Thin-film Solar Cell Model for a Renewable Energy Based Water Supply System.
Tobias Hellgren
Master of Science Thesis in Energy Engineering. Umeå Institute of Technology (löpnr. som tilldelas)
Abstract The aim with the thesis was to calculate the IV curve and power for a thin-film solar module. The purpose is to use the developed model for solar array dimensioning and reliable energy prediction. Several different methods were considered. The diode equation proved to work poorly for polycrystalline materials. Numerical modelling of the solid-state properties of the solar cell showed to be unnecessary complex and very time consuming. Model building from regression analysis on a set of measurements was abandoned due to the lack of flexibility. The applied model was instead the so-called Sandia model. The model scopes the spectral and optical response, the temperature dependence and the beam and diffuse irradiances influence. The model work well for fluctuating operational conditions and can easily be modified to work for another solar cell material. The Sandia model was implemented in Mathcad® together with a solar radiation programme to calculate the energy input to the solar module. Own experiments was conducted to aid the model validation and to transform the model to work for a new solar cell. Measurements were taken of the short circuit current, the open circuit voltage and the entire IV curve. The available equipment limited the possible analysis. The model validation showed that the model has a tendency to over-predict the measured values. The systematic over-prediction suggests that the model cannot explain all of the deterministic information in the solar cell. Comprehensive error analysis suggest an uncertainty < 10 %, so the conclusion is that the model can be used, but with an arbitrary safety factor.
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Modell av tunnfilmssolcell för vattenpump driven med förnybara energikällor.
Tobias Hellgren Sammanfattning Målsättning med examensarbetet var att beräkna IV-kurvan och effekten för en tunnfilmssolcell. Syftet var att använda modellen till dimensionering av solcellssystem och för tillförlitligt uppfattning av energiproduktionen. Flera olika metoder övervägdes. Diodekvationen visade sig fungera dåligt för polykristallina material. Numerisk modellering av elektroniska egenskaper i solcellen var tidskrävande och onödigt komplicerat. Regressionsanalys på mätvärden förkastades på grund av den begränsade anpassningsförmågan. Den tillämpade modellen var istället den så kallade Sandiamodellen. Modellen omfattar solcellens spektrala och optiska respons, dess temperaturberoende och den direkta och diffusa solinstrålningens inverkan. Modellen fungerar tillfredställande under fluktuerande driftvillkor och kan med fördel anpassas till att fungera för en annan godtycklig solcell. Sandia-modellen implementerades i Mathcad® tillsammans med ett solinstrålningsprogram vars syfte var att uppskatta den infallande solenergin. Det genomförda experimentet syftade till att understödja modellvalideringen och att anpassa den befintliga modellen till en ny solcell. Mätta storheter var kortslutningsström, öppen kretsspänning och ström-spännings-kurvan. Den experimentella utrustningen implicerade vissa begränsningar i modellvalideringen. Osäkerhetsanalysen indikerade att modellen hade en tendens att överprediktera de experimentella värdena. Den systematiska överestimeringen tyder på att modellen inte kan förklara all den metodiska informationen i solcellen. Omfattande felanalys föreslår en osäkerhet på < 10 %, så slutsatsen är att modellen kan användas efter att en godtycklig säkerhetsfaktor har lagts på.
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Table of content ABSTRACT .............................................................................................................................. 2 SAMMANFATTNING ............................................................................................................ 3 TABLE OF CONTENT ........................................................................................................... 4 1. INTRODUCTION................................................................................................................ 6 1.1 BACKGROUND AND MOTIVATION....................................................................................... 7 1.2 LITERATURE STUDY ........................................................................................................... 7 1.2.1 Conclusions ............................................................................................................... 8 1.3 AIM ................................................................................................................................... 9 1.4 FORMAT OF THE THESIS ..................................................................................................... 9 1.5 ACKNOWLEDGEMENTS ...................................................................................................... 9 1.6 ABBREVIATIONS AND DEFINITIONS .................................................................................. 10 2. THEORY............................................................................................................................. 12 2.1 SOLAR RADIATION ........................................................................................................... 12 2.1.1 Properties of the sun ............................................................................................... 12 2.1.2 Extraterrestrial radiation........................................................................................ 13 2.1.3 Terrestrial radiation................................................................................................ 14 2.1.4 Geometrical properties ........................................................................................... 18 2.2 PRINCIPLES OF SOLAR CELLS ........................................................................................... 20 2.3 CHARACTERISTICS OF A PHOTOVOLTAIC DEVICE ............................................................. 22 2.3.1 The equivalent circuit.............................................................................................. 22 2.3.2 Design features........................................................................................................ 23 2.4 POLYCRYSTALLINE MATERIALS ....................................................................................... 24 3. METHODS ......................................................................................................................... 26 3.1 EXPLANATION AND MOTIVATION OF TOOLS ..................................................................... 26 3.2 MODEL ............................................................................................................................ 26 3.2.1 Calculation block .................................................................................................... 27 3.2.2 Description of the Sandia equations ....................................................................... 30 3.2.3 IV curve simulation ................................................................................................. 31 3.3 EXPERIMENTS .................................................................................................................. 35 3.3.1 Set-up....................................................................................................................... 35 3.3.2 Description of equipment ........................................................................................ 37 3.3.3 Additional information for repetition of the experiment ......................................... 41 3.3.4 Measurements.......................................................................................................... 43 3.3.5 Theoretically calculations ....................................................................................... 45 3.3.6 Limitations............................................................................................................... 46 3.4 VALIDATION .................................................................................................................... 47 4. RESULTS............................................................................................................................ 49 4.1 MODEL ............................................................................................................................ 49 4.2 EXPERIMENTS .................................................................................................................. 51 4.2.1 Open circuit voltage ................................................................................................ 51 4.2.2 Short circuit current ................................................................................................ 52 4.2.3 IV curves.................................................................................................................. 53
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4.2.4 Analysis of error...................................................................................................... 54 4.3 VALIDATION .................................................................................................................... 56 4.3.1 Analysis indoor experiment ..................................................................................... 56 4.3.2 Analysis Helsinki experiment .................................................................................. 61 4.3.3 Analysis of Denmark experiment............................................................................. 62 4.3.4 Summation of the errors .......................................................................................... 64 4.3.5 New regression coefficients..................................................................................... 65 4.3.6 Comparison with I-VTracer .................................................................................... 66 5. DISCUSSION ..................................................................................................................... 68 5.1 MODEL ............................................................................................................................ 68 5.1.1 Comparison with other studies................................................................................ 68 5.2 EXPERIMENTS .................................................................................................................. 68 5.3 VALIDATION AND CRITICAL ASSESSMENTS ...................................................................... 69 5.4 SUGGESTIONS FOR FURTHER WORK ................................................................................. 71 5.5 CONCLUSIONS ................................................................................................................. 72 6. REFERENCES ................................................................................................................... 73 APPENDIX ............................................................................................................................. 76 A ........................................................................................................................................... 76 B ........................................................................................................................................... 79 C ........................................................................................................................................... 81 D ........................................................................................................................................... 82 E ........................................................................................................................................... 83 F............................................................................................................................................ 86 H ......................................................................................................................................... 101 I........................................................................................................................................... 103
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1. Introduction The transformation from fossil fuels to renewable energy sources is of great importance for a durable society. An agreement such as the Kyoto protocol is a step in the right direction. Renewable energy sources, for example wind power and solar cells, are gaining market shares. Political decisions and guidelines are essential but awareness and enthusiasm from the society to do something is also very important. The motivation to decrease the air pollution and conserve the ecological diversity has benefited the development of clean energy sources. To enhance the utilization of photovoltaic the cost need to be reduced to a level competitive to established energy sources. The growing consciousness of the fossil fuels’ limiting resources and the uncertainties at the oil market can further benefit the transformation to renewable energy sources. The interests and devotion scientist etc. worldwide pay to the subject make the future optimistic for the necessary conversion. During the 1970s crisis in oil industry, an awareness of the importance for not only relying on fossil fuels, but to also have a balanced supply of energy came into the light. Deregulation of the electric market has also favoured the solar cell production. More recently, green certificate or tariffs have been introduced at the electric market in the author’s country (Sweden) and forced the end user to utilize a certain amount of renewable energy sources, Energimyndigheten’s web page. Edmund Bequerel performed the pioneer work of the photovoltaic effect in 1839. Other historical milestones are for example the famous discovery of the photoelectric effect 1905 by Albert Einstein and the properties of a metal- semiconductor interface explained by Walter Schottky, among others, during 1930s. At the 1950s, the development of silicon electronics made it possible to construct a Si pn- junction with efficiency around 6 %. The silicon industry acted as a break through for solar cell applications. They became useful in remote areas, far away from the electric grid and also for purposes where cost is of minor importance, for example for satellites. The key issue for solar cells are to lower the production cost, but this form of energy conversion also has major advantages. The fuel is free and abundant at any location on earth. Photovoltaic have the potential not only to be a primary energy source, but also to produce chemical fuels such as hydrogen for fuel cells. The electricity is also very clean and do not contribute to the global warming and is also nearly maintenance- free. The PV devices are very gentle. They include no moving parts, have no emission and make no sound. The solar array can be integrated in houses and add to the design and save material cost. Solar cells have also a unique scalable possibility and can be connected in series and in parallel to increase the voltage and the current. Other drawbacks than the economy are the mismatch between solar peak and maximum demand and the problem to store the energy. If the solar cell is connected to the grid it can feed the grid with energy during the day and the owner can buy electricity at night. Solar cells also suffer from low power density. The last problem is ignorance. Customers can be reluctant to buy non- standard components to build up a solar array and instead use safe old techniques. The novel purpose for development of thin-film materials was the good power to weight ratio and actually not the lower cost. Thin-film materials are cheaper and make solar cells more
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competitive in comparison with other energy sources. Even though they suffer from lower efficiency than crystalline solar cells they are the future due to lower production cost.
1.1 Background and motivation The pump manufacture Grundfos® A/S has a pump system called SQFlex. It is a renewable energy based water supply system. The applications are mainly for water pumping in remote areas for example in the countryside for cattle or for fresh water use in distant villages. Among other things the system can use solar cells as the energy source. As a step in Grundfos vision of being responsible for the environment they supply their pumps with electricity from solar cells. The solar cells are delivered from First Solar® and are at the moment of a polycrystalline material (CdTe). Due to this, Grundfos is able to reduce the cost of the pump system. This type of material is cheaper to produce than single crystalline materials but suffer from lower efficiency because of recombination processes (Archer and Hill, 2001). It is therefore more difficult to predict how much energy the solar cell can deliver. From an economic point of view Grundfos wants to optimise the dimension of the solar array. The lack of a good model implies that they have to oversize the solar array and the costs increase. Previous attempts at Grundfos include the single diode equation, a two-diode model and some basic curve fitting methods. Since none of them can deliver a reasonable result, a more thoroughly investigation is motivated.
1.2 Literature study A cursory description of models and approaches treated during the literature study together with a discussion of each model and if it suits the thesis aim are performed. For the single crystalline model it is a general knowledge in the solar cell society that the diode equation fails to deliver a realistic result for a polycrystalline material (Nelson (2003), Burgelman, Nollet & Degrave (2000)) so another approach is necessary. Several companies and universities working to develop models based on numerical modelling on polycrystalline materials for describing thin-film solar cells (for example Colorado State University, University of New South Wales and Gent University). Since the 1980s numerous models have been constructed and used for design purposes. The absence of a precise model implies further attempts to explain the complicated structure of polycrystalline materials. Efforts have been made to find an exact analytic expression for the IV- characteristic (Burgelman, Verschraegen, Degrave & Nollet, 2003), but have so far failed. The simulations can, if correctly used and accurately interpreted, be a high-quality complement to experiments and give a better knowledge of the behaviours in thin-film devices (Burgelman et al., 2003). However, the material complexity is cumbersome and requires a lot of computer capacity for the numerical simulation. Additional consequences arise from cells connected in series and in parallel (series and shunt resistance) and non-uniformity in the cell material (personal communication1, Sept 2004). Other aspects are degradation and partial shading that is hard to predict (personal communication, Sept 2004). This leads to an unfeasible method for energy prediction within the scope of this thesis.
Prof. M. Burgelman, Dr. S. Hegedus, Dr G. Agostinelli and graduate student M. Gloeckler
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Based on a large amount of data, a multiple regression analysis may provide an accurate result for a particular case, but deviate for other cases (Eriksson, Johansson, Kettaneh- Wold & Wold, 1999). The method is consequently not suitable for the thesis aim. The last considered model is developed at Sandia National Laboratories over the past twelve years and is described in King, Boyson and Kratochvil (2004). A distinct advantage is the adaptability and resourcefulness to work for almost any solar cell type. Sandia have build up the model based on extensive outdoor measurements in collaboration with other institutes in the United States2. The photovoltaic research is carried out under the Renewable Energy Office with the purpose to increase the accessibility and knowledge of PV devices. University research Prof. Marc Burgelman at Gent University administers a solar cell research group at the ELIS department. The areas of responsibility are modelling issues, CIS, CIGS and CdTe solar cells and thermodynamic behaviour of the cell, ELIS Web page. Prof. JR Sites at Colorado State University (CSU) manage a PV laboratory where the main objective is to understand the difference between single and polycrystalline solar cells. Attention areas is among others thin-film solar cell modelling, see for example Sites (2001). Centre of Excellence in Advanced Silicon Photovoltaics & Photonics at University of New South Wales carry out a comprehensive research divided into five teams with different aims. All of the projects scope properties of silicon-based solar cells and directed by Prof. Martin Green (UNSW web page). Other teams at UNSW perform investigations on GaAs University of Port Elizabeth have a Photovoltaics research group, supervised by Dr Ernest van Dyk. They conduct indoor and outdoor characterization of solar modules, see for example Van Dyk and Meyer (2004) and Meyer and Van Dyk (2000). A review of current models can be viewed in Appendix A. 1.2.1 Conclusions The applied model will be the Sandia model because it has proved to work well for energy prediction, to be flexible and with a reasonable time frame for a Master of Science thesis to construct and modify.
US Department of Energy, National Renewable Energy Laboratory
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1.3 Aim The scope of this thesis work is to develop a model for thin-film solar cells for reliable energy predictions. To achieve the main objective, some sub goals are marked out: • • • • •
The best model from an engineering point of view will be identified and developed according to energy prediction applications. The IV- characteristics and the power will be calculated from the data sheet supplied from the manufacturer. An acceptable model error will be defined. Perform experiment to validate and adopt the model. Implement the model in a simulation tool.
1.4 Format of the thesis After a wide introduction I motivate why a research is necessary and the purpose with the thesis. The theory chapter scopes the solar radiation and the principles of a solar cell. The method, result and discussion chapters are all divided into three sub chapters were the model, the experiment and the validation are discussed. The constructed solar programme can be viewed in Appendix.
1.5 Acknowledgements I would like to grab the opportunity to devote a thought to some persons who helped me personally or professionally during my time in Denmark. On a professional level I’m especially thankful to my supervisor, Hans Stougaard, for giving me freedom to solve the problems independently and always giving me aid and support during the work. I would also like to thank the company Grundfos who financed the stay at their head office. I’m also thankful to Lars Ole that gave me invaluable aid when I carried out the experiments. On a personal level my deepest gratitude is dedicated to Andy and Derek, my “roomies”. Our extensive travelling in Denmark during the weekends, the whisky and poker evenings, football, table tennis, the gym and swimming made the spare time to a great pleasure. I’m really impressed by your ability to find an Irish Pub showing Premier League football no matter where we travel! And I’m still convinced that Swedish humour is funnier than Scottish.
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1.6 Abbreviations and definitions Symbol q kB a b
c T Ns Np n fd E0 Tcell Tamb Gsc Day
h g
s
N z
AM AMa s
H0 KT Hd AOI Rb rd Id rt Ib I0
Definition Elementary charge [C] Boltzmann constant [JK-1] Coefficient for AMa function Temperature calculation constant Coefficient for AOI function Temperature calculation constant Temperature coefficients for voltage [°C-1] Temperature coefficients for current [°C-1] Regression coefficients Temperature difference between back and bulk of solar module [°C] Cells connected in series in one module’s cell string Number of cell strings in parallel in a module Non- ideality factor Ratio of utilized diffuse irradiance by the model Reference irradiance [Wm-2] Cell temperature [°C] Ambient temperature [°C] Solar constant (extraterrestrial) [Wm-2] The day of the year Latitude [rad] Tilt angle [rad] Azimuth angle [rad] Height above sea level [m] Planck’s constant [Js] Ground reflectivity Declination angle [rad] Sunset hour angle [rad] Length of the day [hour] Sun hour angle [rad] Zenith angle of sun [rad] Air Mass (relative path length through atmosphere) Absolute Air Mass (AM adjusted for site altitude) Solar azimuth angle [rad] Daily extraterrestrial radiation on a horizontal surface [kWhm-2] Monthly average clearness index Daily diffuse energy on a horizontal surface [kWhm-2] Angle of incidence [rad /deg] Ratio of beam radiation on a tilted to a horizontal surface Fraction of hourly diffuse and daily diffuse energy on a horizontal surface as function of day length. Hourly average of diffuse radiation on a horizontal surface [kWhm-2] Same as for rd but for the total radiation (I/H) Hourly average beam radiation on horizontal surface [kWhm-2] Extraterrestrial radiation on a horizontal surface over one hour [kWhm-2]
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Ai f IT WS Eb Ed Tback Tcell f1(AMa) f2(AOI) Isc Ee Ix Im Vm Ixx Voc FF Pm MPP Rs Rsh I0 IL, Iph A
Eg Index 0 d b T Global radiation Beam radiation Diffuse radiation Irradiance Irradiation
Anisotropic index Square root of the ratio of beam and total hourly irradiance Total irradiance on the tilted surface [kWhm-2] Wind speed at 10 m [ms-1] Beam irradiance incident on the plane of array [Wm-2] Diffuse irradiance incident on the plane of array [Wm-2] The module temperature at the back surface [°C] The cell temperature, i.e. the temperature of semiconductor [°C] Polynomial of 4th order treating the spectral response Polynomial of 5th order treating the optical response Short circuit current [A] Effective irradiance, fraction of irradiance used by the solar module 4th point on the IV curve where V = 0.5Voc [A] Maximum current [A] Maximum voltage [V] 5th point on the IV curve where V = 0.5(Vm+Voc) [A] Open circuit voltage [V] Fill Factor Maximum power [W] Efficiency [%] Maximum power point Series resistance [ ] Shunt resistance [ ] Saturation current [A] Photo generated or light induced current [A] Solar module area, by convention is the frames included as well [m2] Frequency [s-1] Wave length [m] Band gap [eV] Reference conditions Diffuse Beam Tilted, total Total radiation, beam + diffuse solar radiation on a horizontal surface. [Wm-2] Direct solar radiation [Wm-2] Radiation that have been scattered by the atmosphere [Wm-2] Solar power on a unit area surface [Wm-2] Incident solar energy input on a surface per unit area [Jm-2], correspond to the time integration of the irradiance.
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2. Theory This chapter give a brief description of the sun characteristics, the extraterrestrial and the terrestrial radiation. We will follow a light beams path from the sun to the earth’s surface and try to understand what factors affect its spectral distribution. Finally, some geometrical calculations are performed to determine the sun’s position in the sky that will prove very useful for the solar modelling. The second part of the chapter is devoted to the basics about PV devices and the principles of solar cells. The main features are described and the key words: charge generation, separation and collection is explained and is important to understand how a solar cell works. I also identify the problems with polycrystalline material.
2.1 Solar radiation 2.1.1 Properties of the sun The sun radiates approximately as a black body at a certain temperature, but because of temperature and density gradients along the radius of the sun, a black body cannot fully explain the emitted solar radiation. The energy is converted through a fusion process where hydrogen nuclease (protons) reacts to form helium. The reaction is exoterm and energy is released due to a loss in the mass according to the Einstein formula (2.1), E = (∆m )c 2
(2.1)
where E is the released energy, m the change in mass and c the speed of light. A body that is a perfect absorber of radiation is a so-called black body. Independent of wavelength, all the incident light will be absorbed. The black body is a theoretical subject and do not exist in the real world. But still, it is widely used to explain radiation phenomena. A black body is also an ideal emitter. The sun has a surface temperature close to 6000K and radiate approximately as a black body at the same temperature. To calculate the amount of power emitted from the sun is the StefanBoltzmann law used (2.2), E b = σT 4 ,
where Eb is the radiated power per unit area, temperature of the body.
(2.2) the Stefan-Boltzmann constant and T the
The electromagnetic spectrum The most important radiation for solar cells is thermal radiation. If it exist a temperature difference between the body and the environment, thermal radiation will be exchange. It propagates with the speed of light. When excited electrons or atoms relax to their ground states they emit energy in the form of electromagnetic radiation. The relaxation corresponds to a change in electronic, rotational and vibrational states and range over a large spectrum of wavelength.
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Photon radiation The light can be described by an electromagnetic wave but in some cases it is better explained by an energy package, photons. The photons have almost no mass and have therefore no momentum. The photon has the energy E = hν . Hence it is obvious that longer wavelength, lower frequency, decrease the energy. 2.1.2 Extraterrestrial radiation We have now achieved a basic understanding of the properties of the sun. The next step is to estimate the amount of radiation that hit the earth’s atmosphere. The solar constant is defined as the irradiance incident on a unit surface outside the atmosphere normal to the angle of propagation. The method to determine the nearly fixed solar constant is to use weather balloons and spacecrafts to avoid measuring the attenuation in the atmosphere. Many different values have been proposed and the current value is 1367 Wm2 with an uncertainty of 1 % and is recommended by the World Radiation Centre (WRC). Fluctuations in the extraterrestrial radiation arise from variations explained by the activity of sunspots and because of that the distance between the sun and the earth changes in a yearly cycle. The influences of the sunspots are small and can be neglected for energy prediction for a terrestrial solar cell. However, the elliptic orbital lead to a fluctuation in the solar constant up to ±3% (Duffie & Beckman, 1991) according to, 2πn 1 + 0.033 cos 365
cos θ z
G0 = Gsc
(2.3)
where Gsc is the solar constant, n the day of the year and θz the zenith angle. Substitute the expression for the zenith angle into (2.3) the extraterrestrial radiation on a horizontal surface becomes (2.4), 2πn 1 + 0.033 cos 365
G0 = Gsc
(cos φ cos δ cos ω + sin φ sin δ ) .
(2.4)
where φ is the latitude, δ the declination angle and ω the sun hour angle. If G0 is known then the daily solar radiation on a horizontal surface can be calculated through time integration, H0 = where
24 ⋅ 3600Gsc
π
s
2πn 1 + 0.033 cos 365
(cos φ cos δ sin ω s + ω s sin φ sin δ )
(2.5)
is the sunset hour angle in radians.
The hourly solar radiation (2.6) is given by the time integration of (2.5) between to clock hours,
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I0 =
12 ⋅ 3600
π
2πn (cos φ cos δ (sin ω 2 − sin ω1 ) + (ω 2 − ω1 )sin φ sin δ ) (2.6) 1 + 0.033 cos 365
G sc
where ω2 is the larger and ω1 the smaller sun hour angle. I have now identified a method to calculate the extraterrestrial horizontal radiation and will continue to look at the terrestrial radiation. 2.1.3 Terrestrial radiation To predict future possible utilized energy on a solar module /array, earlier measurements from a nearby location are used. With the aid of the extraterrestrial radiation it is possible to calculate the terrestrial radiation. Therefore, the author needs to understand how the atmosphere influences the solar radiation and finally how to calculate the radiation on a tilted surface on an arbitrary location on earth. Solar radiation data There exist several databases for the specified purpose, the most detailed containing hourly values, but more common is monthly average daily terrestrial radiation on a horizontal surface ( H ). The data is given as global irradiation [Jm-2] and are usually collected with a thermopile pyranometer (Duffie & Beckman, 1991). An extensive monthly average daily solar radiation database is the European Solar Radiation Atlas (ESRA) and more databases can be found elsewhere (Anderson, 2002). Atmospheric attenuation To understand the attenuation phenomenon two new terms are introduced, scattering and absorption. Scattering Detailed research about this topic has been thoroughly performed by Iqbal (1983). The atmospheric scattering arises from interactions with air molecules, water and dust. How large the attenuation is depends on particle size and number. Air molecules have little influence because the wavelength of the solar radiation is much longer than the particle size. Therefore, dust and water particles have the largest influence. The magnitude of the scattering is strongly dependent of the wavelength. A deeper explanation of scattering can be found elsewhere (Fritz, 1958). Absorption The other effect on the emitted radiation in the atmosphere is absorption. Figure 2.1 shows the influence of absorption on the spectral distribution of solar radiation. The smooth black curve corresponds to the extraterrestrial radiation calculated from Planck’s radiation law, (2.7), Sλ =
2πc 2 h
λ
5
1 e
hc
λkT
−1
.
(2.7)
Sλ is the spectral irradiation density, h is the Planck’s constant, λ the wavelength and kB the Boltzmann constant. The spectral irradiation is displayed in figure 2.1, the red curve is AM0 spectrum and the green AM1.5.
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Figure 2.1. Spectral distribution of a black body, AM0 and AM1.5. Taken from Web page, Dec 2004. Available at: http://www.tfp.ethz.ch/Lectures/pv/spectrum.html
Short wavelength radiation, i.e. x-rays and IR, are mostly absorbed in the atmosphere by water vapour and carbon dioxide and UV is absorbed by ozone. This indicates that very little radiation energy from these wavelength regions can be used by a solar cell. Thus, the interesting wavelength region for terrestrial applications is 290nm to 2500nm (Duffie & Beckman, 1991, 67). Reflection The ground can also reflect solar radiation. The effect is specially pronounced in areas with high ground reflectivity such as snow cover or deserts. The reflected radiation can contribute as diffuse radiation to the solar module. Clearness index The monthly average daily clearness index, (2.8), is the ratio of terrestrial to the extraterrestrial radiation, KT =
H . H0
(2.8)
where H is imported from a database and H0 calculated from (2.5). The clearness index provides information of the fraction of diffuse energy to total energy on a daily basis. It is necessary to separate the beam and diffuse component on the horizontal surface to be able to calculate the radiation on a tilted surface. One common technique is to find an empirical relation such as (2.9), Id = f (k T ) , I
(2.9)
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where Id is the hourly diffuse and I the hourly total radiation on a horizontal surface. There exists several correlations, see for example Erbs, Klein and Duffie (1982). The same method is carried out for the daily radiation, Hd = f (k T ) . H
(2.10)
An empirical expression of (2.10) developed by (Erbs et al., 1982) is given by (2.11) and the correlation is used in this thesis, Hd = 1.391 − 3.560 K T + 4.189 K T2 − 2.137 K T3 H ifω s ≤ 81.4 ,0.3 ≤ K T ≤ 0.8
(2.11)
Hd = 1.311 − 3.022 K T + 3.427 K T2 − 1.821K T3 H ifω s > 81.4 ,0.3 ≤ K T ≤ 0.8
where K T is the monthly average clearness index. Calculation of hourly values When I have information of the daily horizontal radiation it can be used to predict the hourly values. The method described here has proved to work best under clear sky conditions. Trough experiments the distribution of radiation over the day have been measured. This leads to a new definition, (2.12), the fraction of hourly to daily radiation on a horizontal surface, rt =
I . H
(2.12)
Collares-Pereira and Rabel (1979) have derived the semi- empirical equation (2.13), rt = (a + b cos(ω ))rd .
(2.13)
The parameters a and b are given by (2.14), a = 0.409 + 0.5016 sin (ω s − 60 )
(2.14)
b = 0.6609 − 0.4767 sin (ω s − 60 )
and rd is the fraction of hourly diffuse to daily diffuse radiation on a horizontal surface (2.15),
rd =
Id π cos ω − cos ω s = . H d 24 sin ω s − ω s cos ω s
(2.15)
Terrestrial radiation on tilted a surface We are now able to calculate and separate the beam and diffuse component on a horizontal surface. The last step is to estimate the radiation on a tilted surface.
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There exist three types of diffuse radiation. It consists of an isotropic part that distribute uniformly over the sky, a circumsolar diffuse part that occurs at the sky near the sun and the last part is a horizon brightening close to the horizon (Duffie & Beckman, 1991, 91-92), figure 2.2.
circumsolar beam isotropic
horizon brightening
Figure 2.2 Description of the different types of radiation, reproduced from Duffie & Beckman (1991). The principles of the beam and the three parts of diffuse radiation are displayed.
To calculate the radiation on a tilted surface a so-called sky model is applied. From database values of radiation on a horizontal surface an empirical expression for the diffuse tilted radiation is developed. The total hourly radiation on the tilted surface is then, I T = I T ,b + I T ,iso + I T ,circ + I T ,hz + I T , refl ,
(2.16)
for the beam, isotropic, circumsolar, horizon brightening and reflective components on the tilted surface. There exist several different methods to calculate the different contributions in (2.16) and the one applied in this thesis is the Anisotropic Sky model. The core of the model has been developed by Hay and Davies (1980) and has been modified by Reindl, Beckman and Duffie (1990), I T = (I b + I d Ai )Rb + I d (1 − Ai )
1 + cos β 2
1 + f sin
3
β
2
1 − cos β , 2
+ Iρ g
(2.17)
where β is the tilt angle of the module, ρg the ground reflectivity and Ai is an anisotropic index, giving the fraction of beam radiation transmitted through the atmosphere (2.18) or the fraction of horizontal diffuse that can be interpreted as forward scattered, Ai =
Ib I0
(2.18)
and f is given by (2.19)
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f =
Ib . I0
(2.19)
Rb is the ratio of beam radiation on a tilted to a horizontal surface, (2.20) Rb =
cosθ . cos θ z
(2.20
where θ is the angle of incidence. If Rb is smaller than one it is of no use to tilt the module surface, see figure 2.3. The first term in (2.17) is the beam and forward scattered diffuse radiation, the second term account for the isotropic part, the third for horizon brightening and the least for ground reflection. Gb
GbT Gbn
z
ß
Figure 2.3. The relation of beam radiation on a tilted surface to that on a horizontal surface. Gb is the normal to the horizontal surface, Gbn the incident beam radiation and GbT the normal to the tilted surface.
Thus, the author have identified the effects influencing the solar radiation and defined an empirical relation to calculate the incident radiation on a tilted surface. 2.1.4 Geometrical properties With a set of geometrical relationships it is possible to describe the sun’s path over the sky and calculate the angles between the sun and the solar module.
The declination angle is the angle between the line of the equator and the beam radiation at solar noon (Markvart, 1994) and can be expressed as, 284 + n δ = 23.45 sin 360 365
(2.21)
where n is the day of the year. The value 23.45 represents the angle between the earth’s rotational axis and the orbital around the sun, figure 2.4. Equation (2.21) states that the declination angle is treated as a constant over a specific day.
18
Summer solstice
Equator
Winter solstice
Equinox
Figure 2.4. The angle between the line of the equator and the beam component of solar radiation at solar noon is the declination angle . The declination angle at winter solstice 21-22 December on the northern hemisphere when is approximately –23.45º. The corresponding summer solstice occurs at 21-22 June with the declination angle of 23.45º. At the equinox dates the declination angel is zero.
Then is the sun hour angle calculated from (2.22),
ω=
2π (hour − 12) 24
(2.22)
where hour is the time in hours. The hour angle describes the position of the sun at a certain moment. Due to the earths rotation around its own axis changes over the day. is zero at solar noon and increases in eastwards direction. An hour corresponds to an angle change of 15 degrees. The zenith angle is then calculated with the relation (2.23),
θ z = arccos(sin φ sin δ + cos φ cos δ cos ω ) The sunrise hour angle
s
(2.23)
is given in (2.24) and occurs at a zenith angel of 90 degrees,
ω s = arccos(− tan φ tan δ ) and the sunset angle is simply –
(2.24) s.
The absolute air mass is given by (2.25), AM a =
[
P −1.634 AM ≈ e − 0.0001184 h AM = e −0.0001184 h cos θ z + 0.5057(96.080 − θ z ) P0
]
−1
. (2.25)
The AMa is the relative path length the solar beam travel trough the atmosphere, adjusted for site altitude h and AM is the air mass. P is the pressure at the specific site and P0 the pressure at sea level. A zenith angle of 0 degrees corresponds to an AM of 1.0. A standard spectrum corresponding to AM 1.5 with a predefined spectral distribution is commonly used for Standard Test Conditions (STC). The principle of the air mass can be seen in figure 2.5.
19
AM 1
~AM 10
Figure 2.5. Schematic illustration of the air mass, describing the relative length the light travels through the atmosphere. Perpendicular incident radiation corresponds to AM 1.0 and radiation from the horizon can increase the air mass to over 10.
The angle of incidence, AOI, is then given by (2.26), sin δ sin φ cos β − sin δ cos φ sin β cos γ + cos δ cos φ cos β cos ω + AOI = arccos cos δ sin φ sin β cos γ cos ω + cos δ sin β sin γ sin ω
where γ is the azimuth angle and when
(2.26)
is zero the last terms dropping out.
Thus, we now know how to determine the relative path length through the atmosphere and the angle of incidence that will prove to be essential for solar cell modelling.
2.2 Principles of solar cells Solar cells are based on semiconductor materials and consist of junction created between two materials or one material but with different doping, n and p- type doping. An n- doped material has abundance of electrons and a p- doped material have more holes, figure 2.6. If put together, the Fermi levels force to coincide at the interface which give rise to a band bending, see figure 2.7. The n- doped material has plenty of electrons and they will attract the holes in the p- layer so that in the vicinity of the interface at the n- side there is a positive charge and on the p- side a negative charge. This is called the depletion layer and has approximately no moving charges. Over the depletion region is an electric field that will affect the band structure and is essential for charge separation.
20
Figure 2.6. Band structure of two semiconductors with different doping. Taken from Web page, May 2004, available at www.dur.ac.uk/~dph0www5/solar.html
Figure 2.7 Heterojunction, due to the doping a depletion region occurs that are able to separate charges. Taken from Web page, May 2004, available at www.dur.ac.uk/~dph0www5/solar.html
When the cell is illuminated with light, the photon flux hit the material. Some of the photons are reflected and some of them tunnel through the whole device and are of no use. But one part of the photons absorbs in the material and if the wavelength is short enough, i.e. the energy is higher than the band gap energy, an electron – hole pair generates. If the charge carriers can diffuse to the depletion layer and not recombine, there is a high probability that they are separated by the electric field, collected by metallic fingers and run an external load, see figure 2.8. Thus, the light energy is converted to a potential energy when the photon is absorbed. If the charge carriers are separated in the junction it give rise to a voltage and the electro magnetic radiation have been converted to electrical energy.
21
Base
Emitter
Figure 2.8. Light absorption, a photon generates an electron- hole pair and are separated over the junction and collected by a circuit and can then run an external load. The semiconductor layer exposed to the sun is usually called emitter and the other layer the base Taken from Web page, May 2004, available at www.dur.ac.uk/~dph0www5/solar.html
2.3 Characteristics of a photovoltaic device 2.3.1 The equivalent circuit A solar cell can be treated as a current generator in parallel with a non-linear resistive element, a diode, from an electrical point of view. A solar cell in the dark behaves as a diode. When a load is connected over the terminals a potential difference occurs, inducing a current flowing against the photo-generated current, Iph. Equation (2.27) describes the IV characteristic of an ideal solar cell,
I (V ) = I ph − I 0 e
qV k BT
−1
(2.27)
where the second term is the dark current and I0 the saturation current, I the current through the load, q the elementary charge, kB the Boltzmann constant and V the voltage over the load. This is called the superposition approximation since the reverse current for an illuminated cell is not the same as the dark current. In a real solar cell the electrical circuit is somewhat more complicated. The cell suffers from series and shunt resistances. The series resistance, Rs, is dependent on the “friction” of the material and the contacts. The shunt resistance, Rsh, is explained by a leakage current and is electrically connected in parallel with the current source. Ideally, we want Rs to approach zero and Rsh infinity. When incorporating the resistance effects equation (2.27) is modified to (2.28), which is the diode equation with parasitic resistance,
22
I (V ) = I ph (V ) − I 0 e
q (V + IRs )
( nk BT )
−1 −
V + IRs . R sh
(2.28)
where n is the non-ideality factor. The equivalent circuit is shown in figure 2.6. Iph
Rs
+
Id
V
Rsh
Figure 2.9. Equivalent circuit for a solar cell. Iph is the light induced or photo-generated current, Id the diode current or the so-called dark current and R are the shunt and series resistance and V the terminal voltage.
2.3.2 Design features The characteristic variables for a solar cell is typically the open circuit voltage, Voc, short circuit current, Isc, the maximum power point, (Vm, Im), which is the voltage and current at maximum power. Two other important variables are the Fill Factor, FF, and the efficiency, .
The open circuit voltage is defined as the voltage at infinite load (no connection between the terminals) and, by definition, no current runs through the circuit. The short circuit current occurs when the terminals are directly connected together, i.e. no resistance in the circuit. The Maximum Power Point, MPP, is the point on the IV curve where the current times the voltage maximize, figure 2.7. IV curve 1 Isc Im
I(A)
0.8
0.6
0.4
0.2
Vm 0
0
0.1
0.2
0.3
0.4 V(V)
0.5
0.6
0.7
Voc 0.8
Figure 2.10. A simulated IV curve for a solar cell where the red dashed curve is the maximum power area and the black dashed curve the area constrained by Isc and Voc.
The fill factor is defines as (2.29), FF =
I mVm . I scVoc
(2.29)
The fill factor is a dimensionless factor between one and zero describing the quality of the solar cell. A geometrical interpretation is that the FF gives the squareness of the IV curve by 23
comparing the two rectangles in figure 2.7. For a practical solar cell the efficiency (2.30) is of paramount importance and is given by,
ε=
Pm I mVm I scVoc FF = = Ps E⋅A E⋅A
(2.30)
where Ps it the incident power, E the plane of array irradiance and A the cell area including frames.
2.4 Polycrystalline materials To read and understand this sub chapter it is required to possess knowledge about characteristics of a solar cell, for instance recombination, diffusion length, mobility and life time. The aim is to describe why it is more difficult to understand the properties of polycrystalline than single crystalline solar cells. Nelson (2003) describe the following complications for polycrystalline materials in general: • • • •
The diffusion lengths Ln,p are shorter than for single crystalline materials, which implies the material to be a strong optical absorber. Large losses in the vicinity of the front surface. Therefore, the emitter is often chosen to be a wide band gap material, i.e. act as a window layer. The impurities make the doping more difficult and can lower the built in bias over the junction. The grain boundaries increase the resistance in the bulk of the material and make the conductivity, the minority carrier lifetime and the diffusion constants carrier density dependent.
Hegedus and Shafarman (2004) define three types of loss mechanisms associated with polycrystalline materials: 1. Recombination that put constrains on the open circuit voltage. 2. Parasitic losses, for example series resistance and shunt conductance. 3. Optical losses, which lower the generated charge carriers and the short circuit current. To outline an approach for polycrystalline materials, knowledge about the different material properties is essential. Grains characterize a polycrystalline material. The grains are small from quantum point of view and have therefore the same optical properties as the single crystal of the same material. But the grains highly affect the transport and recombination effects at the boundaries. The grains have different sizes and position in space and create a lattice mismatch where there are a presence of dislocations, defects, vacancies and different bond angles and distances. The grain boundary disturbance propagates a few atomic layers from the interface. It also exists extrinsic impurities that limit the carrier transport. All this deviation from a single crystal can and will introduce additional electronic states. These states do not have the same pattern as the crystal and can therefore be localized in the band gap. These delocalised intra-band states can act as recombination centres and trap charges. Shallow states usually just “delay” the current while deep traps limits the current.
24
If the current cross the grain boundary perpendicular, the potential barrier will lower the majority charge carrier mobility. The mobility and lifetime of minority charge carriers will be reduced due to the presence of recombination centres at the interface. How rigorous these effects are depends upon the photo-generated carrier density, the density of interface states and on the doping level. Current that flows parallel to the grain boundaries is also affected. The interface attracts the minority carriers and act as sink. The ideal diode equation does not any more describe the IV- characteristic satisfactory. The dark current can be a function of intensity and the photo-generated current is voltage dependent and not well described by the short circuit current (2.31),
I (V ) = I ph (V ) − I 0 exp
qV −1 . nk B T
(2.31)
Landsberg and Abraham (1984) have modelled the behaviour of grain boundaries under illumination. It is proved that a high doping level saturates the trap states. Also, under concentrate illumination level the influence of grain boundaries decrease in strength. We have now seen that the morphology distinctions between single and polycrystalline materials change several properties and that the diode equation no longer can describe the electrical performance of the solar cell. A discussion about the properties of the thin-film material CdTe can be viewed in Appendix B.
25
3. Methods The method chapter gives first an explanation and motivation of tools and continue with a description of the applied model. The base line then set for the experimental set-up and procedure and the chapter ends with the chosen validation method.
3.1 Explanation and motivation of tools MathsoftTM Mathcad® 11 is an adaptable and multi purpose programme for mathematical calculations and programming. One phase of the literature study was to become skilled at using the programme. The programme provides a user-friendly interface but it is still possible to invoke complex analytical expressions and perform iterations and simulations. It is also easy to see each step in the calculation and connect the sheet to other documents, for example an environmental database. To adopt the model to a customer interface, the model will be implemented in a C++ platform to the Grundfos’ product programme, WinCAPS. Mathcad contains a set of math functions, for example 1D PDE solving block, which is useful when solving the semiconductor equations. The last reason for the choice of Mathcad is that it is a known programme at Grundfos and earlier attempts have been made in the programme. The idea is to use a simple, but powerful, programme and focus on the physical and mathematical behaviour of the model instead of spending time on a programming language not familiar to the thesis author. Matlab® 6.5 is another straightforward choice due to its many applications and its accessibility.
3.2 Model This chapter is devoted to the solar cell programme aimed to improve Grundfos’ tools for energy prediction of solar cells. To make it clear and foreseeable the programme is implemented in Mathcad and divided into different block with different tasks. The programme is attached in Appendix F and follows with a comprehensive explanation of each step in the calculation. •
Definition block. The definition block scopes all the constants and coefficients relevant for further calculations.
•
Environmental block. From user defined site and time specific data a set of geometrical relationships is used to calculate the absolute air mass and angle of incidence. With the aid of BP database values, empirical correlations are used to estimate the hourly beam and diffuse radiation on a tilted surface.
•
Calculation block. The calculation block estimates the electrical performance of the solar module or solar array. The equations are semi empirical and use data from the environmental block as input and provide five points on the IV curve as output.
•
Simulation block. Generate the whole IV curve.
26
3.2.1 Calculation block The performance model is based on semi empirical equations, i.e. unknown effects are hidden in the coefficients of the model but nevertheless included and that’s the important issue from an engineering point of view. Despite the lack of a complete analytical expression the Sandia performance equations treat many of the significant effects for a solar cell/module/array under illumination. Each equation has been derived with support of theory for the electrical, thermal and optical behaviour on cell level. During seven years the model has proved its accuracy under comprehensive outdoor testing by Sandia and others. The corresponding coefficients are available at a module database at Sandia’s web page. The electrical characteristic is known at STC (E0, T0, Standard solar spectrum with AMa = 1.5, AOI = 0) and is then scaled due to fluctuations in irradiance, temperature, air mass and angle of incidence.
We will now look more thoroughly on the performance equations and describe the significant effects incorporated in the equations. It is important to understand the fundamentals. The origin is the reference values and according to the prevailing conditions the operational values differ from the reference value with a known (calculated) magnitude. This is illustrated for the short circuit current in (3.1), I sc = f ( AM a )g ( AOI )h(E )i (T ) ⋅ I sc 0
(3.1)
where f, g, h and i are functions. The short circuit current is given by, I sc = I sc 0 f 1 ( AM a )
(E f ( AOI ) + f b
2
E0
d
E diff
)
[1 + α sc (Tcell − T0 )]
(3.2)
where f1(AMa) is a forth order polynomial of the absolute air mass (3.3), Eb the beam irradiance, f2(AOI) a fifth order polynomial of the angle of incidence (3.4), fd the fraction of diffuse irradiance, Ediff, that can be utilized of the solar module and sc the normalized temperature coefficient. Depending on the relative path length the light has to travel through the atmosphere the spectral distribution changes. The longer distance through the atmosphere, the bigger energy loss due to absorption and reflection and the radiation is displaced to longer wavelength. To account for the spectral response of the solar module a function of the absolute air mass is applied to the short circuit current. The general formula is given by f 1 ( AM a ) = a 0 + a1 AM a + a 2 AM a2 + a 3 AM a3 + a 4 AM a4
(3.3)
where ai are coefficients. How to determine the coefficients can be found elsewhere (King, Kratochvil & Boyson, 1997). The AMa function is primarily developed at clear sky conditions and is displayed in figure 3.1.
27
AMa function
1,06
1,05
1,04
f1(AMa)
1,03
1,02
1,01
1
0,99
0,98 0
0,5
1
1,5
2
2,5
3
3,5
4
4,5
5
AMa
Figure 3.1. f1 polynomial as function of the absolute air mass to describe the solar cell’s spectral response.. The function is experimental determined and increases with the AMa.
King, Kratochvil and Boyson (2002) have shown that the spectral response has a quite small influence for annual energy production ( x(2)=1-x(1) F = Ix0.*(x(1).*(Ee)+(1-x(1)).*((Ee).^2)).*(1 + aIsc.*(Tcell - T0)); Maximum voltage function F = Vmax(x,data) %-------------------------------------------------------------------------% Input: guess values x0, data = [Tcell, Isc] % Output: [x(1),x(2)] = [c2 c3] %-------------------------------------------------------------------------Tcell = data(1,:); Isc = data(2,:); % Constants Vmp0 = 65; T0 = 25; Bmp = -0.098; aIsc = 0.00065; n = 1.3; kb = 8.617385*10^-5; Ns = 116; Isc0 = 1; dT = n*kb.*(Tcell+273.15); Ee = Isc./(Isc0.*(1+aIsc.*(Tcell-T0))); F = Vmp0 + x(1).*Ns.*dT.*log(Ee) + x(2).*Ns.*(dT.*log(Ee)).^2+...
84
Bmp.*(Tcell - T0); The fifth IV point function F = Ixx(x,data) %------------------------------------------------------------------------% Input = coefficient guess values and matrix data = [Tcell,Isc] % Return = Vector with the coefficients [x(1), x(2)]=[c6, c7] % Ixx ydata provided from measurements. Defined at (Voc+Vm)/2 %------------------------------------------------------------------------Tcell = data(1,:); Isc = data(2,:); amp = 0.0022; aIsc = 0.00065; T0 = 25; Isc0 = 1; Ixx0 = 0.52; Ee = Isc./(Isc0.*(1+aIsc.*(Tcell-T0))); % Constrain x(1)+x(2)=1 F = Ixx0.*(x(1).*(Ee)+(1-x(1)).*((Ee).^2)).*(1 + amp.*(Tcell - T0)); The open circuit voltage function F = voltoc() %-------------------------------------------------------------------------% Calculate the Voc using feval. % Input: % Output: Voc at given Tcell and Isc. % The calculation can readily be performed in mathcad as well. %-------------------------------------------------------------------------% Tcell = Put in the measured values % Isc = -"Voc0 = 90; Isc0 = 1; T0 = 25; Ns = 116; Boc = -0.22; n = 1.3; kb= 8.617385*10^-5; dT = n.*kb.*(Tcell+273.15); F = Voc0 + Ns.*dT.*log(Isc./Isc0)+Boc.*(Tcell-T0);
85
F Solar programme
86
The solar cell program. Developed by Tobias Hellgren fall 2004. The program consists of a definition, environmental calculation, simulation, reference and database block. The program is connected to a database and the code will be implemented in C.
Extension package: Account for tracking and concentrator solar cells.
Sandia database values, accessible from Internet: http://www.sandia.gov/pv/docs/Database.htm or from the database block below. Change the coefficients if change solar module. The default settings are for the FS 50 D (CdTe) and the database provide information for 125 modules. M-files in Matlab provide support for new coefficients.
Standard values
Isc0 := 1
Voc0 := 90
Im0 := 0.77
Vm0 := 65
Ix0 := 0.91
Ixx0:= 0.52
Area := 0.72
Coefficients for the AMa calculation:
a0 := 0.9196
a1 := 0.07164
a2 := −0.01338
a3 := 0.000961
−5
a4 := −2.46⋅ 10
Coefficients for the AOI function,: Valid for any flat plate module with a glass plate front surface
b0 := 1
b1 := −0.002438
b2 := 0.00031
−5
b3 := −1.25⋅ 10
−7
b4 := 2.11⋅ 10
−9
b5 := −1.36⋅ 10
Temperature coefficients Determined at STC solar spectrum with constant irradiance and normalized by dividing by STC values, i.e. Isc0, Im0, for the current.
α sc := 0.00065 [C-1]
C- coefficients
c0 := 1.073
α m := 0.0022 [C-1]
β oc := −0.22
[VC-1]
β m := −0.098
[VC-1]
Redefine consulting the module database or the Matlab m-files
c1 := −0.073
c2 := −1.25
c3 := −18.544
c4 := 1.051
c5 := −0.051
c6 := 1.155
c7 := −0.155
---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------Temperature calculation
∆T := 3 a := −3.47
Temperature difference between back surface and bulk of the solar module
b := −0.0594
The coefficients are valid for any module with the same design features glass/semiconductor/glass.
Additional Sandia
Ns := 116
Number of cells in series in the module's cell-string
Np := 1
Number of cell-strings in parallel in the module
n := 1.3
Non-ideality factor
fd := 1
Ratio of utilized diffuse irradiance
E0 := 1000
Irradiance at Standard Test Condition (STC)
Figure 1. Solar module with cells connected in series.
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------Solar constant: power per unit area perpendicular to the radiation propagation Constants outside atmosphere. − 19 − 23 T := 25 q := 1.6021773310 ⋅ ⋅ kB := 1.38065810 Gsc := 1.367 0 [kW/m2]
Input: Daily terrestrial radiation (kWhm-2) on horizontal surface total = beam + diffuse and ambient temperature from database. The block is based on Hans Stougaard's programme for the same purpose and the equations can be found elsewhere [1]. User defined input: Day, φ, β, γ, h, (track option) Output: AMa, AOI, Ebeam, Ediff --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------Site and time data:
Day
Latitude(rad)
φ := 55.13⋅
Day := 292
H := 1.16
Tilt angle(rad)
π
β := 58⋅
180
Azimuth angle(rad)
π
γ := 0⋅
180
π 180
height(m)
Track option
h := 25
The daily energy input per unit area on a horizontal surface at the given site (kWh/m-2), monthly average. H collected from BP database.
ρ g := 0.2
Ground reflectivity (albedo). Set to an arbitrary average value (0.2 standard value), no information in database.
--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------Calculation of geometrical relationships between the sun and the solar module.
Declination angle of the sun, i.e. the angle between the plane of the equator and the solar noon (when the sun is on the local meridian), north > 0, south < 0.
δ := 23.45⋅ sin 2⋅ π⋅
( 284 + Day) 365
⋅
π 180
δ = −0.193
Sunset hour angle (at zenith angle = 90)
ωs := acos ( −tan ( δ) ⋅ tan ( φ) )
ωs = 1.287
Length of the day (h)
N :=
24 2⋅ π
⋅ 2⋅ ωs
N = 9.83
Figure 2. The geometrical relationship between the sun and the solar module.
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------Define a vector containing the hours plus the sunrise and sunset
int :=
N 2
intL := floor( int)
intL = 4
Include sunrise and sunset values to the hours, check if sunrise and sunset is on a hour
intL :=
intL if intL
( intL + 1)
int
otherwise
j := 0.. 2⋅ intL
88
Define the solar day where solar noon is 12, i.e. the solar day is symmetrical distributed around solar noon
x := j + 12 − intL j
Sun hour angle
Sunrise
Sunset
x := 12 − int
x
2⋅ intL
0
hour := x
:= 12 + int
x = 7.085 0
x
2⋅ intL
= 16.915
hour vector
--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------New calculations
Define a new hour vector with the midpoint of each hour.
mhour :=
for i ∈ 1.. 2intL − 1 mhour ← hour + 0.5 i
i
mhour Set the first element in mhour to the midpoint between sunrise and the first integer hour.
mhour := hour + 0
(hour 1 − hour 0)
0
2
Set the last element in mhour to the midpoint between last integer hour and sunset
hour mhour
2⋅ intL− 1
:= hour
2⋅ intL−1
+
2intL
− hour
2⋅ intL−1
2
--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Describes the angular displacement of the sun. The solar noon is defined as the time when the sun crosses the local meridian (longitude) and it changes 15 degrees per hour due to the rotation of earth (rotate 360 deg on 24h). 12 positive angle.
ω :=
2⋅ π 24
⋅ ( mhour − 12)
One angle for each middle hour relative to solar noon
Zenith angle of the sun, the angle between the normal of a horizontal surface and the incident solar light. I.e. if the solar module have β = 0 then the AOI an zenith angle coincide.
→ θ z := acos ( cos ( φ) ⋅ cos ( δ) ⋅ cos ( ω) + sin ( φ) ⋅ sin ( δ) ) Check for error values, if the angle if larger than 90 degrees no energy well reach the solar module
j := 0 .. 2⋅ intL − 1 θ z := j
π θ z if θ z ≤ 89.99⋅ 180 j j 0.0001 if θ z j 89.99⋅
π 180
0
To avoid division by zero in the tracking part. Not used at the moment.
otherwise
89
The Air Mass
The relative path length the light has to travel is described by the air mass and is used for the solar modules spectral response. The AMa is 1 if the sun is normal to earth (at sea level) and around 10 at sunrise and sunset. The AMa approaches infinity when the zenith angle approaches zero so and validation check in necessary.
→ → − 1 → → − 1.634 → → → 180 AMa := exp( −0.0001184⋅ h ) ⋅ cos θ z + 0.5057⋅ 96.080 − θ z⋅ π
Assume that the maximum relative path length is AMa = 10.
AMa := j
AMa
j
if AMa < 10 j
10 otherwise Figure 3. Schematic picture of the relative path length through the atmosphere.
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------Solar tracking code: Passive in this version.
Solar Azimuth angle: Angular displacement of the sun's projection on a horizontal surface in relation to south. Used for tracking.
→ sin ( ω) γ s1 := asin cos ( δ) ⋅ sin θ z
( )
γ sdeg :=
180 π
⋅ γ s1
The azimuth angle depends on the quadrant; determine the angle with the aid of the relationship between the hour angle and a new defined hour angle w If the day is longer than 12 hours the angle can be larger than | 90 | degrees.
tan ( δ)
ωew :=
acos
tan ( φ)
if φ > δ
ωew = 1.707
1 otherwise C1 := j
1 if ω j < ωew
1 if ωew
C1 := j
( −1) otherwise
1 if φ⋅ ( φ − δ) ≥ 0
C2 :=
1
( −1) otherwise
C1 otherwise j
→ → → → → 180 1 − ( C1⋅ C2) γ s := C1⋅ C2⋅ γ s1⋅ + C3⋅ ⋅ 180 2 π
C3 := j
1 if ω j ≥ 0
( −1) otherwise
(
)
Daily extraterrestrial radiation on a horizontal surface. The expression include the solar constant's (Gsc) time integration over one day. The next term take care of the variation in the sun to earth distance (the energy emitted from the sun is treated as constant) and the last term count for the incident angle i.e. the zenith angle --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
90
Energy calculation H0 :=
24⋅ Gsc π
⋅
1 + 0.033⋅ cos
2⋅ π⋅ Day
365
(
( )
⋅ cos ( φ) ⋅ cos ( δ) ⋅ sin ωs + ωs ⋅ sin ( φ) ⋅ sin ( δ)
)
Energy for the given day and site outside the atmosphere
H0 = 3.548
[kWh/m2]
Monthly average clearness index is the ratio of monthly average daily radiation on a horizontal surface and the extraterrestrial radiation. Affect how much o the incident energy that can be utilized. H is imported from database, widely available from pyranometer measurements word wide.
H KT := H0
KT = 0.327
One value for each day due to change in H0.
Beam and diffuse components of monthly radiation ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------The purpose to split the total radiation on a horizontal surface in a beam and diffuse component is to be able to calculate the radiation on a tilted surface. The current estimations are based on analysis of measured values. The developed correlation may differ significant for a specific hour but the error will even out over a larger time period, see [1] p81. Diffuse terrestrial energy on a horizontal surface as a function of KT. Uncertainties in the correlation arise due to resolution in measurement and air mass and /or seasonal effects.
π
2
3
2
3
H⋅ 1.391 − 3.560⋅ KT + 4.189⋅ KT − 2.137⋅ KT
Hd :=
if ωs ≤ 81.4⋅ 180
H⋅ 1.311 − 3.022⋅ KT + 3.427⋅ KT − 1.821⋅ KT
The correlation valid for KT = [0.3-0.8]
otherwise
Check for negative or zero value.
Hd :=
0 if Hd ≤ 0
[kWh/m2]
Hd = 0.696
Hd otherwise
Beam energy calculation Angle of incidence (AOI), the angle between the normal of the solar module and the beam radiation
→ → → cos θ := ( sin ( δ) ⋅ sin ( φ) ⋅ cos ( β ) ) − ( sin ( δ) ⋅ cos ( φ) ⋅ sin ( β ) ⋅ cos ( γ ) ) + ( cos ( δ) ⋅ cos ( φ) ⋅ cos ( β ) ⋅ cos ( ω) ) ... → → + ( cos ( δ) ⋅ sin ( φ) ⋅ sin ( β ) ⋅ cos ( γ ) ⋅ cos ( ω) ) + ( cos ( δ) ⋅ sin ( β ) ⋅ sin ( γ ) ⋅ sin ( ω) ) Check for zero or negative
cos θ j :=
cos θ j if cos θ j ≥ 0 0 otherwise
The regression coefficients are fitted to the AOI in degrees, see the b-coefficient in the definition block.
→ → 180 AOI := ⋅ acos ( cos θ) π
Ratio of beam radiation on a tilted surface to a horizontal surface. The database values are for a horizontal surface and for performance calculation is the plane of module radiation needed.
Rb :=
→ cos θ
( )
cos θ z
Rb is the geometric factor describing the relation between the tilted and horizontal radiation. Precaution for high values at sunrise and sunset. Limit Rb to max 10, see figure 4.
91
Rb := j
Rb if Rb ≤ 10 j j 10 otherwise
--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Figure 4. Schematic picture of the geometric factor Rb
Figure 5. Representation of the three different types of diffuse radiation
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------Ratio of hourly diffuse radiation to diffuse daily radiation on a horizontal surface. rd = Id /Hd describe how the diffuse radiation is distributed over the day as an average value for each hour. This will affect the solar module.
→ → → cos ( ω) − cos ωs π rd := ⋅ 24 sin ωs − ωs ⋅ cos ωs
( ) ( )
( )
--------------------------------------------------------
Sum :=
[1/hr]
Sum = 1.009
rd j
The ratio should be close to one
j --------------------------------------------------------
Diffuse radiation on a horizontal surface, hour average. The sky model is a mathematical derivation of diffuse radiation and the beam part can then be extracted.
→ Id := rd ⋅ Hd
(
)
[kWh/m2]
Ratio of hourly total to daily total radiation, days are assumed to center at solar noon. It is calculated from the knowledge of the total daily radiation and the length. Work best for clear days.
Coefficient of rt = I/H. Ratio of hourly total radiation to daily total radiation on a horizontal surface. The equation is empirically based.
π
k := 0.409 + 0.5016sin ⋅ ωs − 60 180
Horizontal hourly total radiation
I := rt⋅ H
π
⋅ l := 0.6609 − 0.4767sin ωs − 60⋅ 180
→ rt := rd ⋅ ( k + l⋅ cos ( ω) )
[1/hr]
Horizontal hourly beam radiation
Ib := j
0 if I ≤ Id j j
[kWh/m2]
I − Id j j
otherwise
Thus, have calculated hourly radiation on a horizontal surface, Id and Ib
--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
92
Calculation on tilted surface, i.e. in the plane of the module.
The diffuse radiation can be split into three components: isotropic part, distributed uniform over the sky dome, circumsolar diffuse, forward scattered and mainly in the same region as the beam radiation and horizon brightening, near the horizon, see figure 5. Applied model: Anisotropic sky
Extraterrestrial radiation on a horizontal surface over one hour. Error detected in old calculations, define I0 as a vector for each hour. It was previously kept constant during the day and the sinus functions were not used in the right way. ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------Old definition
I0 :=
12 π
⋅ Gsc ⋅ 1 + 0.033⋅ cos
I0 = 0.549
2⋅ π⋅ Day
(
365
) (
)
⋅ cos ( φ) ⋅ cos ( δ) ⋅ sin ω6 − ω5 + ω6 − ω5 ⋅ sin ( φ) ⋅ sin ( δ)
[kWh/m2]
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------New calculation
Generate the radiation vector, the radiation is treated as a constant for one hour. The I0 vector is defined such as the first value is the difference between the first integer hour and the sunset, next vector value the difference between the second and first hour and so on. The last vector value is the difference between sunset and the last integer hour, figure 6.
Ψ := I0 :=
2⋅ π 24
⋅ ( hour − 12)
Hour vector, change 15 degrees for each hour
for i ∈ 0.. 2⋅ intL − 1
I0 ← i
12 π
⋅ Gsc ⋅ 1 + 0.033⋅ cos
2⋅ π⋅ Day
365
( (
)
( ) ) + (Ψ i+ 1 − Ψ i)⋅ sin (φ)⋅ sin (δ)
⋅ cos ( φ) ⋅ cos ( δ) ⋅ sin Ψ i+ 1 − sin Ψ i
[kWhm-2]
I0
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Figure 6. Illustration of the sun's movement over the sky. The red crosses represent the hours and the circles the midpoint between two hours used to calculate the radiation.
93
Figure 7. Picture of the extraterrestrial hourly radiation. For clarity the sunrise coincide with E and sunset with W (usually not the case).
--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------Anisotropic index, a function of the transmittance through the atmosphere for beam irradiance. Ai is used to estimate the fraction of diffuse energy on a horizontal surface that is interpreted as forward scattered.
A i :=
→ Ib
Avoid division with zero. I0 cannot be zero. I0 is the radiation per hour and Ib the beam component defined at the midpoint of the hour (new definition), see figure 6.
I0 → Ib
f :=
I
Diffuse radiation on the tilted surface → → → → 3 → β 1 + cos ( β ) 1 − Ai ⋅ ⋅ 1 + f⋅ sin ITd := Id⋅ A i⋅ Rb + 2 2
(
)
(
)
direct diffuse
isotropic
horizon brightening
Reflected radiation on the tilted surface → 1 − cos ( β ) ITr := I⋅ ρ g ⋅ 2
The ground reflectivity is set as a constant and the reflected radiation makes just a small contribution.
Beam radiation on the tilted surface. Detected error in old model, the circumsolar diffuse part (Id*Ai*Rb) was calculated twice
→ ITb := Ib⋅ Rb
(
)
Total radiation on the tilted surface [kWh/m2] per hour
→ IT := ITb + ITd + ITr
(
)
94
Energy plot 0.4 The total energy on the tilted surface for the given time and location is then the sum of the hour values: IT (kWh/m2)
0.3
HT :=
0.2
HT = 2.026
IT j j 3
HTjoule := 3600⋅ 10 HT
0.1
0
[kWhm-2day-1]
6
8
6
HTjoule = 7.295 × 10
[kJday-1]
10
12 14 16 hour (h) Figure 8. Total radiation in the plane of the module.
Input : AMa, AOI, Tamb, ITb, ITdiff
Output : Isc, Im, Vm, Ix, Ixx, Voc
The wind speed and ambient temperature will be scalars representing monthly averaged values. Can be used to calculate the cell temperature.
Ta := 25
WS := 3
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------Represent the ambient temperature as a function of sinus [passive]. The method is to define the ambient temperature as a periodic function and not a constant over the day.
Average daily ambient temperature
Tavg := 20 Difference between daily Tmax and Tmin
Tdelta := 10
(
)
hmod := ω + ωs ⋅
π π 6
−
+ ωs
π
(
)
Tamb := sin hmod ⋅
2
Tdelta 2
+ Tavg
--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------Energy balance approach, Chapter 23.3 [1] [passive]
Absorption factor
α := 0.9
Transmittance of the front surface
τ := 0.9
Constant
K := 0.03125
Efficiency
Eff := 0.07
→
→ → Eff Tbal := Ta + IT⋅ 1000⋅ K ⋅ 1 − τ⋅ α
(
)
The higher efficiency the lower cell temperature.
---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
95
The electrical equations are derived to work for hourly values for AMa, AOI, Tamb and Irr. The equations are taken from [2]. The reflection of solar radiation is treated as a diffuse component.
→ Eb := ITb ⋅ 1000
(
)
Tback :=
Ediff :=
→ ITd + ITr ⋅ 1000
(
)
times 1000 to convert from kW to W
→
(IT⋅ 1000⋅ exp(a + b⋅ WS) ) + Ta
Tc := Tback +
(Eb + Ediff)
→ → Tc + 273.15 δT := n ⋅ kB⋅ q
⋅ ∆T
E0
(
)
Spectral response of the module, empirical based and the coefficient can be found in the definition block
→ → → → → → 2 →3 → 4 f1( AMa ) := a0 + a1⋅ AMa + a2⋅ AMa + a3⋅ AMa + a4⋅ AMa
(
)
(
)
(
)
(
)
Optical response of the module, empirical based and the coefficient can be found in the definition block. The higher AOI the higher optical loss.
→ → → → → → → 2 → 3 → 4 → 5 + b3⋅ AOI + b 4⋅ AOI + b5⋅ AOI f2( AOI) := b0 + b 1⋅ AOI + b 2⋅ AOI
( )
(
)
( )
( )
( )
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------The short circuit current is calculated for a module but can easy be changed to a complete array by adding the modules connected in parallel, Mp.
Number of modules in parallel
Number of modules in series
M p := 1
M s := 1 The adding of modules in series and in parallel add their individual performance and do not account for interconnecting resistance, wire losses or different quality between the modules. According to [2] the error < 5%. Exist program that account for wire losses.
Figure 9. Solar modules connected in series and in parallel to build up an array.
--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------Short-circuit current, one value for each hour (AMa, AOI, Eb, Ediff, Tc)
→ → → → → Eb ⋅ f2( AOI) + fd ⋅ Ediff Isc := M p Isc0 ⋅ f1( AMa ) ⋅ ⋅ 1 + α sc ⋅ Tc − T0 E0
(
)
(
) (
)
(
96
)
Isc := j
Isc if Isc > 0 j j
0 otherwise
Effective irradiance → Isc Ee := → → Isc0 ⋅ 1 + α sc ⋅ Tc − T0
(
)
The fourth IV point → → → → 2 Ix := M p Ix0⋅ c4⋅ Ee + c5⋅ Ee ⋅ 1 + α sc ⋅ Tc − T0
(
)
(
)
Maximum current → → → → → 2 ⋅ 1 + α m⋅ Tc − T0 Im := M p Im0⋅ c0⋅ Ee + c1⋅ Ee
(
)
(
)
The voltage can't be calculated at sunrise and sunset since Ee = 0 and ln(0) is undefined
Vm := j
0 if Ee ≤ 0 j → → → → 2 → → → → → M s ⋅ Vm0 + c2⋅ Ns ⋅ δT ⋅ ln Ee + c3⋅ Ns ⋅ δT ⋅ ln Ee + β m⋅ Tc − T0 j j j j j
Vm := j
otherwise
Vm if Vm ≥ 0 j j 0 otherwise
The fifth IV point → → → → → 2 Ixx:= M p Ixx0⋅ c6⋅ Ee + c7⋅ Ee ⋅ 1 + α m⋅ Tc − T0
(
)
(
)
Open circuit voltage
Voc := j
0 if Ee ≤ 0.0001 j → → → → M s ⋅ Voc0 + Ns ⋅ δT ⋅ ln Ee + β oc ⋅ Tc − T0 j j j
otherwise
Electrical performance parameters
------------------------------------------------------------------------------------------------------------------------------------------------------------------------
FF :=
→ Im⋅ Vm Isc ⋅ Voc
→ Pm := Im⋅ Vm
(
)
ε :=
→ Pm IT⋅ 1000⋅ Area
Utilizability constrains
The power is assumed to be constant over an hour period so the average hour energy is given by time integration of the power. The solar module operates around maximum power point. Not all of the incidence solar energy can be utilized e.g. for photons lower than the band gap can't give any contribution but that is included in the semi- empirical mathematical expression for the Isc etc. Max power at STC
Pm0 := Im0⋅ Vm0
Pm0 = 50.05
97
The objective with this block is to generate the whole IV curve from the five points calculated above. A simple approach is given here and a more comprehensive method is implemented in Matlab. Define Vx and Vxx i.e. the voltage at Ix and Ixx.
Voc Vx := 2
Vxx:=
Vm + Voc 2
Plot of the 5 IV points for each hour
Five IV points 0.4
Select one of the hours and define voltage and current vector to spline interpolation below.
0.3
0
Vx 6
I (A)
Ix 6
Vm 6
vx :=
0.1
0
0
20
40 60 V (V)
80
vy :=
Vxx 6 Voc 6
0.2
Isc 6
Im 6
Ixx 6
0
100
Figure 10. The IV curves for one day One point represents one hour before solar noon and one hour after solar noon i.e. the values are symmetrical around solar noon. The curve represents the operation of one solar module.
Cubic spline interpolation to generate the IV curve
vs := lspline( vx, vy )
lspline returns a vector vs. with second order derivatives that is used as input to interp.
i := 0.. Voc + 1
Generate voltage vector
6
f( i) := interp ( vs , vx, vy , i)
Interpolate I(V) at the generated voltage values.
IV Curve 20 Pmax (W)
I (A)
. 0.2
10
0
10 15 20 hour (h) Figure 12. The maximum power as function of the hour vector.
0
0
20
40
60
80
5
100
V (V) Figure 11. The cubic spline interpolated IV curve for one hour. The cubic spline interpolation contains no information about the characteristics of the solar cell. Can't extract the Iph, I0, Rs or Rsh
98
P( i) := i⋅ f( i)
Calculate the power at each voltage point.
P(W)
20
10
0
0
50 V(V)
100
Figure 13. Plot of the power as a function of voltage with the MPP.
For simulation of the parasitic diode equation see Matlab files
(
q⋅ V+ IRs
I( V)
Iph ( V) − I0⋅ e
)
n⋅ kB⋅ T cell
−1 −
V + IRs Rsh
[1] Duffie, J.A. and Beckman, W. A. (1991). Solar engineering of thermal processes. Chichester: John Wiley & Sons, Inc. [2] King, D.L, Boyson, W.E and Kratochvil, J.A. (2004). Photovoltaic Array Performance Model. Sandia National Laboratories. Available at: www.sandia.gov/pv.
Provide information for the definition block, imported from Microsoft Access.
Sandia :=
See also Mathcad Help for "linking to database" and ODBC Input component.
Provide information about site, latitude, longitude, elevation, monthly average daily radiation and ambient temperature.
BP := ..\..\..\..\Database\Solar data\bp_soldata1.xls
99
G
Consists of m-files for parameterisation of the diode equation and for iteration of the entire IV curve. function F = nonlinIVgen(x) %-------------------------------------------------------------------------% Function: lsqnonlin % Input: Initial values for the coefficient % Output: Regression coefficient (I0, Rs, Rsh) %-------------------------------------------------------------------------% The 5 calculated IV points. The voltage is scaled down to cell level % from module values. Vmodule = [0 43.03 62.04 74.05 86.06]; V = Vmodule./116; I = [0.969 0.881 0.826 0.51 0]; Isc = I(1);
% The first element in the current vector is the Isc
% Constants q = 1.60217733*10^-19; n = 1.3; % Can also be defined as a parameter k = 1.380658*10^-23; T = 307; % Imported from Mathcad F = I-(Isc-x(1).*(exp(q.*(V+x(2).*I)/(n*k*T))-1)-(V+x(2).*I)./x(3));
function f = connect(x) %-------------------------------------------------------------------------% Generation of the IV curve for a solar cell % Function: lsqnonlin % Input: x0, vector of start values for x (current). Proper initial values % are quite tedious since around 50-100 values will be defined (here 77). % Output: x that fulfils the requirements % Aim: plot x vs. V %-------------------------------------------------------------------------Constants Isc = 0.969; % For the prevailing Irr and T. q = 1.60217733*10^-19; n = 1.3; % Can also be defined as a parameter k = 1.380658*10^-23; T = 307; % Imported from Mathcad % Parameters determined by lsqnonlin function I0 = 10^-10; Rs = 0.01; Rsh = 500; V = 0:0.01:0.76; % Generation of the independent variable (voltage) % Define a non-linear least-square problem where the dependent variable x is to be found. % The approach is to find the value of x (current) such as f -> 0 for each values of V.
f = x-(Isc-I0.*(exp(q.*(V+Rs.*x)./(n*k*T))-1)-(V+Rs.*x)./Rsh);
H Appendix G provides support for the model validation and contains error data for the fivemodelled IV points. Table 1. Display the relative error in percent for the modelled points Ix and Ixx. The relative error can be calculated from residual/Ixmeasured*100%. -2 Ixc (%) IxSNL (%) Ixxc (%) IxxSNL (%) Irradiance (Wm ) 300 3.94 1.78 16.61 7.40 450 0.46 2.35 5.18 6.24 530 0.97 3.77 12.03 6.86 670 3.15 4.27 22.43 6.49 740 4.81 4.37 29.10 6.20 1000 6.53 5.79 40.30 5.35 3.31 3.72 20.94 -6.42 Average Table 2. Show the residual with unit and percentage for different irradiance levels. Voc (V) Voc (%) Vmax (V) Vmax (%) Irradiance -2 (Wm ) 300 0.02 0.02 -2.68 -4.34 450 0.20 0.23 -3.24 -5.07 530 0.15 0.17 -3.06 -4.78 670 0.18 0.21 -3.04 -4.78 740 0.17 0.19 -2.70 -4.25 1000 0.17 0.19 -2.66 -4.22 0.17 -4.57 Average
Imax (A)
Imax (%)
0.06 0.00 0.01 0.02 0.03 0.04
6.87 1.39 3.10 3.82 4.42 6.07 4.28
Table 3. Temperature dependent measurement displays the residuals for the new fitted coefficients and the database values. Ixc (%) Ixxc (%) IxSNL(%) IxxSNL(%) T(°C) 31.80 -2.99 -11.26 -2.17 -7.80 34.50 -3.04 -9.56 -2.45 -7.10 39.30 -2.52 -9.16 -1.99 -6.95 43.90 -2.84 -7.18 -2.46 -5.64 48.30 -2.94 -6.82 -2.51 -5.09 52.50 -2.91 -9.24 -2.44 -7.27 56.50 -1.87 -6.52 -1.64 -5.58 60.30 -3.25 -8.91 -2.87 -7.33 64.10 -3.06 -8.44 -2.73 -7.04 67.60 -2.79 -8.31 -2.30 -6.28 70.80 -3.33 -10.91 -2.68 -8.18 73.90 -3.21 -10.49 -2.70 -8.32 77.10 -2.31 -10.06 -2.03 -8.86 80.60 -2.52 -11.45 -2.18 -10.00 83.20 -2.73 -10.63 -2.51 -9.72 85.90 -2.95 -11.42 -2.67 -10.22 89.00 -3.36 -11.58 -3.02 -10.13 91.40 -3.76 -13.70 -3.35 -11.93 -2.91 -9.76 -2.48 -7.97 Average
101
Table 4. The residuals for the open circuit voltage, maximum current and voltage. The errors are shown as the difference between the measured and predicted values and as a relative error in percentage. The numbers in bold represent the absolute average values.. T (°C) Voc (V) Voc (%) Im (A) Im (%) Vm (V) Vm (%) 31.80 1.081 1.204 0.009 1.100 -2.002 -3.224 34.50 1.046 1.177 0.041 5.000 -1.721 -2.757 39.30 1.095 1.246 0.035 4.292 -1.523 -2.449 43.90 0.936 1.081 0.028 3.471 -2.102 -3.434 48.30 0.828 0.968 0.029 3.604 -2.495 -4.133 52.50 0.606 0.717 0.010 1.242 -2.060 -3.412 56.50 0.454 0.544 0.019 2.280 -2.747 -4.627 60.30 0.188 0.229 0.001 0.107 -2.902 -4.936 64.10 -0.031 -0.038 0.002 0.209 -3.438 -5.937 67.60 -0.364 -0.454 0.000 0.016 -4.026 -7.072 70.80 -0.613 -0.773 -0.006 -0.774 -4.422 -7.871 73.90 -0.884 -1.129 -0.017 -2.025 -4.750 -8.545 77.10 -0.992 -1.281 -0.017 -2.143 -5.191 -9.454 80.60 -1.325 -1.736 -0.026 -3.143 -5.946 -11.055 83.20 -1.624 -2.155 -0.024 -3.005 -6.589 -12.449 85.90 -1.823 -2.443 -0.037 -4.573 -7.026 -13.456 89.00 -2.048 -2.778 -0.034 -4.164 -8.004 -15.721 91.40 -2.302 -3.156 -0.041 -5.009 -8.294 -16.467 1.01 1.28 0.02 2.56 4.18 7.61 Table 5. The residuals for the outdoor experiment in Denmark and for a high irradiance level. Residual (A) Residual (%) Isc -0.0790 -8.15585 Ix -0.0334 -3.79183 Im -0.1457 -17.634 Ixx -0.1508 -29.5897 Residual (V) -3.74 -4.18
Vm Voc
Residual (%) -6.42 -5.1
Table 6. The five measured and calculated IV points with the corresponding residuals for a low irradiance level. Measured (A) Calculated (A) Residual (A) Residual (%) Isc 0.1820 0.1832 -0.0012 -0.65 Ix 0.1699 0.1727 -0.0028 -1.62 Im 0.1299 0.1624 -0.0325 -20.01 Ixx 0.0774 0.1122 -0.0348 -31.03
Vm Voc
Measured (V) 60.4 79.58
Calculated (V) 64.9 82.57
102
Residual (V) -4.5 3.0
Residual (%) -7.5 -3.8
I A programme to track the maximum power point (MPP). Figure 1 shows how the power varies with the terminal voltage, there exist only one critical point, a maximum value that corresponds to the MPP. The MPP- code is implemented in MathsoftTM Mathcad ® 11 and the purpose is to compare the calculated maximum voltage with the measured value.
P(W)
Power
V(V)
Figure 1. The power output from a solar cell or module has one single maximum point. Spline comparison
1
cubic spline interp
I(A)
0.8
0.6 Mpp region 0.4
0.2
0 0
10
20
30
40
50
60
70
80
V(V)
Figure 2. The calculated MPP lies to the right of the measured region where the real MPP is situated. The real value is calculated from element wise multiplication of the I and V vector.
103
P :=
i←0 P ← I ⋅V 0
0
0
for i ∈ 1.. length ( I) P ← I ⋅V i
i i
return i − 1 if P < P
i− 1
i
The Mathcad code can be used to find the measured MPP, which corresponds to the largest area under the graph. The limitation is that the MPP only can be one of the measured points and not in the interval between two points. An alternative approach with smaller iteration steps gives a more realistic result and is explained below. Commands and operations in Mathcad vs := lspline( V, I) x := 0 , 0.2.. 82 f( x) := interp ( vs , V, I , x) p( x) := x⋅ f( x)
The voltage variable x can have a smaller step to increase the resolution and the for loop is used to find the MPP of the spline interpolated curve. The code make it possible to find the MPP with higher accuracy and is viewed in figure 2. Vmax:= for i ∈ 1.. 820
pm ← i
return
i 10 i 10
⋅f
i
10
if pm < pm i
i−1
The iteration range from 0.1 to Voc with an increment of 0.1 V and x and f(x) is defined above. The simple but robust MPP tracer based on comparing elements in the generated power vector provides a method to calculate the Vm residual.
104
