International Journal of Energetica (IJECA) https://www.ijeca.info ISSN: 2543-3717 Volume 2. Issue 2. 2017 Page 06-14 IJECA-ISSN: 2543-3717. December 2017 Page 6 Modeling and Parameter Extraction of PV Cell Using Single- and Two-Diode Model B. Benabdelkrim 1,2 , A. Benatillah 2 1 Department of Material Sciences, Institute of Science and Technology, University of Ahmed Draia, Adrar, ALGERIA 2 Laboratory of Energy, Environment and Systems of Information (LEESI), University Ahmed Draia Adrar, ALGERIA benaekbouchra01@gmail.com Abstract – Photovoltaic modules operate under a large range of conditions. This combined with the fact that manufacturers provide electrical parameters at specific conditions (STC). The present study proposes a comparison between single and double diode models of solar PV system and ensures the best suited model under specific environmental condition for accurate performance prediction. An important feature of these models is that its parameters can be determined using data commonly provided by module manufacturers on their published datasheets. Accurate determination of these parameters which arose from a diversification of models and methods dedicated to their estimations is still a challenge for researchers. In this paper the single and two diode models have been studied by mathematical methods based on simulated Newton-Raphson iteration method. Newton-Raphson iteration method is solved by MATLAB simulation. Keywords: PV Module; Single-diode model; Two diode model; Performance I–V Curves, Parameter Extraction Received: 10/12/2017 – Accepted: 25/12/2017 I. Introduction The rapid growth of PV system utilizations is due to its availability everywhere which avoids transmission costs and losses, free, abundant and pollution free. Silicon is the basic material required for the production of solar cells based crystalline or thin film technology. The photovoltaic (PV) modules are generally rated under standard test conditions (STC) with the solar radiation of 1000 W/m2, cell temperature of 25°C, and solar spectrum of 1.5 by the manufacturers. The parameters required for the input of the PV modules are relying on the meteorological conditions of the area. The climatic conditions are unpredictable due to the random nature of their occurrence. These uncertainties lead to either over- or underestimation of energy yield from PV modules. An overestimation up to 40% was reported as compared to the rated power output of PV modules [1, 2]. The growing demand of photovoltaic technologies led to research in the various aspects of its components from cell technology to the modeling, size optimization, and system performance [3–5]. There are various PV cell modules studied by researchers in the literature. One of the simplest is single diode model. [6] In broad sense this model is derived by three parameters: Short Circuit Current (Isc), Open Circuit Voltage (Voc), and Diode Ideality Factor (A). When the parameter series resistance (Rs) is added in this model, the accuracy of model gets improved. One drawback of this model is that it is not capable of temperature (T) variation handling. Parameter shunt resistance (Rsh) significantly improves the model efficiency. [7] This model is having a drawback of reduced accuracy under low irradiance (G) level, especially at open circuit voltage (Voc). Additional diode design is added to the model for the recombination loss in the depletion region of the cell of solar module. [8] This is double- diode model. This model has more parameters to calculate. This model gives more accuracy because this model is more practical especially under low voltages. In this paper, a comparative analysis details the behavioral I-V characteristics of a single-diode using analytical four and five parameter model and two- diode model. The accuracy of the simulation results is verified by comparing it with published data provided by manufacturers of six PV modules of different types (mono-crystalline, poly-crystalline and thin-film). II. Mathematical models of PV module II.1 Single-diode model An electrical circuit with a single diode (single exponential) is considered as the equivalent mailto:benaekbouchra01@gmail.com B. Benabdelkrim et al. IJECA-ISSN: 2543-3717. December 2017 Page 7 photovoltaic cell in the present article. Two different models drawn from the equivalent electrical-circuit are studied: namely four- and five-parameter models. Figure1. PV-cell equivalent-circuit models: single-diode model [9]. An output current equation of I-V characteristic using this model can be written as: . . . exp 1 0 V R I V R I s sI I I pv V R T sh                         (1) Where Ipv Photocurrent I0 Cell saturation current Rsh Shunt resistance Rs Series resistance VT the thermal voltage (VT=a.Ns.k.T/q) Ns Number of cells in series a Ideal factor of the PV diode q Electron charge (1.60281×10 -19 C) k Boltzmann‟s constant=1.38066×10 -23 J/K T Cell operating temperature II.1.1 Four-parameter model The four-parameter model studied in this work has been used elsewhere [10, 11]. Assuming Rsh as infinite and neglecting it in Equation (1), the four- parameter model is obtained as follows: . . exp 1 0 V R I sI I I pv V T               (2) The unknown parameters are denoted at STC as , , a 0 I I pvn n n and R sn ; where the “n” subscript refers to the reference operating conditions. The short circuit current can be found when V=0 I I scn pvn  (3) The following equations are used to calculate the other parameters at STC [10]. 2 3 .( ) . ocn v n n gi Tn pvn n n V K T a EK V I T k T     (4) 0 exp 1 n ocn I pvn I V V Tn          (5) .ln(1 ) V I mpn Tn ocn mpn pvn mpn I V V I R sn     (6) Where Eg is the band gap of the material The parameters can be found at any other operating conditions by using following equations: ( ) pv pvn i n n G I I K T T G      (7) 3 . 1 1 exp 0 0 . q E T g I I n T a k T T n n                         (8) R R s sn  (9) . n n T a a T  (10) This model is implemented as follows: Eqs. (3)–(6) are used to find values of the four parameters under reference conditions. These four parameters are corrected for environmental conditions using Eqs. (7)–(10) and used in Eq. (2), which relates cell current to cell voltage. From Eq. (2) either cell current or voltage could be calculated provided that the other is known. Alternatively, cell current and voltage could both be calculated at the maximum-power point. II.1.2 Five-parameter model As given in Eq. (1), the five-parameter model is an implicit non-linear equation, which can be solved with a numerical iterative method such as Newton Raphson method [12]. However, this requires a close approximation of initial parameter values to attain convergence. Alternatively, the parameters may be extracted by means of analytical methods. Some of the analytical methods are studied elsewhere [12-15]. The five parameters Ipv, Io, Rs, Rsh, and m are calculated at a particular temperature and solar- irradiance level from the limiting conditions of Voc, Isc, Vmp, Imp and using the following definitions of Rso and Rsho: 0 oc s V V dV R dI    (11) 0 sc sh I I dV R dI    (12) Where Rs0 and Rsh0 are the reciprocals of the slopes at the open-circuit point and short-circuit point, respectively. The values of these resistances are not usually provided by module manufacturers. The other parameters are calculated as follows. The following equations are used to calculate the five parameters required. 0 . 1 I . exp 1s sc s pv sc sh T R I R I I R V                    (13) B. Benabdelkrim et al. IJECA-ISSN: 2543-3717. December 2017 Page 8 0 .expoc oc sc sh T V V I I R V              (14) The value of the diode ideality factor (a) may be arbitrarily chosen. Many authors discuss ways to estimate the correct value of this constant. Usually, 1 ≤ a≤ 2 and the chosen value depend on other parameters of the I–V model. As it‟s given in [16], there are different opinions about the best way to choose (a). Because (a) expresses the degree of ideality of the diode and it is totally empirical, any initial value of ( a) can be chosen in order to adjust the model. The Rs and Rsh resistances are calculated by iterative methods. The relation between Rs and Rsh, may be found by making the maximum power calculated by the I–V model, equal to the maximum experimental power from the datasheet (P max,m = Pmax,e) at the (V m; Im) point. In the iterative process, Rs must be slowly incremented starting from Rs = 0 and for every iteration, the value of Rsh is calculated simultaneously: . . . exp 1 max, 0 max,e V R I V R I mp s mp mp s mp P V I I P m mp pv V R T sh                               (15) 0 0 exp ocT s s T VV R R I V         (16) . . max, . exp 1 0 V R I mp s mp R sh V R I P mp s mp e I I pv V V T mp                 (17) The initial condition for the shunt resistance Rsh can be found when considering the initial value of Rs=0 [17, 18] sh,min mp oc mp sc mp mp V V V R I I I     (18) In the proposed iterative method, the series resistance must be slowly incremented starting from a null value. Adjusting the I-V curve to match the cell reference condition requires finding the curve for several values of series and equivalent shunt resistances. The Newton–Raphson method was used in the proposed iterative method due to the ability to overcome undesired behaviors [19]. II.2 Two-diode model The two diode model (Fig.2) equation of the I–V curve is expressed as [20]: . . . . exp 1 . exp 1 01 02 1 2 V R I V R I V R I s s sI I I I pv V V R T T sh                                            (19) Figure2. PV-cell equivalent-circuit models: two-diode model Where the diode factors a1=1 and a2 can be derived from: 1 2 1 a a p   (20) Where, p can be chosen greater than 2.2. The rest of parameters can be deduced from the following equations [20]: I I pv sc  (21)     01 02 1 2 . . . exp 1 . / p i oc v I K T scI I q V K T kT a a            (22) Rs and Rsh are calculated by iterative method, similar to the procedure proposed by [21], where the relation between Rs and Rsh is chosen to verify that the calculated maximum power is equal to the experimental one (P max,m =P max,e) at (Vm, Im) point. The Rs value is found by a slow incrementation by the same manner as the above subsection. The expression of Rsh can be written as: . . . max, . exp exp 2 01 . (p 1) . V R I mp s mp R sh V R I V R I P mp s mp mp s mp e I I q q pv k T k T V mp                           (23) III. Results and discussion The modeling methods described in this paper are validated by measured parameters of selected PV modules. The experimental (V,I) data are extracted from the manufacturer‟s datasheet. Three different modules of different brands/ models are utilized for verification; these include the multi- and mono crystalline as well as thin-film types. The specifications of these modules are summarized in Table 1 B. Benabdelkrim et al. IJECA-ISSN: 2543-3717. December 2017 Page 9 Table1. Specification of the PV modules Modules Isc (A) Voc (V) Imp (A) Vmp (V) Ki(Isc) (mA/°C) Kv(Voc) (mV/°C) Ns Poly-cristallin Kyocera KC200GT 8.21 32.9 7.61 26.3 3.18 -123 54 Shell S70 4.5 21.2 4.12 17 2 -76 36 Mono-cristallin Shell SQ150 4.8 43.4 4.4 34 1.4 -161 72 Shell SP70 4.7 21.4 4.25 16.5 2 -76 36 Thin-Film Shell ST40 2.68 23.3 2.41 16.6 0.35 -100 36 PVL-136 5.1 46.2 4.1 33 5.1 -176 66 Figures (3-5) shows the I-V curves for modules for different levels of irradiance and temperature. It can be seen that for varying irradiance, despite the modeling curves do not match experimental data in all points, the tow diode model strongly agrees to experimental data than the four-parameter and five- parameter models for all types of modules, except for the thin-film (ST40) module at low irradiance of about 200W/m² where the five-parameter modeled curve is closer to the experimental data than four- parameter and tow diode models. In the case of the varying temperature, G is kept constant at 1000 W/m 2 . It can be noted that all three methods show good general agreement with the experimental data. However, a close inspection reveals that the tow-diode model yields the most accurate results at all temperature. 0 5 10 15 20 25 0 1 2 3 4 5 6 Voltage (V) C u rr e n t (A ) Mono-cristallin SP70 I-V Curve experimental data 4-P model 5-P model 2-diode model 600 W/m2 200 W/m2 1000W/m2 800 W/m2 400W/m2 0 5 10 15 20 25 0 1 2 3 4 5 6 Mono-cristallin SP70 I-V Curve Voltage (V) C u rr e n t (A ) experimental data 4-P model 5-P model 2-diode model 60°C 40°C 20°C Figure3. The I-V characteristics of SP70 module at varying irradiance and temperature. 0 5 10 15 20 25 30 35 0 1 2 3 4 5 6 7 8 9 10 Voltage (V) C u rr e n t (A ) poly-cristallin KC200GT I-V Curve experimental data 4-P model 5-P model 2-diode model 1000W/m2 800 W/m2 600 W/m2 400W/m2 0 5 10 15 20 25 30 35 0 1 2 3 4 5 6 7 8 9 10 Voltage (V) C u rr e n t (A ) poly-cristallin KC200GT I-V Curve 50°C 75°C 25°Cexperimental data 4-P model 5-P model 2-diode model Figure4. The I-V characteristics of KC200GT module at varying irradiance and temperature. B. Benabdelkrim et al. IJECA-ISSN: 2543-3717. December 2017 Page 10 0 5 10 15 20 25 0 0.5 1 1.5 2 2.5 3 3.5 Voltage (V) C u rr e n t (A ) Thin-film ST40 I-V Curve experimental data 4-P model 5-P model 2-diode model 800 W/m2 1000W/m2 400W/m2 600W/m2 200W/m2 0 5 10 15 20 25 0 0.5 1 1.5 2 2.5 3 3.5 Voltage (V) C u rr e n t (A ) Thin-film ST40 I-V Curve experimental data 4-P model 5-P model 2-diode model 20°C 40°C 60°C Figure5. The I-V characteristics of ST40 module at varying irradiance and temperature. Table 2-4 shows the parameters used for three models. Four parameters are calculated namely, I0, IPV, ideality factor (a) and Rs for the 4-P model. In the five parameter model, the additional calculated parameter is the shunt resistance; Rsh. and the two- diode model has more variables, the actual number of parameters computed is four because I01=I02=I0 while a1=1 and p can be chosen arbitrarily, i.e. p  2.2. The two-diode model and the five parameter model exhibit similar results at STC. This is to be expected because both models use the similar max power matching algorithm to evaluate the model parameters at STC. However, at low irradiance, more accurate results are obtained from the two-diode model. Table2.Parameters extracted for the four parameter model Poly-crystalline Mono-crystalline Thin-Film Module KC200GT S70 SP70 SQ150-PC ST40 PVL-136 Ipv 8.2100 4.5000 4.7000 4.8000 2.6800 5.1000 a 1.0758 1.0177 1.0222 1.0594 1.3219 1.2573 Rs 0.3541 0.4547 0.6310 1.0296 1.6156 2.3723 Io 2.1954e-9 7.4460e-10 6.9528e-10 1.1570e-9 1.4202e-8 1.9783e-9 Table3. Parameters extracted for the five parameter model Poly-crystalline Mono-crystalline Thin-Film Module KC200GT S70 SP70 SQ150-PC ST40 PVL-136 Ipv 8.2146 4.5055 4.7150 4.8073 2.6961 5.2942 a 1.3000 1.3000 1.3000 1.3000 1.3000 1.3000 Rs 0.2300 0.2200 0.4000 0.6700 1.5100 1.6800 Rsh 601.3368 189.0262 133.1309 466.4639 266.5478 44.1667 Io 9.8252e-8 9.9101e-8 8.7645e-8 6.9745e-8 1.0292e-8 4.0336e-9 Table4. Parameters extracted for the two-diode model Poly-crystalline Mono-crystalline Thin-Film Module KC200GT S70 SP70 SQ150-PC ST40 PVL-136 Ipv 8.2100 4.5000 4.7000 4.8000 2.6800 5.1000 a1 1 1 1 1 1 1 a2 1.2000 1.2000 1.2000 1.2000 1.2000 1.2000 Rs 0.3300 0.3400 0.5100 0.9100 1.7100 1.9600 Rsh 174.1551 119.5882 94.9643 275.2625 204.8492 54.2497 Io1=Io2 4.1280e-10 4.9996e-10 4.2065e-10 3.1059e-10 3.0748e-11 7.5012e-12 Tables 5–8 show the relative errors for Pmax, Voc and Isc at varying irradiance and temperature of SP70 and ST40 modules. The relative error is defined as:    data data *100 calcul relative abs X X E X X        (24) The irradiance is maintained constant at STC. From the data it can be concluded, more accurate results are obtained from the two-diode model for the crystalline silicon technologies. B. Benabdelkrim et al. IJECA-ISSN: 2543-3717. December 2017 Page 11 Table5. Relative errors of three models at different irradiances (T =25°C) for SP70 module. Irradiance (W/m 2 ) Parameters Measured data 4-P model Error % 5-P model Error % 2D model Error % 1000 Pmax Voc Isc 70.07 21.33 4.682 70.5 21.39 4.7 0.61 0.28 0.38 70.11 21.35 4.7 0.057 0.094 0.38 70.22 21.34 4.675 0.21 0.047 0.15 800 Pmax Voc Isc 56.13 21.03 3.752 57.61 21.18 3.76 2.64 0.71 0.21 55.95 21.07 3.76 0.32 0.19 0.21 56.38 21.13 3.74 0.45 0.48 0.32 600 Pmax Voc Isc 41.89 20.5 2.815 43.96 20.91 2.82 4.94 2.00 0.18 41.46 20.72 2.82 1.026 1.073 0.18 41.99 20.84 2.805 0.24 1.66 0.36 400 Pmax Voc Isc 27.53 19.92 1.882 29.62 20.53 1.88 7.59 3.06 0.11 26.76 20.19 1.88 2.79 4.92 0.11 27.12 20.43 1.87 1.49 2.56 0.64 200 Pmax Voc Isc 13.17 19.12 0.9472 14.72 19.81 0.94 11.76 3.61 0.76 12.08 19.25 0.94 8.28 0.68 0.76 11.99 19.65 0.935 8.96 2.77 1.29 Table6: Relative errors of three models at different temperatures (E =1000 W/m 2 ) for SP70 module. Temperature (°C) Parameters Measured data 4-P model Error % 5-P model Error % 2D model Error % 20 Pmax Voc Isc 71.54 21.71 4.743 72.23 21.77 4.69 0.96 0.28 1.12 71.76 21.70 4.69 0.31 0.046 1.12 71.82 21.70 4.665 0.39 0.046 1.64 40 Pmax Voc Isc 64.77 20.18 4.736 65.29 20.26 4.73 0.80 0.39 0.13 65.15 20.25 4.73 0.59 0.35 0.13 65.38 20.24 4.705 0.94 0.29 0.65 60 Pmax Voc Isc 57.94 18.71 4.743 58.34 18.69 4.77 0.69 0.11 0.57 58.54 18.68 4.77 1.036 0.16 0.57 58.86 18.67 4.745 1.59 0.21 0.042 Table7. Relative errors of three models at different irradiances (T =25°C) for ST40 module. Irradiance (W/m 2 ) Parameters Measured data 4-P model Error % 5-P model Error % 2D model Error % 1000 Pmax Voc Isc 40.21 23.29 2.677 40.03 23.30 2.68 0.45 0.04 0.11 39.99 23.27 2.68 0.55 0.086 0.11 40.04 23.26 2.658 |0.42 0.13 0.71 800 Pmax Voc Isc 31.71 22.85 2.149 33.04 23.02 2.144 4.19 0.74 0.23 32.68 22.99 2.144 3.06 0.61 0.23 32.97 23.04 2.126 3.97 0.83 1.07 600 Pmax Voc Isc 23.52 22.33 1.607 25.44 22.67 1.608 8.16 1.52 0.062 24.80 22.62 1.608 5.44 1.30 0.062 25.17 22.76 1.595 7.02 1.92 0.75 400 Pmax Voc Isc 15.34 21.63 1.074 17.26 22.17 1.072 12.52 2.49 0.19 16.4 22.11 1.072 6.91 2.22 0.19 16.7 22.35 1.063 8.86 3.33 1.02 B. Benabdelkrim et al. IJECA-ISSN: 2543-3717. December 2017 Page 12 200 Pmax Voc Isc 6.967 20.28 0.537 8.611 21.33 0.536 23.59 5.18 0.19 7.615 21.17 0.536 9.30 4.39 0.19 7.655 21.61 0.5316 9.87 6.56 1.01 Table8: Relative errors of three models at different temperatures (E =1000 W/m 2 ) for ST40 module. Tempetaure (°C) Parameters Measured data 4-P model Error % 5-P model Error % 2D model Error % 20 Pmax Voc Isc 41.29 23.65 2.702 41.36 23.80 2.678 0.33 0.63 0.89 41.27 23.76 2.678 0.048 0.46 0.89 41.3 23.75 2.656 0.024 0.42 1.70 40 Pmax Voc Isc 36.36 21.7 2.702 36.09 21.79 2.685 0.74 0.41 0.63 36.19 21.77 2.685 0.47 0.32 0.63 36.29 21.75 2.663 0.19 0.23 1.44 60 Pmax Voc Isc 31.49 19.87 2.706 30.93 19.77 2.692 1.78 0.50 0.52 31.21 19.76 2.692 0.89 0.55 0.52 31.34 19.75 2.67 0.48 0.60 1.33 Figure.6 and 8 shows the analysis for relative error of Voc and the Pmax for ST40 module at different irradiance levels. As can be seen at STC irradiance, there is a very small difference in the Voc values among the three models. However as the irradiance is reduced, there is a significant deviation of Voc calculated using the 4-P, 5-P and two-diode models. Similar results can be observed for the Pmax. Figure.7 shows the performance of the three models at different temperature for ST40 module. There is no significant difference between three models for Voc. However the four-parameter model exhibits poor performance for Pmax calculations. Figure.9 shows the performance of the three models at different temperature for SP70 module. We note that the two-diode model and the five- parameter model are the least accurate at the three remarkable points at 60 °C compared to the four- parameter models. This is logical because the value of the ideality factor is assumed to be fixed in the five- parameter model and the two-diode model and in the other hand the values of the recombination and diffusion saturation current are assumed to be equal in the two-diode model. 1 0 5 10 15 20 25 Irradiance (w/m²) E r r o r r e la t iv e ( P m a x % ) ST40(THIN-FILM) 4-P model 5-P model 2D model 1000 800 600 400 200 1 0 1 2 3 4 5 6 7 Irradiance (w/m²) E r r o r r e la t iv e ( V o c % ) ST40(THIN-FILM) 4-P model 5-P model 2D model 800 600 400 2001000 Figure.6. Relative error of Pmax (a) and Voc (b) at varying irradiance for ST40 PV module B. Benabdelkrim et al. IJECA-ISSN: 2543-3717. December 2017 Page 13 1 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 Temperature (°C) E r r o r r e la t iv e ( P m a x % ) ST40(THIN-FILM) 604020 4-P model 5-P model 2D model 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Température (°C) E r r o r r e la t iv e ( V o c % ) ST40(THIN-FILM) 4-P model 5-P model 2D model 20 40 60 Figure.7. Relative error of Pmax (a) and Voc (b) at varying temperature for ST40 PV module 1 0 2 4 6 8 10 12 Irradiance (w/m²) R e la t iv e e r r o r ( P m a x % ) SP70 (mono-cristallin) 4-P model 5-P model 2D model 1000 800 600 400 200 1 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Irradiance (w/m²) R e la t iv e e r r o r ( V o c % ) SP70(mono-cristallin) 4-P model 5-P model 2D model 1000 800 600 400 200 Figure.8. Relative error of Pmax (a) and Voc (b) at varying irradiance for SP70 PV module 1 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 Temperature (°C) R e la t iv e e r r o r ( P m a x % ) SP70 (mono-cristallin) 4-P model 5-P model 2D model 20 40 60 1 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 Température (°C) R e la t iv e e r r o r ( V o c % ) SP70 (mono-cristallin) 4-P model 5-P model 2D model 20 40 60 Figure.9. Relative error of Pmax (a) and Voc (b) at varying temperature for SP70 PV module IV. Conclusion The present paper has proposed the comparison between the four-parameter, five-parameter and tow- diode models. These models used to predict the electrical response of illuminated six PV modules for various operating conditions. The accuracy of the three models is evaluated using practical data from manufacturers of different types of PV modules. Its performances are compared with the experimental values given by the constructors. It has been found that, the tow-diode model is better when subjected to variations in irradiance and temperature. And gives better accuracy for reconstructing the electrical characteristics of mono-crystalline and poly-crystalline PV modules, but for thin-film PV module the five parameter model is closer to the experimental data at the low irradiance. References [1] W. Durisch, D. Tille, A. Worz, and W. Plapp, “Characterisation ¨ of photovoltaic generators,” Applied Energy, vol. 65, no. 1–4, 2000, pp. 273–284. [2] A. Q. Jakhrani, A. K. Othman, A. R. H. Rigit, S. R. Samo, and S. R. Kamboh, “A novel analytical model B. Benabdelkrim et al. IJECA-ISSN: 2543-3717. December 2017 Page 14 for optimal sizing of standalone photovoltaic systems,” Energy, vol. 46, no. 1,2012, pp. 675– 682. [3] A. N. 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