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Relations among photovoltaic cell electrical parameters
Accepted Manuscript Full Length Article Relations among photovoltaic cell electrical parameters Gabriel Cibira PII: DOI: Reference:
S0169-4332(18)31516-2 https://doi.org/10.1016/j.apsusc.2018.05.194 APSUSC 39470
To appear in:
Applied Surface Science
Received Date: Revised Date: Accepted Date:
27 February 2018 11 May 2018 25 May 2018
Please cite this article as: G. Cibira, Relations among photovoltaic cell electrical parameters, Applied Surface Science (2018), doi: https://doi.org/10.1016/j.apsusc.2018.05.194
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
RELATIONS AMONG PHOTOVOLTAIC CELL ELECTRICAL PARAMETERS Gabriel CIBIRA 1
Institute of Aurel Stodola, Faculty of Electrical Engineering, University of Zilina, Univerzitna 1, 01026 Zilina, SK [email protected]
Abstract. Operating ranges of photovoltaic cell electrical parameters strictly depend on internal doped material structures and surfaces properties. Therefore, structure and surface are crucial issue of the research at nanostructure and atomic-charge level. The relations among electrical parameters based on well-known single-diode or double-diode models lead to exact mathematical models. Applying numerical methods, modelled result electrical parameters relations may be obtained, in stated conditions range. Often, simplifications and reduced forms of transcendental differential equations are used. But, not the only reduced differential equations system is applicable for any photovoltaic cell. In this paper, {n, Rs, Rsh, Is, Iph} five-parameter static modelling approach is used for a small-sized general silicon photovoltaic cell. Via computer step-by-step iterative numerical modelling, the five parameters relations are computed, visualized and analyzed.
Keywords Photovoltaic cell, Electrical parameters relations, Reduced forms.
1.
Introduction
A thin film silicon (Si) photovoltaic (PV) cell converts light energy of photons into electricity by the photovoltaic effect. Even if optical and electrical parameters ranges of a photovoltaic cell are well-described by the photovoltaic effect and other known phenomena, they still are not fully convertible as they are non-linearly dependent on several external influences simultaneously. A PV cell represents large-area PN junction of the diode, made of Si semiconductor materials having different chemical electric charges. Photovoltaic handbooks and studies have classified and reviewed PV cells models. Based on the PV cells material properties, general non-linear equivalent electric diagrams have been created: the single-diode and the double-diode one. They are applied with respect to significance of proven currents like generated, saturation, leakage, and other phenomena. The simpler, single-diode equivalent electrical model, Fig. 1, is appropriate enough to get a cleaner general picture of any crystalline Si-based photovoltaic cell electrical parameters, at standard or slightly under standard solar irradiation intensity conditions. When considering significantly lower irradiation intensity, independent recombination losses must be taken into account, thus the double-diode equivalent electrical model should be applied to avoid high currents deviations [1, 2, 8, 15].
Iph
Ish Rsh
Is
Iout
Rs
D1
Vout
Fig. 1: Equivalent single-diode electric diagram of a PV cell.
Non-linear semiconductor junctions’ behavior is strongly dependent on materials homogeneity, surface finish treatment and thus its absorptance and reflectance, material temperature, as well as light characteristics like irradiance
intensity, spectral content and angular distribution. All they directly influence on behavior of main electrical parameters of interest, such as the ideality factor of a diode, n, the series resistance, Rs, the parallel shunt resistance, Rsh, the reverse diode saturation current(s), Is, the photocurrent, Iph. Electrical parameters and relations among them have been investigated and, lot of rather adjusted mathematical models have been designed for the equivalent electric diagrams. Investigation of parameters often has problems with more or less known, even unknown input parameters. Thus, such models are estimative at specified irradiation intensity, temperature and material surface or structure. Applying Ohm’s and Kirchhoffs’ laws, Fig. 1 depicts general relationships between up to five electrical parameters, {n, Rs, Rsh, Is, Iph}. Available solutions’ dimension depend on required electrical parameters number. Single-parameter mathematical models suppose most of parameters to be constant, except of current I and voltage V, thus the only {R s} parameter is investigated to obtain final I-V curves nonlinear behavior. Such case is solvable by ordinary differential equation. Next, supposing parameters’ dynamics, establishing partial differential equations system is needed, trying to find multidimensional system solution over multiple input and multiple output parameters. Two-parameter models investigate {n, Rs} or {Rs, Rsh} parameters, depending on constants, functions or partial derivatives of rest parameters, so the dependence is more complex in this case. Similarly, three-parameter models investigate {n, Rs, Rsh} or {n, Rs, Is} parameters behavior. Next, four-parameter models survey {n, Rs, Rsh, Is} parameters applying more complex numerical iterative methods. The five-parameter models investigate all of the above mentioned parameters, {n, Rs, Rsh, Is, Iph} [1, 2, 3, 4, 8]. Here, partial differential equation system is complex system depending on multivariable functions and their partial derivatives. Hereby, some researchers simplify numerical models supposing that some unknown parameters are temperature independent and thus static, namely {n, Rs, Rsh} in five-parameter models. Supposing this limitation, they apply iterative numerical methods to obtain partial differential equations for the rest of parameters [1, 2, 3, 4]. Generally, the complexity of a circuit model often depends on the desired accuracy. Indeed, many researchers use own system of transition equations at reference conditions. When seeking appropriate approach, they involve there real measurable alternative PV cell input parameters, either in border limit, in expected operational range, or rather at certain operating point. In this manner, open circuit voltage VOC, short circuit current ISC, open circuit resistance ROC, short circuit resistance RSC, maximum power point circuit voltage VMPP, and maximum power point circuit current IMPP are often incorporated [1, 5, 16, 17]. When depicting relations among all of input and output variables, 2D and 3D visualization tools on hands are limited. To view reliable relations dynamics, appropriate range of parameters must be set that might difficulty, sometimes. For reasons given in previous text, the only one accurate model, numerical method, alternative parameters involving approach and testing method might be illusive. In proposed modeling and simulations in this paper, some simplifications are used. Firstly, alternative parameters involving approach is applied for single-diode five-parameter model of a Si PV cell. Secondly, partial differential equations system is step-by-step solved for wide range of input variables, in proposed approach. Profitable, one of the most important advantage is that, numerical methods are applicable over large scale of operating conditions. This allows avoiding small partial gradients vanishing or truncating, since all of the parameters are step-by-step chained. Thirdly, nested loops are repeatedly computed until nearby real results ranges or convergences are obtained. Definitely, such advanced modeling allows simulating any Si PV cell electrical parameters behavior, at stated physics limits. Relations among the above-mentioned five electrical parameters, {n, Rs, Rsh, Is, Iph}, are visualized and discussed in their operating ranges of interest.
2.
Single-diode mathematical model
The energy needed for electrons’ traveling across the material structure and thus current generation is called the band gap energy Eg, Eq. (1). It is temperature dependent parameter. P and N Si materials’ constants α, β have been experimentally found for different materials, thus the Si band gap energy can be approximated as follows [1, 6, 7]: మ
ሺሻ ൌ ሺͲሻ െ ቀ ାஒ ቁ ؆ ͳǤͳ െ
Ǥସଷమ ାଷ
ሾሿ .
(1)
The illuminated Si semiconductor diode produces a photo-generated current, Iph, which will result into I–V curve of a quantity that is related to the surface density of the incident energy. Thus, an ideal PV cell with its I–V curve is depicted as a direct current generator parallel to a diode. Ideal PV cell single-diode model output current I has been defined as follows [1, 2, 3, 4]: ୯
୭୳୲ ൌ ୮୦ െ ୱ ቂ ቀ ୬୩ీ ቁ െ ͳቃ െ ୖ ీ ሾሿ . ౩
(2)
where Iph is generated photocurrent in A, Is is reverse dark saturation current of the non-linear diode in A, q is electric charge of an electron: q = ‒1,6022·10‒19 in C, VD is voltage across the diode in V given as follows: VD = VPV +
RsIPV, VPV is cell output voltage in V, IPV is PV cell output current in A, Rs is series resistance in Ω, n is dimensionless diode ideality factor (generally acceptable at 0.5 … 2.5), k is Boltzmann constant: k = 1,380658·10-23 in J·K-1, T is temperature of a PV cell in K, Rsh is shunt resistance in Ω. If the diode is not illuminated, Iph = 0 and VD/Rsh = 0 thus the Eq. (2) can be reduced. The FF, fill factor of a PV cell, represents power rate among electrical maximum power point PMPP achieved at the tip of particular power-voltage curve i.e. product of voltage VMPP and current IMPP, referred to idealistic maximum electrical power i.e. product of VOC and ISC, Eq. (3). The fill factor is strongly influenced by recombination currents and intrinsic ohmic resistances. The PMPP of a PV cell, FF, the incident light power Pin in W·m-2, the cell surface S in m2, and solar-to-electrical power conversion efficiency η are related by Eq. (4) [1, 13, 15, 16]. ൌ
ౌౌ ୍ౌౌ
Ʉ ൌ
ోి ୍ి
ోి ୍ి ୗ
ሾെሿ ,
ൌ
(3)
ౌౌ ୍ౌౌ ୗ
ሾΨሿ .
(4)
The {n, Rs, Rsh, Is, Iph} parameters of interest for a Si PV cell can be obtained in general forms with respect to standard test conditions STC data as follows [1, 2, 3, 4, 5, 8, 9, 10, 11, 12]: ൌ
୯ీ
୩ሺ୪୬ ୍ି୪୬ ୍౩ሻ
ୱ ൌ ୢ୍ ቚ ୢ
୮୦ ൌ ୗ
ୗ
ୀోి
ሾെሿ ,
(5)
ሾȳሿ ,
(6)
ൣ ୱ̴ୗେ Ƚ ሺ െ ୗେ ሻ൧ሾሿ ,
ి
ୱ ൌ ୱ̴ୗେ ቀ ୱ୦ ൌ ୱ୦ ቀ
ి
ୗి ୗ
ు ଷ ൬ ౝ̴ి ି ుౝ ൰
ቁ
ౡి
ౡ
ቁ ୱ୦̴ୗେ ሾȳሿ .
(7)
ሾሿ ,
(8) (9)
3. Numerical modelling and results Typically, a commercial amorphous, polycrystalline or monocrystalline structural Si cells of a small size of S = 1 cm2 achieve PMPP ≈ 0.010 … 0.018 W·cm-2 at AM1.5G irradiation conditions, the irradiation intensity about 1000 W·m-2, and T ≈ 298.15 K. They achieve FF ≈ 0.8 and η ≈ 10 … 21 %. For such reference PV cell size and irradiation conditions, an amorphous silicon cell typically attack maximum VOC ≈ 0.5 … 3 V and ISC ≈ 0.016 A. A polycrystalline silicon cell typically attacks VOC ≈ 0.5 … 2.5 V, ISC ≈ 0.030 A. Finally, the most efficient monocrystalline silicon cell typically attacks VOC ≈ 0.5 … 3 V, ISC ≈ 35 mA. Mostly, VMPP ≈ (0.8 … 0.9)·VOC in V and IMPP ≈ (0.85 … 0.95)·ISC in A for any of the material structures [16, 17]. For the small-size Si PV cell, {n, Rs, Rsh, Is, Iph} parameters relations are modelled along expected operating ranges and explained in this section. Following the previous explanations and Eq. (2, 5, 6, 7, 8, 9) resulting into Eq. (10, 11, 12, 13, 14), experimental results are obtained using Matlab software static programming. The temperature dependent parameters are chained by step-by-step modelling and simulation. Following previous explanations, involved input alternative parameters are set and posterior parameters range are estimated, both at particular static operational range or point. For all steps, the temperature ranges in T = 273.15 ... 353.15 K interval. ୱ ൌ ൌ
୍ౌౌ
୯ሺିోి ሻ
൰൨
୩ቐ୪୬൬୍ి ି ౌౌ ି୍ౌౌ ൰ି୪୬൬୍ి ି ోి ൰ା ౩
ୱ୦ ൌ ୱ ൌ
ోి షమౌౌ షఽ ቁ ఽ
ͳ ൬െቀ
ా
ሺୖ౩ ୍ౌౌ ିౌౌ ሻሺିోి ିሻୣቀఽቁାୣ ా
ాషోి ౩
ా ൬ ోి ൰ ఽ ିୣቀ ఽቁ
,
୍ౌౌ
,
(10) ,
(11)
ౌౌ ቑ ి ష ోి ౩
൬ ోి ൰ ఽ ൩
൬ ోి ൰ ఽ ൩
ቀ ቁ ୍ౌౌ ሺౌౌ ିୖ౩ ୍ౌౌ ାሻୣ ఽ ିୣ
୍ౌౌ ା ୣ
౩
ౌౌ
,
(12)
(13)
ቀ ోి ቁ
୭୳୲ ൌ ୮୦ െ ୱ
where ൌ
ఽ
୬୩ ୯
,
െ ͳ൨ െ
ోి ୖ౩
,
(14)
ൌ ୱ .
Intrinsic Rs and Rsh resistances influence on a PV cell behavior have been widely discussed in previous works [1, 2, 3]. The series resistance mainly depends on the layer composition. Surface layer thickness and specific material parameters play dominant role. Due to very small IMPP current caused by small cell size and thus huge VMPP/IMPP ratio, huge series resistance Rs can be assumed from Eq. (10). As a temperature dependent variable, the Rs is obtained using Lambert W function. Involved parameters are derived from estimated VOC, ISC, VMPP, IMPP range. Initially, n = 0.5 … 2.5 and T = 273.15 ... 353.15 K are set. Obtained Rs range is depicted in the Fig. 2, as a function of T and n. It starts at the minimum value Rs ~ 38. 5 Ω at T = 273.15 K and n = 1 point. With respect to input parameters, it gently slopes almost linearly up to Rs ~ 42 Ω. Even if the Rs dynamic is very gentle, its dependence on the particular VOC, ISC, VMPP, IMPP measured parameters accuracy is strong. Considering much larger cell area, the Rs would reduce significantly, mainly due to dramatic IMPP increasing and thus rapid decrease of quasi-static VMPP/IMPP ratio. For example, when considering a cell size of S = 20 cm2 with the same homogeneous material structure and operating conditions, it reduces to about Rs ~ 1.92 … 2.1 Ω and, for a size of S = 1 m2 it decreases up to ordinary Rs ~ 0.00385 … 0.0042 Ω. Here, the normalized series resistance rs = Rs.IMPP/VMPP can acting as reliability indicator: for the small-sized cell, rs ~ 1.02 … 1.11 Ω, Fig. 2 below. Finally, let we analyze the Rs influence on FF and η. If the irradiation and temperature conditions does not change, the VMPP/VOC and IMPP/IOC ratios remain constant so the FF and the η does not change too, see Eq. (3, 4).
Fig. 2: Series resistance dependence Rs = f(T, n) (upper) and normalized series dependence rs=f(T, n) (lower) on temperature and ideality factor.
Similarly, n dependence on T and series resistance is depicted in Fig. 3. Here, due to small-sized cell, the normalized r s operating range is used instead of biased Rs. From conventional ratio range Rsh/Rs ≈ 500 … 2000 applicable for most types of Si ideal cells or panels, real posterior value Rsh ≈ 400 Ω was set initially for the Eq. (11) with simulation resulted to Fig. 3 upper. Here, the ideality factor gently increases as opposed to T and r s. Among the
two, more significant is influence of the temperature gradient. In the operational T and r s ranges, posterior n ≈ 1 … 1.35 is obtained that is close to n=1 of the ideal diode. This reinforces the opinion that, single diode model is appropriate for small-sized cells where the Auger or depletion region recombination are less significant. Moreover, the Shockley-ReedHall space charge region recombination must be at low junction level. On the other hand, for R sh ≈ 1000 Ω, the simulated range of local ideality factor tilting rises about twice and n ≈ 1.9 … 2.5, Fig. 3 lower, that is far from the ideal diode. It corresponds with the above mentioned recombination and tunneling significant enhancements, mainly in the deep material space charge regions at junction interface. This analysis allows us to reduce involved n for the rest of numerical calculation steps.
Fig. 3: Ideality factor dependence n = f(T, rs) on temperature and normalized series resistance, at Rsh ≈ 400 [Ω] (upper), Rsh ≈ 1000 [Ω] (lower).
In the third step, Rsh is numerically calculated using Eq. (12). Generally, small Rsh value normally indicates a technical fault in the PV cell. But, higher temperatures cause higher charge mobility thus Rsh may not exceed some practical limit resistance, e.g., Rsh ≈ 1000 Ω. Rsh dependence on T and Rs is depicted in Fig. 4 for the operating ranges of T ≈ 273.15 ... 298.15 K and Rs ≈ 38.3 ... 39.55 Ω, the Rsh ≈ 12 ... 1000 Ω. The Rsh increases very steeply, depending on Rs. Besides, the cell small size plays crucial role of the Rsh values, in this modelling case. The numerical calculations goes above 1000 Ω up to singularity border of numerical computing and, controversially, Rsh decreases rapidly to nonrealistic negative values. In Fig. 4, they are eliminated prevailingly.
Fig. 4: Shunt resistance dependence on temperature and series resistance, Rsh = f(T, Rs).
Next, the absolute value of reverse saturation current behavior is numerically modelled using Eq. (13), Fig. 5. The constant value of radix Is ≈ 10-18 A rapidly non-linearly raises as the temperature does, beginning from the temperature T ≈ 330 K. Additionally, it raises steeply if the Rsh approaches its lower limit Rsh ≈ 50 ... 12 Ω at the T ≥ 340 K.
Fig. 5: Saturation current dependence on temperature and shunt resistance, Is = f(T, Rsh).
In following step, Iph and Iout are numerically calculated using Eq. (2, 7, 14). Considering small-sized cell, Iout achieves fewer than 0.018 A only. In Fig. 6, the Iout strong dependence on T and Is is evident, mainly at higher absolute saturation values and lower temperatures. Its absolute value grows up as the temperature goes down, even more when reverse saturation current value goes down. This is mainly caused by Rsh lowering. Finally, the total output current Iout is depicted in the Fig. 7 as a function of T and V. In keeping with small cell size, this variable parameter achieves I out ≈ 0 ... 0.020 A, with the maximum current values at about ISC values and, with the minimum at VOC points.
Fig. 6: Output current dependence on temperature and saturation current, Iout = f(T, Is).
Fig. 7: Output current dependence on temperature and voltage, Iout = f(T, V).
In this paper, the dynamic change of the irradiation intensity or ISC dependence on the temperature are out of the interest. Therefore, slight controversial Iout dependence on T is not included in this paper, as it was analyzed in previous authors’ work.
4.
Conclusion
In this paper, the single-diode five-parameter model is used to demonstrate applicability of the reduced form approach for electrical parameters {n, Rs, Rsh, Is, Iph} relations study. Involved alternative parameters values {VOC, VMPP, ISC, IMPP} are estimated and used for numerical modelling and simulation, for a small silicon photovoltaic cell of 1 cm2 size. Next, posterior parameters ranges are estimated. In general, due to step-by-step chained numerical modelling as well as visualization needs, both involved and posterior parameters are set at particular static range or point. For all modelling and simulation steps, the temperature ranges in T = 273.15 ... 353.15 K interval. As a result, relations between the five parameters are obtained and visualized for the silicon photovoltaic cell. Besides of temperature dependency of all the five parameters, the most important parameter appears to be the series resistance. It represents intrinsic surface parameters setting and, as proven by visualization results, it influences next simulations steps. Next, shunt resistance indicates charge mobility capability and reveals photovoltaic cell technical state along its life cycle.
The limits of the modeling originate from chemical properties of the doped material structure and physical micro- and nano- surface properties. Here, the material fabrication processes influence the operational ranges, mainly. Therefore, simulated results may differ for real monocrystalline, polycrystalline and amorphous Si cells, even more when surfaces or structures change dramatically. Next simulation limits grow from supposed simple fabrication single PN junction of a Si PV and, from supposed standard or near-standard lighting and temperature conditions. The singlediode-based proposed modelling is appropriate for declared cases, at stated conditions ranges. But, the more limits differences, the more results will differ between simulated and measured parameters. For low light conditions or specified temperature ranges, the double-diode model should be applied. Different types of PV cells by Si-based materials structures or surfaces will be analyzed in future case studies. Proposed electrical parameters analysis using relations visualization is well-promising method for dynamic modelling of a particular silicon PV cell.
Acknowledgements This work was supported by the Slovak research and development agency [APVV-15-0152, APVV-14-0239, APVV0888-11]; the Slovak research grant agency [VEGA 2/0076/15]; EU structural funds, Cohesion fund and European regional development fund [ITMS: 26210120021]; the Tatra banka fund [2016et017]; and the Horizon 2020 MSCARISE-2016 [6260922 SENSIBLE].
References [1] LUQUE, A., HEGEDUS, S.: Handbook of photovoltaic science and engineering. John Willey & Sons Ltd, Chichester, England, 1179 p., ISBN: 0-471-49196-9, 2003. [2] DUFFIE, J. A., BECKMAN, W.A.: Solar engineering of thermal processes. John Willey & Sons Ltd, New Yersey, USA, 928 p., ISBN: 9781-118-41812-49196-2. 2013. [3] REKIOUA, D., MATAGNE, E.: Optimization of photovoltaic power systems: Modelisation, Simulation and Control. Springer Verlag, 283 p., ISBN: 978-1-4471-2348-4. Dec. 2012. [4] HUMADA, A. M., HOJABRI, M., MEKHILEF, S., HAMADA, H. M.: Solar cell parameters extraction based on single and double-diode models: A review. In: Renewable and Sustainable Energy Reviews 56 (2016), pp. 494–509, Dec. 2015. [5] KING, D. I., BOYSON, W. E., KRATOCHVIL, J. A.: Photovoltaic array performance model. Sandia report SAND2004 -3535, Sandia national laboratories, Abuquerque, New Mexico, USA, 43 p., Dec. 2004 [6]
SHOCKLEY, W.: Electrons and holes in semiconductors. 7-th printing, D. Van Nostrad Company, Inc., New York, USA, 558 p., Sept. 1959.
[7] VAN ZEGHBROECK, B.: Principles www.colorado.edu/~bart/book/book/chapter2/ch2_3.htm>.
of
Semiconductor
Devices.
2011,
online:
[8] WOLF, M., RAUSCHENBACH, H.: Series resistance effects on solar cell measurements. In: Advanced Energy Conversion, Pergamon pr ess, GB, pp. 455–479, 1963. [9] LIM, L. H. I., Ye, Z., Ye, J., Yang, D., Du, H.: A linear method to extract diode model parameters of solar panels from a single I-V curve. In: Renewable energy 76 (2015), Elsevier, pp. 135-142, Nov. 2014. [10] LAUDANI, A., DAVID, F. M., FULGINEI, F. R., SALVINI, A.: Reduced-form of the photovoltaic five-parameter model for efficient computation of parameters. In: Solar energy 97 (2013), Elsevier, pp. 122-127, Sept. 2013. [11] LAUDANI, A., FULGINEI, F. R., SALVINI, A.: High performing extraction procedure for the one-diode model of a photovoltaic panel from experimental I–V curves by using reduced forms. In: Solar energy 103 (2014), pp. 316-326, March 2014. [12] LAUDANI, A., FULGINEI, F. R., SALVINI, A.: Identification of the one-diode model for photovoltaic modules from datasheet values. In: Solar energy 108 (2014), Elsevier, pp. 432-446, Aug. 2014. [13]
CIBIRA, G.: Photovoltaic cells electrical parameters. In: ALER 2016, Bobrovec, SES, ISBN 978-80-89456-26-0, CD, pp. 35-42, Nov. 2016.
[14] BAYHAN, H., KAVASOGLU, A. S.: Exact analytical solution of the diode ideality factor of a pn junction device using Lambert W-function model. In: Turk journal Physics, Vol. 31 (2007), pp. 7-10, 2007. [15] KUNZ, G., WAGNER, A.: Internal series resistance determinated of only one IV-curve under illumination. Conference paper No. 5BV 2.70, 19th European photovoltaic solar energy conference, Paris, France, pp. 14, 2004. [16] QARONY, W., HOSSAIN, M. I., HOSSAIN, M. K., UDDIN, M. J., HAQUE, A., SAAD, A. R., TSANG, Y. H.: Efficient amorphous silicon solar cells: characterization, optimization, and optical loss analysis. In: Results in Physics 7 (2017), Elsevier, pp. 4287-4293, Nov. 2017. [17] CHANDER, S., PUROHIT, A., SHARMA, A., ARVIND, NEHRA, S. P., DHAKA, M. S.: A study on photovoltaic parameters of monocrystalline silicon solar cell with cell temperature. In: Energy Reports 1 (2015), Elsevier, pp. 104-109, April 2015
Highlights: 1. Si PV cell simplified single-diode equivalent electrical circuit is applied 2. alternative measured parameters {VOC, ISC, VMPP, IMPP} are involved 3. chained parameters {n, Rs, Rsh, Is, Iph} are numerically computed via reduced PDEs 4. {n, Rs, Rsh, Is, Iph} ranges are iteratively simulated 5. relations among {n, Rs, Rsh, Is, Iph} are computed, visualized and analysed
S0169-4332(18)31516-2 https://doi.org/10.1016/j.apsusc.2018.05.194 APSUSC 39470
To appear in:
Applied Surface Science
Received Date: Revised Date: Accepted Date:
27 February 2018 11 May 2018 25 May 2018
Please cite this article as: G. Cibira, Relations among photovoltaic cell electrical parameters, Applied Surface Science (2018), doi: https://doi.org/10.1016/j.apsusc.2018.05.194
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
RELATIONS AMONG PHOTOVOLTAIC CELL ELECTRICAL PARAMETERS Gabriel CIBIRA 1
Institute of Aurel Stodola, Faculty of Electrical Engineering, University of Zilina, Univerzitna 1, 01026 Zilina, SK [email protected]
Abstract. Operating ranges of photovoltaic cell electrical parameters strictly depend on internal doped material structures and surfaces properties. Therefore, structure and surface are crucial issue of the research at nanostructure and atomic-charge level. The relations among electrical parameters based on well-known single-diode or double-diode models lead to exact mathematical models. Applying numerical methods, modelled result electrical parameters relations may be obtained, in stated conditions range. Often, simplifications and reduced forms of transcendental differential equations are used. But, not the only reduced differential equations system is applicable for any photovoltaic cell. In this paper, {n, Rs, Rsh, Is, Iph} five-parameter static modelling approach is used for a small-sized general silicon photovoltaic cell. Via computer step-by-step iterative numerical modelling, the five parameters relations are computed, visualized and analyzed.
Keywords Photovoltaic cell, Electrical parameters relations, Reduced forms.
1.
Introduction
A thin film silicon (Si) photovoltaic (PV) cell converts light energy of photons into electricity by the photovoltaic effect. Even if optical and electrical parameters ranges of a photovoltaic cell are well-described by the photovoltaic effect and other known phenomena, they still are not fully convertible as they are non-linearly dependent on several external influences simultaneously. A PV cell represents large-area PN junction of the diode, made of Si semiconductor materials having different chemical electric charges. Photovoltaic handbooks and studies have classified and reviewed PV cells models. Based on the PV cells material properties, general non-linear equivalent electric diagrams have been created: the single-diode and the double-diode one. They are applied with respect to significance of proven currents like generated, saturation, leakage, and other phenomena. The simpler, single-diode equivalent electrical model, Fig. 1, is appropriate enough to get a cleaner general picture of any crystalline Si-based photovoltaic cell electrical parameters, at standard or slightly under standard solar irradiation intensity conditions. When considering significantly lower irradiation intensity, independent recombination losses must be taken into account, thus the double-diode equivalent electrical model should be applied to avoid high currents deviations [1, 2, 8, 15].
Iph
Ish Rsh
Is
Iout
Rs
D1
Vout
Fig. 1: Equivalent single-diode electric diagram of a PV cell.
Non-linear semiconductor junctions’ behavior is strongly dependent on materials homogeneity, surface finish treatment and thus its absorptance and reflectance, material temperature, as well as light characteristics like irradiance
intensity, spectral content and angular distribution. All they directly influence on behavior of main electrical parameters of interest, such as the ideality factor of a diode, n, the series resistance, Rs, the parallel shunt resistance, Rsh, the reverse diode saturation current(s), Is, the photocurrent, Iph. Electrical parameters and relations among them have been investigated and, lot of rather adjusted mathematical models have been designed for the equivalent electric diagrams. Investigation of parameters often has problems with more or less known, even unknown input parameters. Thus, such models are estimative at specified irradiation intensity, temperature and material surface or structure. Applying Ohm’s and Kirchhoffs’ laws, Fig. 1 depicts general relationships between up to five electrical parameters, {n, Rs, Rsh, Is, Iph}. Available solutions’ dimension depend on required electrical parameters number. Single-parameter mathematical models suppose most of parameters to be constant, except of current I and voltage V, thus the only {R s} parameter is investigated to obtain final I-V curves nonlinear behavior. Such case is solvable by ordinary differential equation. Next, supposing parameters’ dynamics, establishing partial differential equations system is needed, trying to find multidimensional system solution over multiple input and multiple output parameters. Two-parameter models investigate {n, Rs} or {Rs, Rsh} parameters, depending on constants, functions or partial derivatives of rest parameters, so the dependence is more complex in this case. Similarly, three-parameter models investigate {n, Rs, Rsh} or {n, Rs, Is} parameters behavior. Next, four-parameter models survey {n, Rs, Rsh, Is} parameters applying more complex numerical iterative methods. The five-parameter models investigate all of the above mentioned parameters, {n, Rs, Rsh, Is, Iph} [1, 2, 3, 4, 8]. Here, partial differential equation system is complex system depending on multivariable functions and their partial derivatives. Hereby, some researchers simplify numerical models supposing that some unknown parameters are temperature independent and thus static, namely {n, Rs, Rsh} in five-parameter models. Supposing this limitation, they apply iterative numerical methods to obtain partial differential equations for the rest of parameters [1, 2, 3, 4]. Generally, the complexity of a circuit model often depends on the desired accuracy. Indeed, many researchers use own system of transition equations at reference conditions. When seeking appropriate approach, they involve there real measurable alternative PV cell input parameters, either in border limit, in expected operational range, or rather at certain operating point. In this manner, open circuit voltage VOC, short circuit current ISC, open circuit resistance ROC, short circuit resistance RSC, maximum power point circuit voltage VMPP, and maximum power point circuit current IMPP are often incorporated [1, 5, 16, 17]. When depicting relations among all of input and output variables, 2D and 3D visualization tools on hands are limited. To view reliable relations dynamics, appropriate range of parameters must be set that might difficulty, sometimes. For reasons given in previous text, the only one accurate model, numerical method, alternative parameters involving approach and testing method might be illusive. In proposed modeling and simulations in this paper, some simplifications are used. Firstly, alternative parameters involving approach is applied for single-diode five-parameter model of a Si PV cell. Secondly, partial differential equations system is step-by-step solved for wide range of input variables, in proposed approach. Profitable, one of the most important advantage is that, numerical methods are applicable over large scale of operating conditions. This allows avoiding small partial gradients vanishing or truncating, since all of the parameters are step-by-step chained. Thirdly, nested loops are repeatedly computed until nearby real results ranges or convergences are obtained. Definitely, such advanced modeling allows simulating any Si PV cell electrical parameters behavior, at stated physics limits. Relations among the above-mentioned five electrical parameters, {n, Rs, Rsh, Is, Iph}, are visualized and discussed in their operating ranges of interest.
2.
Single-diode mathematical model
The energy needed for electrons’ traveling across the material structure and thus current generation is called the band gap energy Eg, Eq. (1). It is temperature dependent parameter. P and N Si materials’ constants α, β have been experimentally found for different materials, thus the Si band gap energy can be approximated as follows [1, 6, 7]: మ
ሺሻ ൌ ሺͲሻ െ ቀ ାஒ ቁ ؆ ͳǤͳ െ
Ǥସଷమ ାଷ
ሾሿ .
(1)
The illuminated Si semiconductor diode produces a photo-generated current, Iph, which will result into I–V curve of a quantity that is related to the surface density of the incident energy. Thus, an ideal PV cell with its I–V curve is depicted as a direct current generator parallel to a diode. Ideal PV cell single-diode model output current I has been defined as follows [1, 2, 3, 4]: ୯
୭୳୲ ൌ ୮୦ െ ୱ ቂ ቀ ୬୩ీ ቁ െ ͳቃ െ ୖ ీ ሾሿ . ౩
(2)
where Iph is generated photocurrent in A, Is is reverse dark saturation current of the non-linear diode in A, q is electric charge of an electron: q = ‒1,6022·10‒19 in C, VD is voltage across the diode in V given as follows: VD = VPV +
RsIPV, VPV is cell output voltage in V, IPV is PV cell output current in A, Rs is series resistance in Ω, n is dimensionless diode ideality factor (generally acceptable at 0.5 … 2.5), k is Boltzmann constant: k = 1,380658·10-23 in J·K-1, T is temperature of a PV cell in K, Rsh is shunt resistance in Ω. If the diode is not illuminated, Iph = 0 and VD/Rsh = 0 thus the Eq. (2) can be reduced. The FF, fill factor of a PV cell, represents power rate among electrical maximum power point PMPP achieved at the tip of particular power-voltage curve i.e. product of voltage VMPP and current IMPP, referred to idealistic maximum electrical power i.e. product of VOC and ISC, Eq. (3). The fill factor is strongly influenced by recombination currents and intrinsic ohmic resistances. The PMPP of a PV cell, FF, the incident light power Pin in W·m-2, the cell surface S in m2, and solar-to-electrical power conversion efficiency η are related by Eq. (4) [1, 13, 15, 16]. ൌ
ౌౌ ୍ౌౌ
Ʉ ൌ
ోి ୍ి
ోి ୍ి ୗ
ሾെሿ ,
ൌ
(3)
ౌౌ ୍ౌౌ ୗ
ሾΨሿ .
(4)
The {n, Rs, Rsh, Is, Iph} parameters of interest for a Si PV cell can be obtained in general forms with respect to standard test conditions STC data as follows [1, 2, 3, 4, 5, 8, 9, 10, 11, 12]: ൌ
୯ీ
୩ሺ୪୬ ୍ି୪୬ ୍౩ሻ
ୱ ൌ ୢ୍ ቚ ୢ
୮୦ ൌ ୗ
ୗ
ୀోి
ሾെሿ ,
(5)
ሾȳሿ ,
(6)
ൣ ୱ̴ୗେ Ƚ ሺ െ ୗେ ሻ൧ሾሿ ,
ి
ୱ ൌ ୱ̴ୗେ ቀ ୱ୦ ൌ ୱ୦ ቀ
ి
ୗి ୗ
ు ଷ ൬ ౝ̴ి ି ుౝ ൰
ቁ
ౡి
ౡ
ቁ ୱ୦̴ୗେ ሾȳሿ .
(7)
ሾሿ ,
(8) (9)
3. Numerical modelling and results Typically, a commercial amorphous, polycrystalline or monocrystalline structural Si cells of a small size of S = 1 cm2 achieve PMPP ≈ 0.010 … 0.018 W·cm-2 at AM1.5G irradiation conditions, the irradiation intensity about 1000 W·m-2, and T ≈ 298.15 K. They achieve FF ≈ 0.8 and η ≈ 10 … 21 %. For such reference PV cell size and irradiation conditions, an amorphous silicon cell typically attack maximum VOC ≈ 0.5 … 3 V and ISC ≈ 0.016 A. A polycrystalline silicon cell typically attacks VOC ≈ 0.5 … 2.5 V, ISC ≈ 0.030 A. Finally, the most efficient monocrystalline silicon cell typically attacks VOC ≈ 0.5 … 3 V, ISC ≈ 35 mA. Mostly, VMPP ≈ (0.8 … 0.9)·VOC in V and IMPP ≈ (0.85 … 0.95)·ISC in A for any of the material structures [16, 17]. For the small-size Si PV cell, {n, Rs, Rsh, Is, Iph} parameters relations are modelled along expected operating ranges and explained in this section. Following the previous explanations and Eq. (2, 5, 6, 7, 8, 9) resulting into Eq. (10, 11, 12, 13, 14), experimental results are obtained using Matlab software static programming. The temperature dependent parameters are chained by step-by-step modelling and simulation. Following previous explanations, involved input alternative parameters are set and posterior parameters range are estimated, both at particular static operational range or point. For all steps, the temperature ranges in T = 273.15 ... 353.15 K interval. ୱ ൌ ൌ
୍ౌౌ
୯ሺିోి ሻ
൰൨
୩ቐ୪୬൬୍ి ି ౌౌ ି୍ౌౌ ൰ି୪୬൬୍ి ି ోి ൰ା ౩
ୱ୦ ൌ ୱ ൌ
ోి షమౌౌ షఽ ቁ ఽ
ͳ ൬െቀ
ా
ሺୖ౩ ୍ౌౌ ିౌౌ ሻሺିోి ିሻୣቀఽቁାୣ ా
ాషోి ౩
ా ൬ ోి ൰ ఽ ିୣቀ ఽቁ
,
୍ౌౌ
,
(10) ,
(11)
ౌౌ ቑ ి ష ోి ౩
൬ ోి ൰ ఽ ൩
൬ ోి ൰ ఽ ൩
ቀ ቁ ୍ౌౌ ሺౌౌ ିୖ౩ ୍ౌౌ ାሻୣ ఽ ିୣ
୍ౌౌ ା ୣ
౩
ౌౌ
,
(12)
(13)
ቀ ోి ቁ
୭୳୲ ൌ ୮୦ െ ୱ
where ൌ
ఽ
୬୩ ୯
,
െ ͳ൨ െ
ోి ୖ౩
,
(14)
ൌ ୱ .
Intrinsic Rs and Rsh resistances influence on a PV cell behavior have been widely discussed in previous works [1, 2, 3]. The series resistance mainly depends on the layer composition. Surface layer thickness and specific material parameters play dominant role. Due to very small IMPP current caused by small cell size and thus huge VMPP/IMPP ratio, huge series resistance Rs can be assumed from Eq. (10). As a temperature dependent variable, the Rs is obtained using Lambert W function. Involved parameters are derived from estimated VOC, ISC, VMPP, IMPP range. Initially, n = 0.5 … 2.5 and T = 273.15 ... 353.15 K are set. Obtained Rs range is depicted in the Fig. 2, as a function of T and n. It starts at the minimum value Rs ~ 38. 5 Ω at T = 273.15 K and n = 1 point. With respect to input parameters, it gently slopes almost linearly up to Rs ~ 42 Ω. Even if the Rs dynamic is very gentle, its dependence on the particular VOC, ISC, VMPP, IMPP measured parameters accuracy is strong. Considering much larger cell area, the Rs would reduce significantly, mainly due to dramatic IMPP increasing and thus rapid decrease of quasi-static VMPP/IMPP ratio. For example, when considering a cell size of S = 20 cm2 with the same homogeneous material structure and operating conditions, it reduces to about Rs ~ 1.92 … 2.1 Ω and, for a size of S = 1 m2 it decreases up to ordinary Rs ~ 0.00385 … 0.0042 Ω. Here, the normalized series resistance rs = Rs.IMPP/VMPP can acting as reliability indicator: for the small-sized cell, rs ~ 1.02 … 1.11 Ω, Fig. 2 below. Finally, let we analyze the Rs influence on FF and η. If the irradiation and temperature conditions does not change, the VMPP/VOC and IMPP/IOC ratios remain constant so the FF and the η does not change too, see Eq. (3, 4).
Fig. 2: Series resistance dependence Rs = f(T, n) (upper) and normalized series dependence rs=f(T, n) (lower) on temperature and ideality factor.
Similarly, n dependence on T and series resistance is depicted in Fig. 3. Here, due to small-sized cell, the normalized r s operating range is used instead of biased Rs. From conventional ratio range Rsh/Rs ≈ 500 … 2000 applicable for most types of Si ideal cells or panels, real posterior value Rsh ≈ 400 Ω was set initially for the Eq. (11) with simulation resulted to Fig. 3 upper. Here, the ideality factor gently increases as opposed to T and r s. Among the
two, more significant is influence of the temperature gradient. In the operational T and r s ranges, posterior n ≈ 1 … 1.35 is obtained that is close to n=1 of the ideal diode. This reinforces the opinion that, single diode model is appropriate for small-sized cells where the Auger or depletion region recombination are less significant. Moreover, the Shockley-ReedHall space charge region recombination must be at low junction level. On the other hand, for R sh ≈ 1000 Ω, the simulated range of local ideality factor tilting rises about twice and n ≈ 1.9 … 2.5, Fig. 3 lower, that is far from the ideal diode. It corresponds with the above mentioned recombination and tunneling significant enhancements, mainly in the deep material space charge regions at junction interface. This analysis allows us to reduce involved n for the rest of numerical calculation steps.
Fig. 3: Ideality factor dependence n = f(T, rs) on temperature and normalized series resistance, at Rsh ≈ 400 [Ω] (upper), Rsh ≈ 1000 [Ω] (lower).
In the third step, Rsh is numerically calculated using Eq. (12). Generally, small Rsh value normally indicates a technical fault in the PV cell. But, higher temperatures cause higher charge mobility thus Rsh may not exceed some practical limit resistance, e.g., Rsh ≈ 1000 Ω. Rsh dependence on T and Rs is depicted in Fig. 4 for the operating ranges of T ≈ 273.15 ... 298.15 K and Rs ≈ 38.3 ... 39.55 Ω, the Rsh ≈ 12 ... 1000 Ω. The Rsh increases very steeply, depending on Rs. Besides, the cell small size plays crucial role of the Rsh values, in this modelling case. The numerical calculations goes above 1000 Ω up to singularity border of numerical computing and, controversially, Rsh decreases rapidly to nonrealistic negative values. In Fig. 4, they are eliminated prevailingly.
Fig. 4: Shunt resistance dependence on temperature and series resistance, Rsh = f(T, Rs).
Next, the absolute value of reverse saturation current behavior is numerically modelled using Eq. (13), Fig. 5. The constant value of radix Is ≈ 10-18 A rapidly non-linearly raises as the temperature does, beginning from the temperature T ≈ 330 K. Additionally, it raises steeply if the Rsh approaches its lower limit Rsh ≈ 50 ... 12 Ω at the T ≥ 340 K.
Fig. 5: Saturation current dependence on temperature and shunt resistance, Is = f(T, Rsh).
In following step, Iph and Iout are numerically calculated using Eq. (2, 7, 14). Considering small-sized cell, Iout achieves fewer than 0.018 A only. In Fig. 6, the Iout strong dependence on T and Is is evident, mainly at higher absolute saturation values and lower temperatures. Its absolute value grows up as the temperature goes down, even more when reverse saturation current value goes down. This is mainly caused by Rsh lowering. Finally, the total output current Iout is depicted in the Fig. 7 as a function of T and V. In keeping with small cell size, this variable parameter achieves I out ≈ 0 ... 0.020 A, with the maximum current values at about ISC values and, with the minimum at VOC points.
Fig. 6: Output current dependence on temperature and saturation current, Iout = f(T, Is).
Fig. 7: Output current dependence on temperature and voltage, Iout = f(T, V).
In this paper, the dynamic change of the irradiation intensity or ISC dependence on the temperature are out of the interest. Therefore, slight controversial Iout dependence on T is not included in this paper, as it was analyzed in previous authors’ work.
4.
Conclusion
In this paper, the single-diode five-parameter model is used to demonstrate applicability of the reduced form approach for electrical parameters {n, Rs, Rsh, Is, Iph} relations study. Involved alternative parameters values {VOC, VMPP, ISC, IMPP} are estimated and used for numerical modelling and simulation, for a small silicon photovoltaic cell of 1 cm2 size. Next, posterior parameters ranges are estimated. In general, due to step-by-step chained numerical modelling as well as visualization needs, both involved and posterior parameters are set at particular static range or point. For all modelling and simulation steps, the temperature ranges in T = 273.15 ... 353.15 K interval. As a result, relations between the five parameters are obtained and visualized for the silicon photovoltaic cell. Besides of temperature dependency of all the five parameters, the most important parameter appears to be the series resistance. It represents intrinsic surface parameters setting and, as proven by visualization results, it influences next simulations steps. Next, shunt resistance indicates charge mobility capability and reveals photovoltaic cell technical state along its life cycle.
The limits of the modeling originate from chemical properties of the doped material structure and physical micro- and nano- surface properties. Here, the material fabrication processes influence the operational ranges, mainly. Therefore, simulated results may differ for real monocrystalline, polycrystalline and amorphous Si cells, even more when surfaces or structures change dramatically. Next simulation limits grow from supposed simple fabrication single PN junction of a Si PV and, from supposed standard or near-standard lighting and temperature conditions. The singlediode-based proposed modelling is appropriate for declared cases, at stated conditions ranges. But, the more limits differences, the more results will differ between simulated and measured parameters. For low light conditions or specified temperature ranges, the double-diode model should be applied. Different types of PV cells by Si-based materials structures or surfaces will be analyzed in future case studies. Proposed electrical parameters analysis using relations visualization is well-promising method for dynamic modelling of a particular silicon PV cell.
Acknowledgements This work was supported by the Slovak research and development agency [APVV-15-0152, APVV-14-0239, APVV0888-11]; the Slovak research grant agency [VEGA 2/0076/15]; EU structural funds, Cohesion fund and European regional development fund [ITMS: 26210120021]; the Tatra banka fund [2016et017]; and the Horizon 2020 MSCARISE-2016 [6260922 SENSIBLE].
References [1] LUQUE, A., HEGEDUS, S.: Handbook of photovoltaic science and engineering. John Willey & Sons Ltd, Chichester, England, 1179 p., ISBN: 0-471-49196-9, 2003. [2] DUFFIE, J. A., BECKMAN, W.A.: Solar engineering of thermal processes. John Willey & Sons Ltd, New Yersey, USA, 928 p., ISBN: 9781-118-41812-49196-2. 2013. [3] REKIOUA, D., MATAGNE, E.: Optimization of photovoltaic power systems: Modelisation, Simulation and Control. Springer Verlag, 283 p., ISBN: 978-1-4471-2348-4. Dec. 2012. [4] HUMADA, A. M., HOJABRI, M., MEKHILEF, S., HAMADA, H. M.: Solar cell parameters extraction based on single and double-diode models: A review. In: Renewable and Sustainable Energy Reviews 56 (2016), pp. 494–509, Dec. 2015. [5] KING, D. I., BOYSON, W. E., KRATOCHVIL, J. A.: Photovoltaic array performance model. Sandia report SAND2004 -3535, Sandia national laboratories, Abuquerque, New Mexico, USA, 43 p., Dec. 2004 [6]
SHOCKLEY, W.: Electrons and holes in semiconductors. 7-th printing, D. Van Nostrad Company, Inc., New York, USA, 558 p., Sept. 1959.
[7] VAN ZEGHBROECK, B.: Principles www.colorado.edu/~bart/book/book/chapter2/ch2_3.htm>.
of
Semiconductor
Devices.
2011,
online:
[8] WOLF, M., RAUSCHENBACH, H.: Series resistance effects on solar cell measurements. In: Advanced Energy Conversion, Pergamon pr ess, GB, pp. 455–479, 1963. [9] LIM, L. H. I., Ye, Z., Ye, J., Yang, D., Du, H.: A linear method to extract diode model parameters of solar panels from a single I-V curve. In: Renewable energy 76 (2015), Elsevier, pp. 135-142, Nov. 2014. [10] LAUDANI, A., DAVID, F. M., FULGINEI, F. R., SALVINI, A.: Reduced-form of the photovoltaic five-parameter model for efficient computation of parameters. In: Solar energy 97 (2013), Elsevier, pp. 122-127, Sept. 2013. [11] LAUDANI, A., FULGINEI, F. R., SALVINI, A.: High performing extraction procedure for the one-diode model of a photovoltaic panel from experimental I–V curves by using reduced forms. In: Solar energy 103 (2014), pp. 316-326, March 2014. [12] LAUDANI, A., FULGINEI, F. R., SALVINI, A.: Identification of the one-diode model for photovoltaic modules from datasheet values. In: Solar energy 108 (2014), Elsevier, pp. 432-446, Aug. 2014. [13]
CIBIRA, G.: Photovoltaic cells electrical parameters. In: ALER 2016, Bobrovec, SES, ISBN 978-80-89456-26-0, CD, pp. 35-42, Nov. 2016.
[14] BAYHAN, H., KAVASOGLU, A. S.: Exact analytical solution of the diode ideality factor of a pn junction device using Lambert W-function model. In: Turk journal Physics, Vol. 31 (2007), pp. 7-10, 2007. [15] KUNZ, G., WAGNER, A.: Internal series resistance determinated of only one IV-curve under illumination. Conference paper No. 5BV 2.70, 19th European photovoltaic solar energy conference, Paris, France, pp. 14, 2004. [16] QARONY, W., HOSSAIN, M. I., HOSSAIN, M. K., UDDIN, M. J., HAQUE, A., SAAD, A. R., TSANG, Y. H.: Efficient amorphous silicon solar cells: characterization, optimization, and optical loss analysis. In: Results in Physics 7 (2017), Elsevier, pp. 4287-4293, Nov. 2017. [17] CHANDER, S., PUROHIT, A., SHARMA, A., ARVIND, NEHRA, S. P., DHAKA, M. S.: A study on photovoltaic parameters of monocrystalline silicon solar cell with cell temperature. In: Energy Reports 1 (2015), Elsevier, pp. 104-109, April 2015
Highlights: 1. Si PV cell simplified single-diode equivalent electrical circuit is applied 2. alternative measured parameters {VOC, ISC, VMPP, IMPP} are involved 3. chained parameters {n, Rs, Rsh, Is, Iph} are numerically computed via reduced PDEs 4. {n, Rs, Rsh, Is, Iph} ranges are iteratively simulated 5. relations among {n, Rs, Rsh, Is, Iph} are computed, visualized and analysed