the Eq. 2.53. The most known are called two-stream approximations. In those methods, the variables are the upward and downward hemispherical diffuse irradiation: Hhþ ðmz Þ ¼ 2p Z1 Ih ðmz ; lÞl dl ð2:55aÞ Ih ðmz ; lÞl dl ð2:55bÞ 0 and Hh ðmz Þ ¼ 2p Z1 0 H+h and Hh are the only variables of the differential system if we use an approximation of Ih(l) completely determined by H+h and Hh . That approximation can thus have only two freedom degrees. 2.1.3.6 Spherical Harmonics Decomposition The function Ih(mZ, l) can be decomposed in Legendre polynomial series. The first Legendre polynomials are P0 ðlÞ ¼ 1 P1 ðlÞ ¼ l 1 P2 ðlÞ ¼ ð3l2 1Þ 2 ð2:56Þ 2.1 Irradiance Modeling 49 These polynomials obey the orthogonallity condition 2i þ 1 2 Z1 Pi ðlÞPj ðlÞdl ¼ dij ð2:57Þ 1 And the recurrence formula [37] ð2i þ 1Þl Pi ðlÞ ¼ ði þ 1ÞPiþ1 ðlÞ þ i Pi1 ðlÞ ð2:58Þ We can consider decompositions Ih ðmz ; lÞ ¼ I0 ðmz ÞP0 ðlÞ þ 3 I1 ðmz ÞP1 ðlÞ þ 5 I2 ðmz ÞP2 ðlÞ þ ð2:59Þ and Ph ðlÞ ¼ -0 P0 ðlÞ þ -1 P1 ðlÞ þ -2 P2 ðlÞ þ ð2:60Þ The Legendre polynomials have the property [25]: 1 2p Z2p Pi ðcos HÞdu ¼ Pi ðlÞPi ðl0 Þ ð2:61Þ 0 Using this property and Eqs. 2.56, 2.57, 2.53 can be decomposed as: d d ði þ 1Þ Iiþ1 þ i Ii1 ¼ k0 ð2i þ 1ÞIi k0 x0 -i Ii dmz dmz k0 x0 -i 0 Pi ðl0 ÞH0 ek mz =l0 4p ð2:62Þ Applying the analog of Eqs. 2.42 and 2.43. for Ph(l), we have always [24] -0 ¼ 1 - 1 ¼ 3 gh ð2:63aÞ and, by reference to the Henyey-Greenstein phase function, there is a reason [23] to take -2 ¼ 0 ð2:63bÞ We assume that -i ¼ 0 for all ii1. In the particular case where gh = 0, the hemispherical diffusion is isotropic, i.e. Ph(l) = 1. In that case, the identification of that Ih (mz, l, u) with Ih (mz, l) is exact, and thus also the hemispherical hypothesis. In order to obtain a system of two differential equations, the decomposition of Eq. 2.59 can only involve two degrees of freedom. The Eddington approximation consists to keep only the first two terms of Eq. 2.59. However, empirical investigations [16] and accurate numerical solutions [26] show that an approximation of the first degree in l is not sufficient for a correct modeling of the hemispherical radiance. 50 2 Modeling of Solar Irradiance and Cells 2.1.3.7 Taking into Account the Second Order Term The following analysis is equivalent to the method developed in [27]. That approach is to consider the three first terms of the series Eq. 2.59 with an additional condition in order to keep only two freedom degrees. Equation 2.62 yields [28]: 2 dI1 0 ¼ a0 k0 I0 b0 k0 H0 ek mz =l0 dmz ð2:64aÞ dI2 dI0 0 þ ¼ a1 k0 I1 b1 k0 H0 ek mz =l0 dmz dmz ð2:64bÞ dI1 0 ¼ a2 k0 I2 b2 k0 H0 ek mz =l0 dmz ð2:64cÞ and, assuming I3 = 0, 2 where ai ¼ ð2i þ 1Þ x0 -i ð2:65aÞ 1 0 x -i Pi ðl0 Þ 4p ð2:65bÞ and bi ¼ Comparing Eqs. 2.64a and 2.64c, we see that the system is degenerated and can only have a solution if 0 I2 ¼ c1 I0 þ c2 H0 ek mz =l0 ð2:66aÞ with c1 ¼ 2 a0 a2 and c2 ¼ b2 2b0 a2 ð2:66bÞ Constraint Eq. 2.66 allows for replacing Eq. 2.64b by dI0 k0 mz =l0 ¼ a01 I1 b01 H0 e dmz ð2:67Þ where a01 ¼ a1 1 þ 2c1 and b01 ¼ b1 2c2 =l0 1 þ 2c1 ð2:68Þ The system Eqs. 2.64a and 2.67 is easy to solve. Its solution, completed with Eq. 2.66, is of the form 2.1 Irradiance Modeling 51 I0 ¼ qﬃﬃﬃﬃﬃ qﬃﬃﬃﬃﬃ kk0 mz kk0 mz k0 mz =l0 a01 c1 e þ a01 c2 e þ c3 H0 e ð2:69aÞ I1 ¼ pﬃﬃﬃﬃﬃ kk0 mz pﬃﬃﬃﬃﬃ kk0 mz k0 mz =l0 a0 c 1 e a0 c 2 e þ c4 H0 e ð2:69bÞ qﬃﬃﬃﬃﬃ qﬃﬃﬃﬃﬃ 0 0 0 a01 c1 eks þ c1 a01 c2 eks þ ðc1 c3 þ c2 ÞH0 es =l0 ð2:69cÞ I2 ¼ c1 where k¼ qﬃﬃﬃﬃﬃﬃﬃﬃﬃ a0 a01 ð2:70Þ c3 ¼ l0 ðb01 a01 b0 l0 Þ 1 a0 a01 l20 ð2:71aÞ c4 ¼ l0 ðb0 a0 b01 l0 Þ 1 a0 a01 l20 ð2:71bÞ and The coefficients c1 and c2 are to be determined using boundary conditions. For that, we use the upward and downward horizontal irradiance Eq. 2.55 whose expression is [28], by assuming coefficient I3 and the next ones equal to 0: 5 ¼ p I0 þ 2I1 þ I2 4 ð2:72aÞ 5 Hh ¼ p I0 2I1 þ I2 4 ð2:72bÞ Hhþ and Thus, introducing Eq. 2.69 into Eq. 2.72, qﬃﬃﬃﬃﬃ qﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃ 5 5 0 0 a01 1þ c1 2 a0 ekk mz c1 þp a01 1þ c1 2 a0 ekk mz c2 Hh ¼ p 4 4 5 5 0 þp 1þ c1 c3 þ c2 2c4 H0 ek mz =l0 ð2:73Þ 4 4 The boundary conditions are then Hh ¼ 0 at the TOA ðmz ¼ 0Þ Hhþ ¼ Rreg ðHh þ Hd0 n l0 Þ at the ground level where Rreg is the regional soil albedo. ðmz ¼ mz sfc Þ ð2:74aÞ ð2:74bÞ 52 2 Modeling of Solar Irradiance and Cells Fig. 2.7 Hemispherical radiance computed by the present model (with k0 mz = 0.220 x0 = 0.4141, gh = 0 and Rreg = 0.2779) There results a linear system of two equations in c1 and c2, which can thus be computed. Then, the expression Eq. 2.59 is completely determined. Integrating Eq. 2.59 similarly to Eq. 2.2, we obtain hemispherical irradiance on any plane. By example, considering the case of Fig. 2.5, we obtain good results using k0 mz = 0.220, x0 = 0.4141 gh = 0 and Rreg = 0.2779, behind the fact that the number of freedom degrees is much smaller. Figure 2.7 shows the corresponding radiance as a function of the zenith angle. We can see that this second order approximation is not realistic since it leads to negative values of the radiance. In fact, the function Ih(s0 ,l) thus obtained is a smooth function unable to correctly describe the directional properties of radiance near the ground level, because at that boundary the sky and soil radiances do not have the same form. 2.1.3.8 Enhanced Approximation for Hemispherical Radiance Although the second order model do not lead to realistic values of the radiance, it seems to give an acceptable approximation of the global state of radiation (see Fig. 2.5). Another advantage of that second order model is that the solution includes only three exponentials functions of mz: 0 ekk mz 0 ekk mz 0 ek mz =l0 ð2:75Þ Then, if we assume that the second term of Eq. 2.53 can be computed using that model, Eq. 2.53 becomes very simple and can be solved analytically for l \ 0. That enhanced solution contains only one more exponential, that is to say 0 ek mz =l ð2:76Þ 2.1 Irradiance Modeling 53 Fig. 2.8 Hemispherical radiance computed by the second ordre model and the enhanced model (with k0 mz = 0.220 x0 = 0.288, gh = 0 and Rloc = Rreg = 0.302) with that enhanced model, new parameter identification has been needed in order to keep the good correspondence of Fig. 2.5. We obtain then x0 = 0.288 and Rreg = 0.302. The result is shown at Fig. 2.8 assuming that the soil radiance is isotropic and that Rloc = Rreg The present model can take into account the fact that sky brightness is often higher near the horizon. In addition to the soil albedos Rreg and Rloc, it makes use of only three atmospheric parameters with well defined physical significance: k0 , x0 and gh. So, a few number of global irradiance measurements on tilted surfaces is sufficient to determine those parameters. Further investigations are needed to obtain the value of gh in function of k0 and x0 , and even a relation between k0 and x0 , in order to reduce the number of measurements needed for the determination of the model. 2.2 PV Array Modeling In literature, there are several mathematical models that describe the operation and behavior of the photovoltaic generator. For example, Borowy and Salameh [29] have given a simplified model, with which the maximum power output can be calculated for a module once photovoltaic solar irradiance on the photovoltaic module and the temperature is found, and Jones and Underwood [30] also introduced a simplified model of the maximum power output which has a reciprocal relationship with the temperature module and logarithmic relationship with the solar radiation absorbed by the photovoltaic module. In addition, Jones and Underwood have given the thermal model of the temperature module photovoltaic through the evaluation of many factors. These models differ in the calculation procedure, accuracy and the number of parameters involved in the calculation of the current–voltage characteristic. 54 2 Modeling of Solar Irradiance and Cells 2.2.1 Ideal Model The simplified equivalent circuit of a solar cell consists of a diode and a current source connected in parallel (Fig. 2.9). The current source produces the photocurrent Iph, which is directly proportional to solar irradiance G. The two key parameters often used to characterize a PV cell are its short-circuit current and its open-circuit voltage which are provided by the manufacturer’s data sheet. The equation of the current voltage Ipv–Vpv simplified equivalent circuit is derived from Kirchhoff’s law. We have Ipv ¼ Iph Id ð2:77Þ where 3 Vpv q 6 AKT 7 j 17 Id ¼ I0 6 4e 5 2 Thus 3 Vpv q 6 AKT 7 j 17 ¼ Iph I0 6 4e 5 ð2:78Þ 2 Ipv ð2:79Þ with Iph (A) is the photocurrent that is equal to short-circuit current, I0 (A) is the reverse saturation current of the diode, q is the electron charge (1.602910-19 C), K Botzman’s constant (1.381910-23 J/K), A is diode ideality factor, Tj is junction temperature of the panels (K), Id is the current shunted through the intrinsic diode, Vpv is the voltage across the PV cell. 2 3 Vpv q 6 AKT 7 j 17 ð2:80Þ Ipv ¼ Isc I0 6 4e 5 We can determine the reverse saturation current I0 by setting Ipv=0 (case when no output current). Ipv ¼ 0 Vpv ¼ Voc 2 3 ðVoc Þ 6 7 0 ¼ Iph I0 4e AKT j 15 q ð2:81Þ 2.2 PV Array Modeling 55 Fig. 2.9 Simplified equivalent circuit of solar cell Iph Ipv Id G,T j Vpv Table 2.1 Parameter of the PV panel Siemens SM110-24 [31] Parameter Value Pmpp Impp Vmpp Isc Voc asc boc 110 W 3.15 A 35 V 3.45 A 43.5 V 1.4 mA/C -152 mV/C Thus we obtain, taking into account the fact that, with this model, the photocurrent is equal to the short-circuit current: I0 ¼ Isc ðVoc Þ q AKT j e ð2:82Þ 1 Application: The solar cell is modeled and simulated using Matlab software. The simulation is based on the datasheet of Siemens SM110-24 photovoltaic module. The parameters of this solar module are given in Table 2.1 The module is made of 72 solar cells connected in series to give a maximum power output of 110 W (Figs. 2.10, 2.11) 2.2.1.1 Model With Ohmic Losses To obtain a better representation of the electrical behavior of the cell of the ideal model, the second model takes account of material resistivity and the ohmic losses due to levels of contact. These losses are represented by a series resistance Rs in the equivalent circuit (Fig. 2.12). The current voltage equation is given as follows: ðVpv þIpv Rs Þ q AKT j Ipv ¼ Iph I0 e 1 ð2:83Þ or, making the approximation that Iph & Isc, 56 2 Modeling of Solar Irradiance and Cells Fig. 2.10 Example of PV array structure Fig. 2.11 Bloc diagram of ideal model [195] Ipv ¼ Isc I0 e ðVpv þIpv Rs Þ q AKT j 1 ð2:84Þ The short-circuit Isc can be calculated at a given temperature Tj: IscGref ¼ Iscref ½1 þ asc DT ð2:85Þ DT ¼ Tj Tjref ð2:86Þ 2.2 PV Array Modeling 57 Fig. 2.12 Simplified equivalent circuit of solar cell with Rs where Isc-ref is measured under irradiance Gref = 1000 W/m2 and Tj-ref = 25C and is given on the datasheet, asc is the temperature coefficient of short-current (/K) and found on the data sheet, Tjref is the reference temperature of the PV cell (K), Tj is the junction temperature (K). The short-current generated at any other irradiance G (W/m2) can be obtained by: IscG ¼ IscGref G Gref ð2:87Þ Applying Eq. 2.84 to the case where Ipv = 0 (open-circuit case), one sees that the reverse saturation current at a reference temperature (Tjref) is given by: IscTjref # VocTjref Þ ð q AKT jref e 1 I0Tjref ¼ " ð2:88Þ Defining VthTjref ¼ A K Tjref q ð2:89Þ one can write Eq. 2.88 in the form: IscTjref I0Tjref ¼ e VocTjref VthTjref ð2:90Þ 1 The reverse saturation current at any other temperature Tj (K) can be obtained by: " !# 3 E q AKg Tj A ð2:91Þ I0 ¼ I0Tjref exp 1 1 Tjref Tj Tjref 58 2 Modeling of Solar Irradiance and Cells G Iscref 1 þ asc Tj Tref Gref 0 1 " !# A3 Eg q I T scref j A exp 1 AK @ 1 Vocref Tjref exp q AKT 1 Tj Tjref jref Vpv þ Rs Ipv exp q 1 A K Tj Ipv ¼ ð2:92Þ where Eg represents the gap energy. The simple relationship of power for a photovoltaic module is Ppv ¼Ipv Vpv ( G ¼ Isc 1þasc Tj Tref Gref 0 1 " !# ) A3 Eg q V þR I I T pv s pv scref j AK A exp q exp 1 1 1 @ Vocref Tjref AK Tj exp q AKT 1 Tj Tjref jref Vpv ð2:93Þ Neglecting the term ‘‘-1’’ added to the exponential in Eq. 2.84, the value of Rs can be obtained by dVpv 1 Rs ¼ dIpv Vpv ¼Voc w ð2:94Þ Isc w¼q A K Tj The first term of Eq. 2.94 ðdVpv =dIpv jVpv ¼Voc Þ can be determined either by experimental or by measures of Ipv–Vpv characteristic of the manufacturer (datasheets). Eq. 2.84 can be solved using Matlab/simulink or writing a function in Matlab. We can for example Newton’s method which is described as [32]: xnþ1 ¼ xn f ðxn Þ f 0 ðxn Þ with f ðxn Þis the function, f 0 ðxn Þ is the derivate of the function. We have f ðxn Þ ¼ 0 xn is the first value xnþ1 is the next value We obtain ð2:95Þ 2.2 PV Array Modeling 59 Fig. 2.13 Bloc diagram of model with ohmic losses ðVpv þIpv Rs Þ q f ðIpv Þ ¼ Isc Ipv I0 e AKTj 1 ¼ 0 ð2:96Þ and then Isc Ipv I0 e Ipv nþ1 ¼ Ipv n ðVpv þIpv :Rs Þ q AKT j 1 q 1 I0 AKT Rs eqðVpv þIpv Rs Þ=ðAKTj Þ j ð2:97Þ Defining Vth ¼ A K Tj q ð2:98aÞ we obtain Isc Ipv I0 e Ipvnþ1 ¼ Ipvn ðVpv þIpv :Rs Þ Vth 1 1 I0 VRths eðVpv þIpv Rs Þ=ðVth Þ ð2:98bÞ The resolution under Matlab/simulink is given in Figs. 2.13 and 2.14. 2.2.1.2 Other One Exponential PV Array Models In the equations of models above described (Sects. 2.2.1 and 2.2.2), there is only one exponential function. In this section, we present and explain four commonly used models which exhibit the same property. 60 2 Modeling of Solar Irradiance and Cells Fig. 2.14 PV subsystem bloc diagram of model with ohmic losses [195] Fig. 2.15 Simplified equivalent circuit of solar cell Rs Iph Id Ipv IRsh G Rsh Vpv Tj 2.2.1.3 Model No. 1 The first model studied in this section is defined, as one of these Sects. 2.2.1 and 2.2.2, by an equivalent circuit. This one consists of a single diode for the phenomena of cell polarization and two resistors (series and shunt) for the losses (Fig. 2.15). It can thus be named ‘‘one diode model’’. This model is used by manufacturers by giving the technical characteristics of their solar cells (data sheets). Ipv(Vpv) characteristic of this model is given by the following equation [33]: Ipv ¼ Iph Id IRsh or, developing the terms Id and IRsh: qðVpv þ Rs Ipv Þ Vpv þ Rs Ipv Ipv ¼ Iph I0 exp 1 Nscell KTj Rsh ð2:99Þ ð2:100Þ There are different methods to solve Eq. 2.100 resulting in different approximation mathematical models. The different mathematical models generally include parameters that are provided by photovoltaic modules manufacturers. For this, several methods have been proposed in the literature to determine different 2.2 PV Array Modeling 61 parameters. Wolf and Rauschenbach [34] suggest that the current–voltage characteristics of photovoltaic cells can be determined by three different methods. The three characteristics that results are the photovoltaic output characteristic, the pn junction characteristic, and the rectifier forward characteristic. These methods give different results because of the effects of the cell internal series resistance. In Ref. [35], authors propose simple approximate analytical expressions for calculating the values of current and voltage at the maximum power point and the fill factor of a solar cell. While in reference [36], authors use dynamic measurements for integration procedures based on computation of the area under the current–voltage curves. Recent methods [37] have applied to extract the intrinsic and extrinsic model parameters of illuminated solar cells containing parasitic series resistance and shunt conductance. The approach is based on calculating the Co-content function (CC) from the exact explicit analytical solutions of the illuminated current–voltage (Ipv–Vpv) characteristics. The resulting CC is expressed as a purely algebraic function of current and voltage from whose coefficients the intrinsic and extrinsic model parameters are then readily determined by bidimensional fitting. Ref [33] proposes an accurate method using Lambert W-function to study different parameters of organic solar cells. The method proposed by [38] uses separate fitting in two different zones in the Ipv–Vpv curve. In the first one, near short circuit, current fitting is used because the error in current dominates. In the second one, near open-circuit, voltage fitting is used because this is the dominant error. The method overcomes some drawbacks of common procedures: voltage errors are properly managed and no accurate initial guesses for the parameters are needed. This approach is a combination of lateral and vertical optimization to extract the parameters of an illuminated solar cell. The suggested technique in Ref [35] deals with the extraction of bias independent parameters. It is based on the current–voltage (Ipv–Vpv) characteristics and the voltage-dependent differential slope a = d(ln I)/d(lnV) in order to extract the relevant device parameters of the nonideal Schottky barrier, p-n and p-i-n diodes. In Ref [39], authors propose an approach for photovoltaic (PV) sources modeling based on robust least squares linear regression (LSR) parameter identification method. The parameter extraction is performed starting by the consideration of the temperature and MPPs voltage and current measured values distributions versus solar irradiance. Such distributions show a data placement along a straight line that suggests the possibility to obtain such data by a linear least squares (LSR) [39]. In Ref [40], authors present a method to determine the five solar cell parameters of the single diode lumped circuit model. These parameters are usually the saturation current, the series resistance, the ideality factor, the shunt conductance and the photocurrent. The method includes the presentation of the standard Ipv = f (Vpv) function as Vpv = f (Ipv) and the determination of the factors of this function that provide the calculation of the illuminated solar cell parameters [40–41]. • The simplified model with four parameters ðIph ; I0 ; Rs ; aÞ 62 2 Modeling of Solar Irradiance and Cells This model assumes the shunt resistance as infinite and thus neglects it. The model becomes then equivalent to the model of Sect. 2.2.2, but the development presented here is different. Equation 2.100 (:Eq. 2.83) will be written, using again the definition Eq. 2.98a, as: qðVpv þ Rs Ipv Þ Ipv ¼ Iph I0 exp 1 AKTj ð2:101Þ Vpv þ Rs Ipv ¼ Iph I0 exp 1 AVth To identify the four parameters required for Eq. 2.101, a method is proposed by [42]. They propose to treat the product AVth in Eq. 2.101 as a single parameter denoted a. a ¼ A Vth Vpv þ Rs Ipv ¼ Iph I0 exp 1 a Ipv ð2:102Þ The following approximations Eqs. 2.103a–2.103b are used to find values of the four parameters under reference conditions Iphref ¼ Iscref The other parameters are calculated by the following equations: 8 b Tjref Vocref þ Eg > aref ¼ oc > > > Tjref asc Iphref 3 > > > < Iphref I0 ref ¼ exp ½ ð V > ocref =aref Þ 1 > > > > a ln 1 Imppref Iphref Vmppref þ Vocref > > : Rs ¼ ref Imppref ð2:103aÞ ð2:103bÞ We can then find the cell parameters at the operating temperature of cells and solar irradiance from: 8 Iph ¼ GGref Iphref þ asc Tj Tref > > > > h i 3 > < T E T I0 ¼ I0 ref exp ag 1 Tjrefj Tjrefj ð2:104Þ > > R ¼ R > s sref > > : a ¼ aref Tj =Tjref These four parameters (Iph, I0, Rs, a) are corrected for environmental conditions using Eq. 2.104 and used in Eq. 2.102. • The implicit model with five parameters We can note that Eq. 2.100 is an implicit nonlinear equation, which can be solved with a numerical iterative method such as Levenberg–Marquardt algorithm 2.2 PV Array Modeling 63 which requires a close approximation of initial parameter values to attain convergence. Different analytical methods can be used to extract parameters. 1. Proposed analytical method The five parameters IL, I0, Rs, Rsh, and A are calculated at a particular temperature and solar-irradiance level from the limiting conditions of Voc, Isc, Vmpp, Impp and using the following definitions of Rso and Rsho. Ipv(ipv) characteristic of this model is given by Eq. 2.100. rewritten using again definition Eq. 2.98a: Vpv þ Rs Ipv Vpv þ Rs Ipv ð2:105Þ Ipv ¼ Iph I0 exp 1 AVth Rsh For Vpv = Voc, Ipv = 0, we have: Voc Voc 0 ¼ Iph I0 exp 1 AVth Rsh For Vpv = 0, Ipv = Isc, we have: Isc Rs Isc Rs Isc ¼ Iph I0 exp 1 AVth Rsh ð2:106Þ ð2:107Þ Replacing Iph in Eq. 2.107 by its value extracted from Eq. 2.106, we have: Voc Voc Isc Rs Isc Rs Isc ¼ I0 exp 1 þ I0 exp ð2:108Þ 1 AVth Rsh AVth Rsh Voc Isc Rs Voc Isc Rs exp ð2:109Þ Isc ¼ I0 exp þ AVth AVth Rsh Rsh Thus I0 Voc Isc Rs exp exp AVth AVth Voc Rs Isc 1 þ þ ¼0 Rsh Rsh ð2:110Þ We derive Eq. 2.105 with respect to the current: dVpv 1 Vpv þ Ipv Rs Rs 1 dVpv Rs þ 1 ¼ I0 exp Rsh dIpv Rsh dIpv AVth AVth AVth Vpv þ Ipv Rs dVpv I0 Rs Vpv þ Ipv Rs I0 1 dVpv Rs ¼ exp exp Rsh dIpv Rsh AVth AVth dIpv AVth AVth ð2:111Þ dVpv I0 Vpv þ Ipv Rs 1 exp þ Rsh dIpv AVth AVth Vpv þ Ipv Rs Rs I 0 Rs exp 1¼0 AVth AVth Rsh ð2:112Þ For Vpv = Voc we define 64 2 Modeling of Solar Irradiance and Cells dVpv ¼ Rs0 dIpv Vpv ¼VOC Replacing Eq. 2.113 in Eq. 2.112, we find I0 Voc 1 I 0 Rs Voc Rs Rs0 exp þ exp 1¼0 AVth AVth Rsh AVth AVth Rsh I0 Voc 1 ðRs0 Rs Þ exp þ 1¼0 AVth AVth Rsh ð2:113Þ ð2:114Þ For Ipv = Isc, we define: dVpv ¼ Rsh0 dIpv Ipv ¼ISC Replacing Eq. 2.115 in Eq. 2.112 we find: I0 Isc Rs 1 I0 Rs Isc Rs Rs Rsh0 exp þ exp 1¼0 AVth AVth Rsh AVth AVth Rsh I0 Isc Rs 1 ðRsh0 RS Þ exp þ 1¼0 AVth AVth Rsh ð2:115Þ ð2:116Þ Dividing Eq. 2.116 by (Rsh0-Rs), we find 1 1 I0 Isc Rs þ exp ¼0 Rsh Rsh0 Rs AVth AVth At maximum power point, we have Vmpp þ Impp Rs Vmpp þ Impp Rs 1 Impp ¼ Iph I0 exp AVth Rsh ð2:117Þ ð2:118Þ From Eq. 2.106, we obtain Voc Voc Iph ¼ I0 exp 1 þ AVth Rsh ð2:119Þ Substituting Eq. 2.119 into Eq. 2.118, we get Vmpp þ Impp Rs Vmpp þ Impp Rs Voc Voc Impp ¼ I0 exp 1 þ I0 exp 1 AVth Rsh AVth Rsh ð2:120Þ Voc Vmpp Vmpp þ Impp Rs Voc Rs I0 exp þ I0 þ Impp ¼ 0 ð2:121Þ I0 exp AVth Rsh Rsh AVth Kennerud and Charles showed that the four parameters A, Rs, I0, and Rsh can be determined by the Newton-Raphson solving simultaneous nonlinear equations 2.2 PV Array Modeling 65 Eqs. 2.110, 2.114, 2.117 and 2.121. However, this method requires long calculations and initial conditions for strict convergence hence, it is difficult to determine these parameters. So it is necessary to have analytical expressions for determining these parameters directly. As Rsh [ [ Rs , we assume for the parameters determination that 1 þ Rs =Rsh 1 In Eq. 2.110, we assume also: exp Voc Isc Rs ii exp AVth AVth In Eq. 2.114, we assume: I0 Voc 1 exp ii AVth AVth RSh V þI R Finally, in Eq. 2.117, we assume: AVI0th exp pv AVpvth serial hh10% of the remaining terms. With these simplifications, we get from Eqs. 2.110, 2.114, 2.117 and 2.121: I0 exp Voc Voc Isc þ ¼0 AVth Rsh I0 Voc exp 1¼0 AVth AVth ð2:123Þ Rsh ¼ Rsh0 ð2:124Þ Voc Vmpp Vmpp þ Impp Rs Voc þ Impp I0 exp ¼0 AVth Rsh AVth ð2:125Þ ðRs0 RS Þ I0 exp ð2:122Þ From these last four equations, we obtain an analytic expression of A. For that, from Eq. 2.122, we have: Voc Voc I0 ¼ Isc exp ð2:126Þ Rsh AVth From Eq. 2.123, we have Rs0 Rs ¼ Rs ¼ Rs0 I0 AVth 1 Voc ¼ Rs0 exp AV th 1 AVth I0 AVth 1 Voc exp AV th 1 Isc VRocsh ð2:127Þ 66 2 Modeling of Solar Irradiance and Cells Substituting Eq. 2.126 into Eq. 2.125, we obtain: Vmpp þ Impp Rs Voc Voc Vmpp þ Impp ¼ I0 exp Rsh Rsh AVth Vmpp Vmpp þ Impp Rs Voc Voc Isc Impp ¼ Isc þ exp Rsh Rsh AVth AVth Isc or, substituting Eq. 2.127 in that equation: Isc Vmpp Rsh Impp Isc VRocsh ln Isc Vmpp Rsh I ¼ exp Impp Isc VRocsh AV th Voc þ Vmpp þ Impp Rs0 Iscmpp Voc =Rsh ! AVth ! ¼ Voc þ Vmpp þ Impp Rs0 Impp AVth Isc VRoc sh Finally, we obtain the expression of A A¼ Vmpp þ Impp Rs0 Voc V Impp Voc Vth ln Isc Rmpp I ln I þ mpp sc Rsh sh I Voc sc ð2:128Þ Rsh and I0, Rs and Iph are obtained by 8 Voc Voc > > I0 ¼ Isc exp > > Rsh AVth > > > < AVth Voc Rs ¼ Rs0 exp > I0 AVth > > > > > R I R > : Iph ¼ Isc 1 þ s þ I0 exp sc s 1 RSh AVth ð2:129Þ Once these parameters (A, Iph, Rs, and I0) are determined, the Ipv–Vpv characteristic will be calculated by Eq. 2.100 using the Newton Raphson method. 2. Second method (Method of Pi constants) The model can be enhanced by releasing the constraint of linearity between the photocurrent Iph and the irradiance G. For that, we make choice of an expression for Iph with a term quadratic in G: Iph ¼ P1 G ½1 þ P2 ðG Gref Þ þ P3 ðTj Tref Þ ð2:130Þ where G is irradiance on the panel plane (W/m2); Gref corresponds to the reference irradiance of 1000 W/m2 and Tjref to the reference panel temperature of 25C. P1, P2 and P3 are constant parameters. 2.2 PV Array Modeling 67 The polarization current Id of junction PN, is given by the expression: q ðVpv þ Rs Ipv Þ Id ¼ I0 exp 1 ð2:131Þ A Nscell K Tj Eg I0 ¼ P4 Tj3 exp ð2:132Þ K Tj where I0 (A) is the saturation current, q is the elementary charge, K Botzman’s constant, A ideality factor of the junction, Tj junction temperature of the panels (K), Rs and Rsh (X) are resistors (series and shunt) and ns the number of cells in series. The shunt current is given by: Vpv þ Rs Ipv IRsh ¼ ð2:133Þ Rsh Thus Ipv ¼ P1 G 1 þ P2 ðG Gref Þ þ P3 Tj Tref Vpv þ Rs Ipv Vpv þ Rs Ipv Eg P4 Tj3 exp exp q 1 A ns K T j Rsh K Tj ð2:134Þ Different methods exist to determine the constant parameters P1, P2, P3, and P4. • First approach We determine the seven constant parameters P1, P2, P3, P4, the coefficient A and the resistance Rs and Rsh of PV model with a numerical resolution and the use of data sheets PV panels. Parameters commonly provided by module manufacturers are values of short-circuit current Isc, the open-circuit voltage Voc and the point of optimum power (Impp, Vmpp). We determine the system of nonlinear equation as follows: 8 Ipv ðVoc Þ ¼ 0 > > > > > > < Ipv ð0Þ ¼ Isc Ipv ðVmpp Þ ¼ Impp ð2:135Þ > I¼Impp > > > dPpv dIpv > > ¼ Impp þ ¼0 : dV dV pv P¼Pmpp pv V¼Vmpp In order to determine all the seven parameters, three additional equations are needed. In order to determine P2, points with different irradiance are necessary. Similarly, in order to determine P3, points with different temperatures are needed. The system resolution of the nonlinear Eq. 2.135 is done by running the function ‘fsolve’ contained in toolboxes of Matlab which is based on the least squares method (Fig. 2.16). 68 2 Modeling of Solar Irradiance and Cells Beginning Write the program executing the statement of nonlinear Eq.2.134 Introduction of initial conditions X o = [P1 P2 P3 Resolution of P4 F(X) A Rserial Rsh ] by the function: a = fsolve ( F ( X ), X o ) No Convergence Test: F (a) ≈ ε ? Yes a = X(i) = [P1 P2 P3 P4 A R serial R sh ] End Fig. 2.16 Flowchart of resolution with parameters Pi • Second approach To determine the parameters of the panels (Pi, Rs, Rsh), we developed a method based on optimization techniques to find the extrema for the function indicated. Its principle is simple; it consists in the first step to define an objective function, called a criterion. This criterion E depends on a set of parameters grouped in a vector p. This objective function is the error between the practical results and those of the simulation. It is given by: 2.2 PV Array Modeling 69 Begining Introduction Ipv(Vpv) characteristics Value of i Experimental values of Ipvi-exp and Vpvi-exp Calcul of Ipvi-sim (Eq 2.100) Calcul of E i I=i-1 No i Yes Maximiser E i Display Pi parameters End Fig. 2.17 Flowchart to determine the parameters of the panels (Pi, Rs, Rsh) EðpÞ ¼ Ipv exp Ipvsim Ipv exp ð2:136Þ with Ipv-sim is calculated from Eq. 2.134 for the model with a diode, using a wide range of variation of the illumination received by the photovoltaic panel, p contains various parameters to determine (P1, P2, P3, P4, A, Rs, Rsh). We use the least 70 2 Modeling of Solar Irradiance and Cells Fig. 2.18 Bloc diagram of the PV model [195] Fig. 2.19 PV subsystem model [195] squares method to find the value of P that minimizes the function R (E(p))2. We present the proposed flowchart (Figs. 2.17, 2.18, 2.19). Application in Matlab/simulink (see CD program-model 1) The simulated current–voltage (Ipv–Vpv) characteristic and power-voltage (Ppv– Vpv) of the PV Module is shown in Fig. 2.20. The characteristic is obtained at a constant level of irradiance and by maintaining a constant cell temperature. The variation in both the Ipv–Vpv and Ppv–Vpv characteristics with irradiance level are simulated and the results are shown in Fig. 2.21. 2.2 PV Array Modeling Fig. 2.20 Characteristics Ipv = f (Vpv) and Ppv = f (Vpv) in STC conditions Fig. 2.21 Effects of solar irradiance changing Fig. 2.22 Effects of temperature changing 71 72 2 Modeling of Solar Irradiance and Cells Fig. 2.23 Experimental bench [9] Fig. 2.24 Comparison of experimental results with simulation ones [195] The simulated Ipv–Vpv and Ppv–Vpv characteristics of the solar cell at different cell temperatures are shown in Fig. 2.22. We make validation through the following experimental bench: We can use the method of resistance variation or the charging and discharging capacitor method. First, we measure the irradiation after we close the switch K1. The capacitor charges, we measure the different values of current and voltage panel. To discharge the capacitor, we open the switch K1 and close K2 (Figs. 2.23, 2.24, 2.25, 2.26). 2.2.1.4 Model No. 2 The second model here presented has no physical meaning, but is characterized by a very simple resolution. It requires only four parameters namely Isc, Voc, Vmp, and Imp. The form of the Ipv–Vpv characteristic of this model is loosely based on that of Model No. 1, that is to say: 2.2 PV Array Modeling 73 Fig. 2.25 Bloc diagram of Model No. 2 [195] Fig. 2.26 PV subsystem (Model No. 2) [195] Ipv ¼ Isc Vpv 1 C1 exp 1 C2 Voc ð2:137Þ That equation fits exactly the short-circuit point. The open-circuit and maximum power points are approximately restituted with: Vmpp 1 Voc C2 ¼ I ln 1 mpp Isc ð2:138Þ Impp Vmpp C1 ¼ 1 exp Isc C2 Voc The simple relationship of power for a photovoltaic module is 74 2 Modeling of Solar Irradiance and Cells Fig. 2.27 Characteristics Ipv = f (Vpv) and Ppv = f (Vpv) under the STC conditions (Model No. 2) Fig. 2.28 Effects of irradiance changing (Model No. 2) Ppv ¼ Isc Vpv 1 C1 exp 1 Vpv C2 Voc ð2:139Þ The constant parameters can be determined directly by Eq. 2.138. Application in Matlab/simulink(see CD program-model2) Figure 2.27 below shows the characteristic power/voltage and current/voltage obtained under the STC conditions (Standard Test Condition G = 1000 W/m2, Tj = 25C) The Figs. 2.28, 2.29 and 2.30 show the influence of irradiance G and temperature Tj on the electrical characteristics. 2.2.1.5 Model No. 3 The third model offers one more freedom degree than Model No. 2. The PV array current Ipv obeys to the expression: 2.2 PV Array Modeling 75 Fig. 2.29 Effect of temperature changing (model No. 2) Fig. 2.30 Effects of irradiance and temperature changing (model No. 2) m Ipv ¼ Isc f1 C1 ½expðC2 Vpv Þ 1g ð2:140Þ All the three rated points are exactly fitted if the coefficients C1, C2 and m verify: C2 ¼ C4 m Voc I ð1 þ C1 Þ Impp C3 ¼ ln sc C1 Isc 1 þ C1 C4 ¼ ln C1 h i C ln C3 m ¼ hV 4 i ln Vmpp oc 76 2 Modeling of Solar Irradiance and Cells with Vmpp voltage at maximum power point; Voc open-circuit voltage; Impp current at maximum power point; Isc short-circuit current. The parameters determination is achieved with the arbitrary condition C1 ¼ 0:01175 Equation 2.140 is only applicable at one particular irradiance level G and cell temperature Tj, at standard test conditions (STC) (G = 1000 W/m2, Tj = 25C). When irradiance and temperature vary, the parameters change according to the following equations: DIpv DTj ¼ Tj Tjref G G ¼ asc 1 Isc;ref DTj þ Gref Gref ð2:141Þ DVpv ¼ boc DTj Rs DIpv where asc is the current temperature coefficient and boc the voltage temperature coefficient. The new values of the photovoltaic voltage and the current are given by Vpv; new ¼ Vpv þ DVpv Ipv; new ¼ Ipv þ DIpv ð2:142Þ The simple relationship of power for a photovoltaic module is (Fig. 2.31) h i m Þ 1 g Vpv ð2:143Þ Ppv ¼ Isc f1 C1 expðC2 Vpv Application under Matlab/simulink (see CD program-model 3) The Fig. 2.32 shows the characteristic power/voltage and current/voltage obtained under the STC conditions (Figs. 2.33, 2.34, 2.35). We make validation through the experimental bench given in Fig. 2.26. We obtain the following results (Fig. 2.36) comparing to the simulation ones. 2.2.1.6 Model No. 4 The fourth model is based on references [39, 40] and [43]. The advantage of this model is that it can be established using only standard data for the module and cells, provided by the manufacturer (data sheets). This model is simple, because it is independent of the saturation current I0. The current delivered by the solar module (Ipv) in any conditions is given by Vpv Vocpv þ Rspv Ipv Vocpv Ipv ¼ Iscpv 1 exp exp A Vthpv A Vthpv ð2:144Þ 2.2 PV Array Modeling 77 Fig. 2.31 Bloc diagram of Model No. 3 [195] Fig. 2.32 Characteristics Ipv = f (Vpv) and Ppv = f (Vpv) under the STC conditions(Model No. 3) [195] 78 2 Modeling of Solar Irradiance and Cells Fig. 2.33 Effects of solar irradiance changing (Model No. 3) [195] Fig. 2.34 Effects of temperature changing (Model No. 3) [195] 120 5 4.5 G=900W/m², Tc=35°C 80 Power(W) Current(A) G=900W/m², Tc=35°C 100 4 3.5 3 G=650W/m², Tc=33°C 2.5 2 G=450W/m², Tc=25°C 1.5 1 G=650W/m², Tc=33°C 60 G=450W/m², Tc=25°C 40 20 0.5 0 0 0 5 10 15 20 25 Voltage(V) 30 35 40 45 50 0 5 10 15 20 25 30 Voltage(V) Fig. 2.35 Characteristics Ipv = f (Vpv) and Ppv = f (Vpv) (Model No. 3) [195] 35 40 45 50 2.2 PV Array Modeling 79 Fig. 2.36 Equivalent circuit for two diode model [44] Iph G, Tj Rs Id1 Id2 Ipv IRsh Rsh Vpv with Iscpv ¼ Npcell Isccell Vocpv ¼ Nscell Voccell ð2:145Þ where Ns-cell and Np-cell are the number of series and parallel cells, respectively. The open-circuit voltage Voc-cell is given by Voccell ¼ Voccellref þ boc ðTj Tjref Þ ð2:146Þ with Voc-cell-ref and Tjref are the open-circuit voltage and cell temperature at standard conditions, boc is the temperature coefficient of Voc-cell-ref Voccellref ¼ Vocpvref =Ns ð2:147Þ The thermodynamic voltage of the cell Vth-cell is given by: Vthpv ¼ Nscell Vthcell Vthcell ¼ K Tj q ð2:148Þ The resistance of the module is calculated by the following equation: Rspv ¼ Rscell Nscell Npcell ð2:149Þ Rs-cell is calculated by reference to model of Sect. 2.2.2 Eq. 2.94: dVpvcell A K Tj Rscell ¼ dIpvcell Vpv ¼Vocpv q Isccell ð2:150Þ The temperature Tj can be taken equal to Ta, and the thermodynamic voltage Vth-cell can be easily calculated, using the coordinates of the maximum power point of the cell (Vmpp-cell and Impp-cell). The expression of Vth-cell can have the following form: Vthcell ¼ Vmppcell þ Rserialcell Imppcell Voccell ln 1 Imppcell Isccell ð2:151Þ 80 2 Modeling of Solar Irradiance and Cells 2.2.2 Two Diode PV Array Models The ‘‘two diodes’’ model uses an equivalent circuit and takes into account the mechanism of electric transport of charges inside the cell. In this model, the two diodes represent the PN junction polarization phenomena. These diodes represent the recombination of the minority carriers, which are located both at the surface of the material and within the volume of the material (Fig. 2.36). The following equation is then obtained: Ipv ¼ Iph ðId2 þ Id2 Þ IRsh ð2:152Þ with Iph and IRsh maintaining the same expressions as above Eqs. 2.130 and 2.133. For the recombination currents, we have: h i 8 < Id1 ¼ I01 exp qðVpv þRs Ipv Þ 1 ANscell KTj h i ð2:153Þ : Id2 ¼ I02 exp qðVpv þRs Ipv Þ 1 2ANscell KTj The saturation currents are written as: 8 < I01 ¼ P4 Tj3 exp Eg kTj : I02 ¼ P5 T 3 exp Eg j 2kTj ð2:154Þ with Ns is the number of cells in branched series. The final equation of the model is thereby written as (Figs. 2.37, 2.38): Vpv þ Rs Ipv Ipv ¼ P1 G 1 þ P2 ðG Gref Þ þ P3 Tj Tref R sh E V þ R I pv s pv g P4 Tj3 exp exp q 1 k Tj A Nscell K Tj Eg Vpv þ Rs Ipv P5 Tj3 exp exp q 1 ð2:155Þ 2 k Tj 2 A Nscell K Tj Application under Matlab/simulink(see CD program-modele2D) We make validation through the experimental bench given in Fig. 2.25. We obtain the following results (Fig. 2.39) comparing to the simulation ones. 2.2.3 Power Models 2.2.3.1 Model No. 1: Polynomial Model This model can give the same power of solar modules operating at MPP (Maximum Power Point). It is intended to polycrystalline silicon technology. The maximum power Ppvmax can be given by [44] (Figs. 2.40, 2.41, 2.42): 2.2 PV Array Modeling 81 Fig. 2.37 Block diagram of two diode model [195] Fig. 2.38 Characteristics Ipv = f (Vpv) and Ppv = f (Vpv) simulation results (two diode model) [195] Ppvmax ¼ K1 ð1 þ K2 ðTj Tjref ÞÞ:ðK3 þ GÞ ð2:156Þ where K1, K2, and K3 are constants to be determined (data sheets) We can obtain the maximum power for a given irradiance G and panel temperature Tj with only three constant parameters and then solve a simple equation Eq. 2.156. 82 2 Modeling of Solar Irradiance and Cells Fig. 2.39 Characteristics Ipv = f (Vpv) and Ppv = f (Vpv) comparison of experimental results with simulation ones(two diode model) The identification parameter was carried from the maxima characteristic voltage/power panels with experimental measurements performed on site. Actual values used for several surveys covering a wide range of variation of sunshine actions were taken as: • K1 between 0.095 and 0.105 for a panel represents the dispersion characteristics of the panels. • K2 = -0.47%/C drift in temperature of the panels. • a parameter (K3) added to the characteristic of the manufacturer, to obtain results much more satisfying: Application under Matlab (see CD program Power model 1) For this model, the optimization is given only on the maximum power. It is possible to represent the curves corresponding to the variations of the maximum power for different days in the year (summer, spring, winter). 2.2.3.2 Model No. 2 The following benchmark model can determine the maximum power provided by a PV module for given irradiance and temperature with only four constant parameters to determine a, b, c, and d. These parameters are obtained solving a simple equation system for a resulting set of measurement points sufficiently extended [30]. We have Ppvmax ¼ ða G þ bÞ T j þ c G þ d ð2:157Þ where Ppv-max is the maximum power output and where a, b, c, and d are positive constants which can be obtained experimentally. Application under Matlab (see CD program Power model 2) 2.2 PV Array Modeling 83 100 journée Summerd'été day Spring day journée de printemps journée Winter d'hiver day 90 80 70 Power (W) 60 Summer day 50 40 30 Spring day 20 10 Winter day 0 07h 08h 09h 10h 11h 12h 13h 14h 15h 16h 17h Time (hours) Fig. 2.40 Variations of power for different days of the year (Model No. 1) [9] Fig. 2.41 Variations of power (Model No. 3) [195] Fig. 2.42 Variations of power (Model No. 4) [195] 18h 19h 84 2 Modeling of Solar Irradiance and Cells 2.2.3.3 Model No. 3 The energy produced by a photovoltaic generator is estimated from the data the global irradiation on inclined plane, the ambient temperature data manufacturer for the PV module used. The power output of PV array can be calculated from the following equation [1]: Ppvmax ¼ gpv :Apv Nm G ð2:158Þ where Apv is the total area of the photovoltaic generator and ggen the efficiency of the photovoltaic generator. ð2:159Þ gpv ¼ gr gpc 1 asc ðTj Tjref Þ where G is a solar radiation on tilted plane module, gr is the reference efficiency of the photovoltaic generator, gpc is the power conditioning efficiency which is equal to 1 if a perfect maximum power tracker (MPPT) is used, asc is the temperature coefficient of short-current (/K) as found on the datasheet, Tj is cell temperature and Tjref is the reference cell temperature. Application under Matlab (see CD program Power model 3) 2.2.3.4 Model No. 4 Jones and Underwood developed the following practical model in 2002 for the production of optimal output power of a photovoltaic module [30, 45]: G lnðP1 GÞ Tjref Ppv max ¼ FF Isc : Voc Gref lnðP1 Gref Þ Tj ð2:160Þ where P1 is a constant coefficient, which can be calculated by the following formula: P1 ¼ Isc G FF is the Filling factor. FF ¼ Ppv max Vmpp Impp ¼ Voc Isc Voc Isc with Ppv-max the maximum power under STC conditions. Application under Matlab (see CD program Power model 4) ð2:161Þ 2.2 PV Array Modeling 85 Fig. 2.43 General equivalent circuit for PV cell or module modeling 2.2.4 General Remarks on PV Arrays Models 2.2.4.1 The Forgotten Equation Some of the above described model use for the parameters identification the values of Vmpp and Impp. However, only one of them (see Eq. 2.135) makes use of the fact that the point (Vmpp, Impp) is the point at maximum power. And yet, for that point, it is easy to prove (see Eq. 1.7) that, whatever the used model may be, dIpv Impp ¼ ð2:162Þ dVpv Vpv ¼ Vmpp Vmpp That additional equation allows determining one more parameter using the nominal values of a photovoltaic cell or module. 2.2.4.2 General Structure and Consequences All the models above described which are based on an equivalent circuit have the same general structure shown at Fig. 2.43. The above described models differ only due to the nature of the nonlinear element J in parallel on the current source Iph. Following the complexity of the model, that element can be formed with a variable number of ideal elements: one or two diodes and sometime a resistance Rsh. The consequence of that general structure is that all the characteristics Ipv–Vpv corresponding to the same junction temperature but different irradiance can be obtained by translation of one of them [13]. The translation has a vertical component DIph and a horizontal component –Rs DIph. It is then possible to deduce the value of Rs from the Ipv–Vpv characteristics without reference to one particular model. That method is thus an interesting alternative to Eqs. 2.94, 2.103b and 2.129 for determining the series resistance Rs. It has the additional advantage to consider globally the characteristics and not one particular point. With the general structure of Fig. 2.43, we have, for a variation of Vpv 86 2 Modeling of Solar Irradiance and Cells Fig. 2.44 General model for multijunction cell or module dIpv ¼ dIJ ¼ dIj ½dVpv þ Rs dIpv dVJ ð2:163Þ and thus dIpv dIJ =dVJ ¼ dVpv 1 þ Rs ðdIJ =dVJ Þ ð2:164Þ Reporting (2.164) for the maximum power point into (2.161), we have: ½1 þ Rs ðdIJ =dVJ Þj Vpv ¼ Vmpp Impp ¼ ½ðdIJ =dVJ Þj Vpv ¼ Vmpp Vmpp ð2:165Þ or Impp ¼ ½ðdIJ =dVJ j Vpv ¼ Vmpp ðVmpp Rs Impp Þ ð2:166Þ 2.2 PV Array Modeling 87 2.2.4.3 Partial Linearity of the Models Starting with the values of open-circuit voltage, maximum power current and voltage, and short-circuit current, we obtain in all case a system of four equations (three equations for the three nominal points and Eq. 2.165). That system is linear in Iph, 1/Rsh, Id1 and Id2. Then, no approximation formula is necessary for determining that parameters. The only parameters which arrive in the equations system in a nonlinear way are Rs and the non ideality factor A. If one uses the two diodes model, the simplest way is to assume that the factor A is unity. Then, beside Rs which has to be determined by another way, all the parameters of the model are determined using the three nominal points. 2.2.4.4 Remark About the Multijunction Cells All the above mentioned models are suited for single junction cells. In case of multijunction cells, the model of each cell corresponds to the series connection of two or three circuits with the structure of Fig. 2.43, but with different parameters. Summing all the Rs resistance in only one, we obtain the general model of Fig. 2.44. Of course, that model contains many too parameters. However, if the manufacturer has well equilibrated the junctions for the standard spectrum, we have (for standard spectrum only) equality of the three current sources. In only that case we can combine the three nonlinear elements of Fig. 2.44 and retrieve the simpler structure of Fig. 2.43. http://www.springer.com/978-1-4471-2348-4

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