A NEW METHOD OF DEVELOPING THE PARAMETERS FOR THE TWO-DIODE MODEL FROM DATA SHEET INFORMATION FOR MORE ACCURATE YIELD SIMULATION OF PV MODULES Bernhard Gatzka Martin Hofmann Valentin Software, Stralauer Platz 34 D-1243Berlin bernhard.gatzka@valentin.de martin.hofmann@valentin.de ABSTRACT The two-diode model (fig. 1) is considered as one of the most accurate models to describe the electrical behavior of polycrystalline and monocrystalline PV modules. Based on the diode equations of Shockley, it reproduces the physical processes in the cell material and combines them into an electrical equivalent circuit diagram, from which formulas can be derived for calculating the electrical behavior of cells. Fig. 1: Equivalent circuit diagram of the two-diode model So far, the two-diode model has been applied above all in the scientific field, as the necessary input parameters must first be obtained by means of multidimensional fit algorithms from measurements of existing cells (1). This paper shows how the input parameters for the twodiode model can be calculated solely on the basis of the information in PV module data sheets and a number of material-specific values, which allows the use of the model in simulation programs and leads to an improvement in the mapping accuracy of polycrystalline and monocrystalline materials. Based on present measurement data for some modules, the method is validated and the differences from the onediode model are discussed. 1. PARAMETERS FOR THE TWO-DIODE MODEL The data sheets of PV modules usually show the electrical parameters for the operating point under standard test conditions (STC): open circuit voltage V OC, STC, short circuit current I SC, STC, and voltage V MPP, STC and current I MPP,STC at maximum power point (MPP). Information about the thermal behavior is provided by the temperature coefficients of voltage, current and power α V_OC, α I_SC and α P_MPP. These module-specific parameters are supplemented by materialspecific parameters such as the band gap of cell material (2). The balance of current for the two-diode model leads to an equation with the variables V and I, which cannot be resolved explicitly by I or V (3): If the individual terms are further resolved, you get in addition to the five material-specific parameters (E gap, m 1, m 2, ) six to be determined, module-dependent parameters (C 1, C 2, C S1, C S2, R S, R sh ). The module temperature T and irradiation E remain as further variables: 1
< 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 1 1 11 11 12 12 13 13 14 14 15 15 16 16 17 17 18 18 19 19 2 > 2 Anzahl in PV*Sol-Datenbank < 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 1 1 11 11 12 12 13 13 14 14 15 15 16 16 17 17 18 18 19 19 2 > 2 Anzahl in PV*Sol-Datenbank The following conditions can be established for the calculation of the six open parameters: In the STC short-circuit, the temperature-induced change of the photocurrent corresponds with sufficient accuracy to the temperature-induced change in the short-circuit current. In the STC short-circuit, the terms of the serial resistance of the one-diode model and the two-diode model must deliver the same value. If the STC open-circuit voltage is used in the main equation, the resulting current must be zero. If the STC short-circuit current is used in the main equation, the resulting voltage must be zero. If the STC MPP voltage is used in the main equation, the current must amount to the STC MPP current. The derivation of the performance equation in MPP must be equal to zero. From these conditions it is possible to formulate a nonlinear system of equations, which allow to determine the parameters for the two-diode model. Table 1 shows, using a number of examples (4), that input parameters calculated in this way for the two-diode model also then show realistic magnitudes when it comes to new crystalline cell concepts. TABLE 1: CALCULATED PARAMETER SETS * MAIN cell concept ** acid texture Atersa Schott Solar World Solon mono poly* poly poly** 4.51 3.69 4.45 7.62 2. 5.3 2.4 2.31 93.6 89.5 13.3 1151 81.1 77. 94.8 129 7.43 6.82 6.84 4.33 4.48 1.58 3.16 3.62 To test the stability of the calculations, the resistances R S und R sh were calculated for all poly- or mono-crystalline modules in the PV*SOL database. Figures 2 and 3 show the frequency distributions that are consistent with the values listed in the literature. 9 8 7 6 5 4 3 2 1 Fig. 2: Frequency of serial resistance R S (m 12 1 8 6 4 2 Fig. 3: Frequency of shunt resistance R sh ( 2. VALIDATION WITH REAL PV MODULES With the help of measurement data for the electrical behavior of real PV modules, which were recorded using a wide range of temperature and irradiance values, it is possible to verify whether the parameters calculated in this manner enable the two-diode model to represent the electrical behavior of PV modules. This paper presents a number of examples to show that the mapping accuracy of the two-diode model achieves a very high quality using the calculated parameters. 2.1 Validation of Outdoor Measurements Measured data from October 29 to September 212 recorded in Southern Bavaria (48 north latitude) serve as the basis for the outdoor validation. The characteristic curve and the other key data for the module (SolarWorld SW165 mono) were recorded every minute, otherwise it was kept in the MPP. At the same time all the relevant meteorological parameters were recorded. In the following, three variants of the modeling are compared against the measurement. All three are based on the specifications under STC conditions and model III uses an additional specification: 2
Energy Yield in kwh I: PV*Sol model with standard part-load performance l II: two-diode model, calculation of parameters according to the above assumptions l III: PV*Sol model with additional measurement point in part-load range The module (165 W peak ) was measured over a year in minute values. The following table shows the deviations of each model from the measured annual energy yield (192.4 kwh). TABLE 2: DEVIATIONS FOR THE DIFFERENT MODELS Deviation I II III absolute kwh 13.6 4.66.51 percentaged % 7.1 2.4.26 rmbe -.251.32.5 rrmse -.639.42.387 The simulation with the two-diode model, whose parameters were calculated only from the data sheet information, shows results, with a 2.4% shortfall in output over a year, that are much closer to reality than model I. With a.26% variation in annual energy yield, model III provides very accurate results. Essential to the precision of the models are the areas of low irradiation the largest differences arise here. Figure 4, using the example of the performance data from June 21, shows qualitatively that the deviations from the measurement data depend strongly on the irradiation. Figure 5 shows the distribution of the annual yield on the irradiation. At the present location approx. 4% of the total yield will be generated in the irradiation range, but 9% of the deviations arise. 6 5 4 3 2 1-25 25-5 Measurement II 5-75 Fig. 5: Deviations of the annual yield 75-1 Irradiation in W/m² I III >1 2.2 Validation of Indoor Measurements Indoor measurements from German test institutes (Fraunhofer ISE Freiburg, PI Berlin, TÜV Rheinland) were used for the validation of further modules. At a constant temperature of 25 C, the irradiation was increased gradually from 1 W/m² to 1 W/m² and the module measured electrically. For this study the simulation results based on the three models were compared with the measured values. While the mapping accuracy of all models is very high, for the currents with errors <3% (short circuit) and <5% (MPP), in most cases over the entire irradiation range, the mapping accuracy of the voltage is considerably less and thus contributes to a decisive degree to the that of the MPP power (fig. 6). Fig. 4: Relative mean bias error (rmbe) of the simulation results to the measurement data As with the validation of the outdoor measurement data, here model III, with an additional part-load point, also reaches the best approximation to the measured values. In addition, it is shown that the two-diode model also achieves very accurate approximations to the measured values in respect of 3
performance (fig. 7). Both for the model with additional measurement point in partload range and for the two-diode model, the deviations in the determination of the MPP power are thus within the measurement tolerances, which are generally given as ±3% for STC performance for this type of flasher measurement and increase further in the direction of lower irradiation. 3. DIFFERENCES FROM THE ONE-DIODE MODEL In yield simulation software tools the one-diode model is often used. But, in the case of using the two-diode model, differences can indeed arise. In the measurements described in 2.1, modules with microand monocrystalline silicon as cell material were studied. In order to eliminate differences due to the module tolerances, the data sheet information for the model calculations was checked and adjusted. The MPP performance deviations from the measured values for both model calculations are shown in Table 3. Fig. 6: Percentaged deviations of open circuit and MPP voltage TABLE 3: DEVIATIONS ONE-DIODE AND TWO-DIODE MODEL Deviation Micro- Crystalline Silicon Mono- Crystalline Silicon One-Diode Two-Diode rmbe rrmse rmbe rrmse.88.41.126.152.36.17.71.14 The analysis of the results (see fig. 5 and 7) showed that, for the most part, the deviations from the measured values result from the measured values in the part-load range. Overall, these results support the conclusion that the use of the two-diode model compared to the one-diode model leads to more accurate simulation results. This is particularly true for higher degrees of latitude, since there the importance of the part-load range is particularly great. 4. CONCLUSION Fig. 7: Percentaged deviations of MPP power and fill factor It was shown with which assumptions the parameter determination for the two-diode model of data sheet information is possible. The results were validated against measured values from both indoor and outdoor measurements. With the calculated parameters, the two-diode model shows a 4
high level of mapping accuracy for silicon-based crystalline PV modules of different technologies over a wide range of temperatures and irradiation. The differences between the two-diode model and measurement are in the range of measurement uncertainty of flashers and outdoor test systems. The two-diode model based on the data for standard test conditions shows, particularly in the part-load range, a better agreement with the actual module behavior than the one-diode model. However, in order to further increase the mapping accuracy in the part-load range, it is necessary to specify at least one measurement value in the part-load range for the parameterization of the model. 6. REFERENCES (1) Elies, Hermle, Burger: Neue Mathematische le für Solarzellenkennlinien; Proceedings of the 24. PV Symposiums, OTTI, 29. (2) Varshni: Temperature dependance of the energy gap in semiconductors; S. 149-154 in Physica 34, Elsevier Science B.V.; 1967. (3) Quaschning: Regenerative Energiesysteme; Hanser Verlag, 29. (4) Zinßer: Jahresenergieerträge unterschiedlicher Photovoltaik-Technologien bei verschiedenen klimatischen Bedingungen; dissertation, IPE Universität Stuttgart, 21. 5