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Output characteristics of PV module considering partially reverse biased conditions
Available online at www.sciencedirect.com
Solar Energy 92 (2013) 214–220 www.elsevier.com/locate/solener
Output characteristics of PV module considering partially reverse biased conditions Tae Hee Jung a, Jae Woo Ko b, Gi Hwan Kang a, Hyung Keun Ahn b,⇑ a
Photovoltaic Research Group, Korea Institute of Energy Research (KIER), 71-2, Jang-Dong, Yuseong-Gu, Daejeon 305-343, South Korea b Department of Electrical Engineering, Konkuk University, 1 Hwayang-Dong, Gwangjin-Gu, Seoul 143-701, South Korea Received 19 September 2012; received in revised form 26 January 2013; accepted 24 March 2013 Available online 21 April 2013 Communicated by: Associate Editor Igor Tyukhov
Abstract In this paper, a mathematical model for the output characteristics of a photovoltaic (PV) module including shaded solar cells in series is proposed. The proposed model was developed using a general one diode solar cell model and is useful for predicting the output of a partially reverse biased module. To obtain an appropriate model for output of a PV module under partially shaded conditions, we applied three key factors to the one diode solar cell model. First, the equation for the avalanche effect is altered in order to include a parameter to relate the number of shaded solar cells to the number of non-shaded solar cells. Second, a resistance component is added in order to express the variable current originating from the avalanche effect depending on the different resistances in the PV module. Third, a compensational voltage source is included to reduce the difference in current near open circuit voltage. A special PV module that can easily change the number of solar cells in series was fabricated to validate the proposed model. By comparing the theoretical results with the measured data, we confirmed that the newly proposed equation is valid showing a root mean square error (RMSE) of less than 2.37%. Ó 2013 Elsevier Ltd. All rights reserved. Keywords: PV module; Partial shading; Avalanche effect; Mathematical model; Reverse bias
1. Introduction The low output of partially shaded solar cells in a photovoltaic (PV) module leads to the operation of a bypass diode and results in the reduction of maximum power in the PV module. This is attributed to changing generation into reverse bias state for shaded solar cell. The reverse biased condition of shaded solar cells leads to power dissipation resulting in an increase in the surface temperature, which, for example, results in the evolution of hot spots. If the reverse bias cross the shaded solar cells is not limited by a bypass diode, the solar cell will be irreversibly damaged. Pre⇑ Corresponding author. Tel.: +82 2 450 3481; fax: +82 2 447 9186.
E-mail address: [email protected] (H.K. Ahn). 0038-092X/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.solener.2013.03.015
vious literature has attributed hot spots in solar cells to the relationship of the operation of a bypass diode and the reverse biased condition of the solar cells (Herrmann et al., 1997; Yang et al., 2012). The different maximum temperature in crystalline solar cell including hot spot is empirically analyzed by the relationships with the number of solar cells per string, shading rate and shading location in PV module (Zhang and Li, 2012). Many models have been developed to analyze the electrical output of solar cell, attributed to the reverse bias of the partially shaded solar cells. According to Bishop’s theory, a current voltage (I–V) curve of the solar cell in both the forward and the reverse bias regions is described by multiplying the current through the shunt resistance and non-linear avalanche multiplication factor (Bishop, 1998). Quaschning and Hanitsch’s theory includes
T.H. Jung et al. / Solar Energy 92 (2013) 214–220
215
Nomenclature I
total current of partially reverse biased PV module (A) Iph photocurrent of solar cell without shadow (A) Iph(shade) photocurrent of solar cell with partially shadow (A) I0 reverse saturation current of solar cell (A) I1 current flowing through reverse biased solar cell by avalanche effect (A) I2 current flowing through reverse biased solar cell by the series and the shunt resistance (A) I3 current flowing through reverse biased solar cell by potential difference between shading and low irradiation conditions (A) M number of partially shaded solar cells in PV module (ea) n ideality factor N number of solar cells in PV module (ea) q electric charge (C) Rs series resistance of PV module (X) Rsh shunt resistance of PV module (X)
an extension term for the avalanche effect in order to express the characteristic curve of solar cells in the negative breakdown region (Quaschning and Hanitsch, 1996). However, their theories are used to estimate the electrical characteristics for only one solar cell and would be very hard calculated to predict the output power of a partially reverse biased PV module or array (Karatepe et al., 2007). To account for a PV module, simulation tools such as Matlab (Silvestre and Chouder, 2008; Mathur et al., 2010) and PSPICE (Silvestre et al., 2009; Paraskevadaki and Papathanassiou, 2011) have been utilized to easily predict the output of a PV module with reverse biased solar cells based on those models using the reverse biased characteristic of one solar cell. In some literatures, one diode models would be modified easily to calculate output of the partially shaded PV module. However, the output characteristic of PV module in reverse bias region would be difficult to describe accurately due to an absence of the equation about the output characteristic of partially reverse biased PV module (Guo et al., 2011; Ishaque et al., 2011). Sample diode model including reverse biased mode constituted by a diode in the opposite direction and reverse biased breakdown voltage in series is released to quickly simulate approximate output of partially reverse biased PV module (Zegaoui et al., 2011). Some literatures neglect the reverse biased effects of shaded solar cells in the PV module or array due to the bypass diode operation (Li and Zheng, 2011; Moballegh and Jiang, 2011). Therefore, it is difficult to predict the exact electrical output for a partially reverse biased PV module. It is necessary for the mathematical model for output of partially reverse biased PV module based on one diode model easily to predict the output of the module by consid-
T V Vb Voc V1,V2
absolute temperature (K) total output voltage of partially reverse biased PV module (V) breakdown voltage of solar cell (V) open circuit voltage of solar cell (V) output voltages of the solar cell under different irradiation conditions (V)
Greek letters a, b constant for related current due to avalanche effect of shaded solar cell c constant for related current of resistance in PV module k Boltzmann constant (J/K) Subscripts MPP maximum power point RMSE maximum root mean square error RME relative mean error
ering PV module conditions, such as the number of shaded solar cells, the number of solar cells in the module, the shading ratio, and the breakdown voltage of solar cells. In this paper, we propose a versatile model considering various module conditions. This model was derived from the one diode model of the PV module. Three factors to reflect the actual electrical output of the partially reverse biased PV module were added to the one diode model. First, the number of shaded solar cells and the total number of solar cells in the PV module are applied to the model to express the avalanche effect of the partially reverse biased PV module. Second, an additional resistance component is considered to reflect the variations of the reverse-biased current. Third, the current simulated using the one diode model is gradually reduced near the open circuit voltage; however, the measurement data from the partial reverse biased PV module is dramatically reduced near the open circuit voltage. The difference could be reduced by adding a dependant voltage source which could be done by connecting a diode of the one diode model in series. This model is validated by comparing the Matlab simulation results with the experimental data obtained from the partially shaded PV module. 2. Mathematical model 2.1. Avalanche effect of shaded solar cells Generally, a one diode or a two diode model is used as an equivalent circuit of a PV module. In the two diode model, the first diode accounts for the electrical characteristics of the solar cells in the PV module. The second diode
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includes the recombination of the carriers in the PV module. The effect for the recombination of carriers in the PV module and not in the solar cells was so small that it could be ignored. Therefore, the one diode model will be used in this study. Many authors’ theories are based on the avalanche effect, which expresses the current voltage characteristics of the forward and reverse bias of only one solar cell. It is not easy to show the various I–V characteristics for the generated conditions such as the total number of solar cells in a module, the number of shaded solar cells in the actual PV module, resistance, and the shading ratio. In this paper, the current (I1) flowing through reverse biased solar cell by avalanche effect is proposed by applying the generated conditions to the general avalanche equation. I1 is expressed as follows: ðI ph I phðshadeÞ Þ ðN MÞV oc ðV þ NIRs Þ I1 ¼ a MðRs þ Rsh Þ I ph b MV b V b ðN MÞV oc þ V þ NIRs
2.3. Potential difference between shading and low irradiation conditions Additional current flows through the partially reverse biased solar cell in the PV module under the photocurrent region, with the exception of the photocurrent from PV modules with entire low irradiation. The current measurement shows dramatic changes near the open circuit voltage when compared to the PV module under low irradiation. To express this output characteristics of PV module by using one diode model. The supplementary current is added to the theoretical current obtained from the one diode model. The total current under the photocurrent region is defined by I3 in Eq. (3). qðV =N þ IRs þ V c Þ ðV =N þ IRs Þ I 3 ¼ I ph I 0 exp 1 nkT Rsh ð3Þ
ð1Þ
The number of solar cells in the PV module in series and the number of shaded solar cells are represented by N and M, respectively. The photo current in the non-shaded solar cells (irradiance of 1 kW/m2) is represented by Iph, and Iph(shade) is the photo current for a different shading ratio. The breakdown voltage and open circuit voltage of a solar cell in the module are Vb and Voc, respectively. The values for a and b are constants. Term Iph Iph(shade)/Iph in Eq. (1) is included to express the different slopes in the I–V characteristics according to the variable ratio of the partially shaded PV module. 2.2. Resistance effect in PV module Resistance in the PV module is variable because the solar cell is wider than the cross section of the general diodes, and the solar cells are produced using diverse methods such as single crystalline, polycrystalline, and thin film. In particular, the resistance of the module continually increases after the installation of the solar cells in the field. Therefore, the variable resistance in the PV module has an effect on the non-avalanche current flowing through the reverse biased solar cells. The various current (I2) produced by the series and the shunt resistance are described in the following equation: ðI ph I phðshadeÞ Þ ðN MÞV oc V NIRs I2 ¼ c ð2Þ I ph MðRsh þ Rs Þ In this equation, Rs and Rsh denote the average series and shunt resistance of a solar cell inside the PV module, respectively, and c is a constant. Term Iph Iph(shade)/Iph is included to express the different slopes in the I–V graph according to the variable ratio of the partially shaded PV module.
where Vc denotes the potential difference at an identical photocurrent point between the non-shaded (irradiance of 1 kW/m2) and low irradiation conditions. qðV 1 =N þ IRs Þ ðV 1 =N þ IRs Þ I ¼ I ph I 0 exp 1 nkT Rsh qðV 1 =N þ IRs Þ ffi I ph I 0 exp ð4Þ nkT qðV 2 =N þ IRs Þ ðV 2 =N þ IRs Þ 1 I ¼ I phðshadeÞ I 0 exp nkT Rsh qðV 2 =N þ IRs Þ ffi I phðshadeÞ I 0 exp ð5Þ nkT The current equations could be approximately expressed as Eqs. (4) and (5) for the conditions stated above. The current due to large shunt resistance can be ignored, as discussed previously. The respective voltages at identical photocurrent points of the solar cell under different shading ratios are represented by V1 and V2, respectively. The voltage difference (Vc) can be expressed as Vc = V1 V2 and it is rearranged as shown in the following equation: Vc ¼
nkT I ph I ln q I phðshadeÞ I
ð6Þ
Substitution of Eq. (6) into Eq. (3) gives Eq. (7); by securing an absolute value, the equation can be expanded to include voltages higher than the open circuit voltage. I phðshadeÞ I exp qðV =N þ IRs Þ 1 I 3 ¼ I ph I 0 nkT I ph I ðV =N þ IRs Þ ð7Þ Rsh The sudden variation in the photocurrent near the open circuit voltage due to the application of a potential difference between different shading ratios of solar cells in the one diode model is expressed by Eq. (7).
T.H. Jung et al. / Solar Energy 92 (2013) 214–220
2.4. Proposed model The total current obtained from the sum of I1, I2, and I3 can be written as shown in the following equation: h i I I qðV =NþIRs Þ ðV =NþIRs Þ 1 I ¼ I ph I 0 phðshadeÞ exp nkT Rsh I ph I h ih ib ðI I Þ ðNMÞV oc ðVþNIRs Þ MV b þa ph IphðshadeÞ V b ðNMÞV oc þVþNIRs MðRs þRsh Þ ph h i ðI ph I phðshadeÞ Þ ðNMÞV oc ðVþNIRs Þ þc I ph MðRsh þRs Þ ð8Þ A proposed equivalent circuit, as shown in Fig. 1, can be represented by the one diode model, which includes Vc, R(M), Vm(N, M), and S(N, M), as defined by the following equations. RðMÞ ¼ MðRs þ Rsh Þ
ð9Þ
V m ðN ; MÞ ¼ ðN MÞV oc
SðN ; MÞ ¼ a
ð10Þ b
ðI ph I phðshadeÞ Þ MV b I ph V b ðN MÞV oc þ V þ NIRs
þc
ðI ph I phðshadeÞ Þ I ph ð11Þ
3. Experiment and simulation The procedures of measurements and simulations in this study are showed to validate the proposed equation and the output of a partially reverse biased PV module, as shown in Fig. 2. Fig. 3 shows a PV module fabricated with identical thirty polycrystalline solar cells that have 16.8% efficiency and contains individual external connections between the solar cells. It would be possible to reduce the measurement data error by changing the number of measuring solar cells due to the usage of one PV module. The tape of paper material is used to intentionally protect light exposure to the designed parts of PV module as shown in Fig. 4. The I–V curves of the whole PV module are measured by using Nisshinbo PV module simulator
217
having Xenon light source with a maximum power accuracy of ±5%. Fig. 2 shows block diagram of the experimental method under the diverse conditions. First, the I–V curves of PV module under the different shaded areas for one solar cell in PV module conditions are measured to confirm the variable outputs of the whole PV module according to avalanche effect of solar cell. Next, three different numbers, 22, 26 and 30, of solar cells are selected to obtain their module performances. All the measurements used one fixed solar cell having half shaded area. The outputs of the whole PV module are also measured by including 1, 2 and 3 solar cells with shading ratio of 33% for each cell. On the other hand, different values of resistors (0.5, 0.25 and 0 X) are connected in series to the external electrode of the module to trace the avalanche effect. Last, one and 30 solar cells, shaded by one third of each, are used in the module (5 6 cells) respectively to find the effect of potential differences near open circuit voltage of the module. The proposed Eq. (8) was solved by using MATLAB and it was validated by comparing simulation data with experiment results. The values shown in Table 1 were used to simulate Eq. (8). 4. Results and discussion The simulation results obtained with Eq. (8) were compared to the experimental data shown in Fig. 5 by increasing the shading ratio of 1/3 for one non-shaded solar cell in the PV module. The simulation results were obtained using a different photo current Iph in Eq. (8). It was proved that the output of a PV module with different shading can be predicted by the proposed Eq. (8). The maximum root mean square error (RMSE) between the simulated and measured currents for the PV module with a completely shaded solar cell is approximately 2.37%. Fig. 6 shows the output of module with a half shaded solar cell that was fabricated with 30, 26, and 22 individual solar cells. The short circuit current of the PV module increases with the number of total solar cells. This is attributed to the voltage reaching the breakdown point of the shaded solar cells. However, although the number of total
Fig. 1. Proposed equivalent circuit.
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Fig. 2. The block diagram for measurement and simulation method.
Fig. 3. PV module structure with an individual external plug between two adjacent solar cells.
Fig. 4. PV module including partially shaded solar cell.
solar cells in the module is the same, the increase in the number of shaded solar cells brings about reducing the
short circuit current of module, as illustrated in Fig. 7. The current obtained when more than two solar cells are
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Table 1 Simulation parameters for PV module. Parameter
Value
Reverse saturation current Ideality factor Temperature of PV module Boltzmann’s constant Series resistance Shunt resistance Open circuit voltage Photocurrent (irradiance of 1 kW/m2) a b c Breakdown voltage
1.9 108 (A) 1 300 (K) 1.39 1023 (J/K) 6.0 103 (X) 10 (X) 0.6 (V) 8.33 (A) 3.5 103 3 3.0 103 20.5 (V)
Fig. 7. I–V curves of PV modules with different numbers of shaded solar cells.
Fig. 5. I–V curves of a PV module with a variety of shading ratios of one solar cell.
Fig. 6. I–V curves of PV modules with different numbers of solar cells.
shaded is almost identical to the current of module with two shaded solar cells, because the generated voltage of the non-shaded module is lower than the breakdown voltage (51 V) of two shaded solar cells. Eq. (8) is used to validate these module conditions. The simulation results
of the proposed equation closely match the measured output under the various module conditions such as the number of shaded solar cells and the number of total solar cells in the PV module. The different outputs of the PV module attributed to the variation in the series and shunt resistance are shown in Fig. 8. In this experiment, the output of the PV module was measured using the different values of the series resistance by adding different resistances to the module. The measured short circuit current increases with a decrease in the series resistance, and it is illustrated in Fig. 8. The simulated results are obtained by using Eq. (8) according to only the variable series resistance (Rs) under the assumption that each solar cell has an identical shunt resistance. The simulation results obtained with Eq. (8) were compared with the measured data of the PV module with additional series resistance variations. The results show that the proposed model could predict the reduced avalanche effect by increasing the series resistance. The output of the PV module having shading ratio of 1/3 for one solar cell is shown in Fig. 9. Simulation results of the module under non-shaded and overall low irradiation conditions could also be obtained by using the general one diode model in Eq. (7), without the supplementary current parameter (Vc). The experimental results obtained by the module with one shaded solar cell are distinctly different from the calculated results of a general one diode model under overall low irradiation conditions. Specifically, the current of the PV module with the shaded solar cell decreased rapidly near the open circuit voltage, and the current calculated by using the general one diode model for overall low irradiated module was less than that of the partially shaded module. The modified one diode model as Eq. (8) could express sharp decrease in the current for partially shaded PV module near open circuit voltage. The simulation results for PV module with one shaded solar cell were obtained using Eq. (8) that included the potential difference between a partially shaded and a fully
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by the proposed one diode model considering a potential difference between the partially shaded PV module and the overall low irradiated PV module. This model was validated by comparing the simulation results with the measured data acquired from a partially shaded PV module. Acknowledgments This work was supported by the New & Renewable Energy of the Korea Institute of Energy Technology Evaluation and Planning (KETEP) Grant funded by the Korea government Ministry of Knowledge Economy (No. 2009T100100579) (No. 20113010010010-11-1-000). Fig. 8. I–V curves of PV module with variable series resistances.
Fig. 9. I–V curves of PV module under low irradiation and partially shaded conditions.
low irradiated PV module. The relative mean error (RME) of the short circuit current and the maximum power point (MPP) of the PV module with a shaded solar cell are 8.4% and 1.33%, respectively. These results prove that the effect of the potential difference, as expressed in Eq. (8), provides good agreement with the data obtained experimentally. 5. Conclusion The various outputs of a PV module with a partially reverse biased solar cell were determined using the shading method. A newly proposed equation for the reverse bias effect in a PV module contains three factors accurately to predict the output characteristics of partial shaded PV module. First, the new equation could reflect the module conditions such as the number of shaded and non-shaded solar cells. Second, the proposed model could express the output of the PV module with variable resistances. Third, the rapidly declining current of the partially shaded PV module near an open circuit voltage could be expressed
References Bishop, J.W., 1998. Computer simulation of the effects of electrical mismatches in photovoltaic cell interconnection circuits. Solar Cells 25, 73–89. Guo, S., Walsh, T.M., Armin, G.A., Peters, M., 2011. Analysing partial shading of PV modules by circuit modelling. In: Proceedings of IEEE Photovoltaic Specialists Conference, 2012, pp. 2957–2960. Herrmann, W., Wiesner, W., Vaaß, W., 1997. Hot spots investigations on PV modules-new concepts for a test standard and consequences for module design with respect to by-pass diodes. In: Proceedings of IEEE Photovoltaic Specialists Conference, pp. 1129–1132. Ishaque, K., Salam, Z., Syafaruddin, 2011. A comprehensive MATLAB Simulink PV system simulator with partial shading capability based on two-diode model. Solar Energy 85, 2217–2227. Karatepe, E., Boztepe, M., Colak, M., 2007. Development of a suitable model for characterizing photovoltaic arrays with shaded solar cells. Solar Energy 81, 977–992. Li, S., Zheng, H., 2011. Energy extraction characteristic study of solar photovoltaic cell and modules. In: Proceedings of Power and Energy Society General Meeting, 2011, pp. 1–7. Mathur, B., Santhosh, K., Sathyanarayanan, S., 2010. MALAB based modelling of SPVA characterization under reverse bias condition. Proceedings of Emerging Trends in Engineering and Technology 2010, 334–339. Moballegh, S., Jiang, J., 2011. Partial shading modeling of photovoltaic system with experimental validations. In: Proceedings of Power and Energy Society General Meeting, 2011, pp. 1–9. Paraskevadaki, E.V., Papathanassiou, S.A., 2011. Evaluation of MPP voltage and power of mc-Si PV modules in partial shading conditions. IEEE Transactions on Energy Conversion 26, 923–932. Quaschning, V., Hanitsch, R., 1996. Numerical simulation of current– voltage characteristics of photovoltaic systems with shaded solar cells. Solar Energy 56, 513–520. Silvestre, S., Chouder, A., 2008. Effects of shadowing on photovoltaic module performance. Progress in Photovoltaics: Research Applications 16, 141–149. Silvestre, S., Boronat, A., Chouder, A., 2009. Study of bypass diodes configuration on PV modules. Applied Energy 86, 1632–1640. Yang, H., Wang, H., Wang, M., 2012. Investigation of the relationship between reverse current of crystalline silicon solar cells and conduction of bypass diode. International Journal of Photoenergy, 1–5. Zegaoui, A., Petit, P., Aillerie, M., Sawicki, J.P., Belarbi, A.W., Krachai, M.D., Charles, J.P., 2011. Photovoltaic cell/panel/array characterizations and modeling considering both reverse and direct modes. Energy Procedia 6, 695–703. Zhang, Q., Li, Q., 2012. Temperature and reverse voltage across a partially shaded Si PV cell under hot spot test condition. In: Proceedings of IEEE Photovoltaic Specialists Conference, 2012, pp. 1344–1347.
Solar Energy 92 (2013) 214–220 www.elsevier.com/locate/solener
Output characteristics of PV module considering partially reverse biased conditions Tae Hee Jung a, Jae Woo Ko b, Gi Hwan Kang a, Hyung Keun Ahn b,⇑ a
Photovoltaic Research Group, Korea Institute of Energy Research (KIER), 71-2, Jang-Dong, Yuseong-Gu, Daejeon 305-343, South Korea b Department of Electrical Engineering, Konkuk University, 1 Hwayang-Dong, Gwangjin-Gu, Seoul 143-701, South Korea Received 19 September 2012; received in revised form 26 January 2013; accepted 24 March 2013 Available online 21 April 2013 Communicated by: Associate Editor Igor Tyukhov
Abstract In this paper, a mathematical model for the output characteristics of a photovoltaic (PV) module including shaded solar cells in series is proposed. The proposed model was developed using a general one diode solar cell model and is useful for predicting the output of a partially reverse biased module. To obtain an appropriate model for output of a PV module under partially shaded conditions, we applied three key factors to the one diode solar cell model. First, the equation for the avalanche effect is altered in order to include a parameter to relate the number of shaded solar cells to the number of non-shaded solar cells. Second, a resistance component is added in order to express the variable current originating from the avalanche effect depending on the different resistances in the PV module. Third, a compensational voltage source is included to reduce the difference in current near open circuit voltage. A special PV module that can easily change the number of solar cells in series was fabricated to validate the proposed model. By comparing the theoretical results with the measured data, we confirmed that the newly proposed equation is valid showing a root mean square error (RMSE) of less than 2.37%. Ó 2013 Elsevier Ltd. All rights reserved. Keywords: PV module; Partial shading; Avalanche effect; Mathematical model; Reverse bias
1. Introduction The low output of partially shaded solar cells in a photovoltaic (PV) module leads to the operation of a bypass diode and results in the reduction of maximum power in the PV module. This is attributed to changing generation into reverse bias state for shaded solar cell. The reverse biased condition of shaded solar cells leads to power dissipation resulting in an increase in the surface temperature, which, for example, results in the evolution of hot spots. If the reverse bias cross the shaded solar cells is not limited by a bypass diode, the solar cell will be irreversibly damaged. Pre⇑ Corresponding author. Tel.: +82 2 450 3481; fax: +82 2 447 9186.
E-mail address: [email protected] (H.K. Ahn). 0038-092X/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.solener.2013.03.015
vious literature has attributed hot spots in solar cells to the relationship of the operation of a bypass diode and the reverse biased condition of the solar cells (Herrmann et al., 1997; Yang et al., 2012). The different maximum temperature in crystalline solar cell including hot spot is empirically analyzed by the relationships with the number of solar cells per string, shading rate and shading location in PV module (Zhang and Li, 2012). Many models have been developed to analyze the electrical output of solar cell, attributed to the reverse bias of the partially shaded solar cells. According to Bishop’s theory, a current voltage (I–V) curve of the solar cell in both the forward and the reverse bias regions is described by multiplying the current through the shunt resistance and non-linear avalanche multiplication factor (Bishop, 1998). Quaschning and Hanitsch’s theory includes
T.H. Jung et al. / Solar Energy 92 (2013) 214–220
215
Nomenclature I
total current of partially reverse biased PV module (A) Iph photocurrent of solar cell without shadow (A) Iph(shade) photocurrent of solar cell with partially shadow (A) I0 reverse saturation current of solar cell (A) I1 current flowing through reverse biased solar cell by avalanche effect (A) I2 current flowing through reverse biased solar cell by the series and the shunt resistance (A) I3 current flowing through reverse biased solar cell by potential difference between shading and low irradiation conditions (A) M number of partially shaded solar cells in PV module (ea) n ideality factor N number of solar cells in PV module (ea) q electric charge (C) Rs series resistance of PV module (X) Rsh shunt resistance of PV module (X)
an extension term for the avalanche effect in order to express the characteristic curve of solar cells in the negative breakdown region (Quaschning and Hanitsch, 1996). However, their theories are used to estimate the electrical characteristics for only one solar cell and would be very hard calculated to predict the output power of a partially reverse biased PV module or array (Karatepe et al., 2007). To account for a PV module, simulation tools such as Matlab (Silvestre and Chouder, 2008; Mathur et al., 2010) and PSPICE (Silvestre et al., 2009; Paraskevadaki and Papathanassiou, 2011) have been utilized to easily predict the output of a PV module with reverse biased solar cells based on those models using the reverse biased characteristic of one solar cell. In some literatures, one diode models would be modified easily to calculate output of the partially shaded PV module. However, the output characteristic of PV module in reverse bias region would be difficult to describe accurately due to an absence of the equation about the output characteristic of partially reverse biased PV module (Guo et al., 2011; Ishaque et al., 2011). Sample diode model including reverse biased mode constituted by a diode in the opposite direction and reverse biased breakdown voltage in series is released to quickly simulate approximate output of partially reverse biased PV module (Zegaoui et al., 2011). Some literatures neglect the reverse biased effects of shaded solar cells in the PV module or array due to the bypass diode operation (Li and Zheng, 2011; Moballegh and Jiang, 2011). Therefore, it is difficult to predict the exact electrical output for a partially reverse biased PV module. It is necessary for the mathematical model for output of partially reverse biased PV module based on one diode model easily to predict the output of the module by consid-
T V Vb Voc V1,V2
absolute temperature (K) total output voltage of partially reverse biased PV module (V) breakdown voltage of solar cell (V) open circuit voltage of solar cell (V) output voltages of the solar cell under different irradiation conditions (V)
Greek letters a, b constant for related current due to avalanche effect of shaded solar cell c constant for related current of resistance in PV module k Boltzmann constant (J/K) Subscripts MPP maximum power point RMSE maximum root mean square error RME relative mean error
ering PV module conditions, such as the number of shaded solar cells, the number of solar cells in the module, the shading ratio, and the breakdown voltage of solar cells. In this paper, we propose a versatile model considering various module conditions. This model was derived from the one diode model of the PV module. Three factors to reflect the actual electrical output of the partially reverse biased PV module were added to the one diode model. First, the number of shaded solar cells and the total number of solar cells in the PV module are applied to the model to express the avalanche effect of the partially reverse biased PV module. Second, an additional resistance component is considered to reflect the variations of the reverse-biased current. Third, the current simulated using the one diode model is gradually reduced near the open circuit voltage; however, the measurement data from the partial reverse biased PV module is dramatically reduced near the open circuit voltage. The difference could be reduced by adding a dependant voltage source which could be done by connecting a diode of the one diode model in series. This model is validated by comparing the Matlab simulation results with the experimental data obtained from the partially shaded PV module. 2. Mathematical model 2.1. Avalanche effect of shaded solar cells Generally, a one diode or a two diode model is used as an equivalent circuit of a PV module. In the two diode model, the first diode accounts for the electrical characteristics of the solar cells in the PV module. The second diode
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includes the recombination of the carriers in the PV module. The effect for the recombination of carriers in the PV module and not in the solar cells was so small that it could be ignored. Therefore, the one diode model will be used in this study. Many authors’ theories are based on the avalanche effect, which expresses the current voltage characteristics of the forward and reverse bias of only one solar cell. It is not easy to show the various I–V characteristics for the generated conditions such as the total number of solar cells in a module, the number of shaded solar cells in the actual PV module, resistance, and the shading ratio. In this paper, the current (I1) flowing through reverse biased solar cell by avalanche effect is proposed by applying the generated conditions to the general avalanche equation. I1 is expressed as follows: ðI ph I phðshadeÞ Þ ðN MÞV oc ðV þ NIRs Þ I1 ¼ a MðRs þ Rsh Þ I ph b MV b V b ðN MÞV oc þ V þ NIRs
2.3. Potential difference between shading and low irradiation conditions Additional current flows through the partially reverse biased solar cell in the PV module under the photocurrent region, with the exception of the photocurrent from PV modules with entire low irradiation. The current measurement shows dramatic changes near the open circuit voltage when compared to the PV module under low irradiation. To express this output characteristics of PV module by using one diode model. The supplementary current is added to the theoretical current obtained from the one diode model. The total current under the photocurrent region is defined by I3 in Eq. (3). qðV =N þ IRs þ V c Þ ðV =N þ IRs Þ I 3 ¼ I ph I 0 exp 1 nkT Rsh ð3Þ
ð1Þ
The number of solar cells in the PV module in series and the number of shaded solar cells are represented by N and M, respectively. The photo current in the non-shaded solar cells (irradiance of 1 kW/m2) is represented by Iph, and Iph(shade) is the photo current for a different shading ratio. The breakdown voltage and open circuit voltage of a solar cell in the module are Vb and Voc, respectively. The values for a and b are constants. Term Iph Iph(shade)/Iph in Eq. (1) is included to express the different slopes in the I–V characteristics according to the variable ratio of the partially shaded PV module. 2.2. Resistance effect in PV module Resistance in the PV module is variable because the solar cell is wider than the cross section of the general diodes, and the solar cells are produced using diverse methods such as single crystalline, polycrystalline, and thin film. In particular, the resistance of the module continually increases after the installation of the solar cells in the field. Therefore, the variable resistance in the PV module has an effect on the non-avalanche current flowing through the reverse biased solar cells. The various current (I2) produced by the series and the shunt resistance are described in the following equation: ðI ph I phðshadeÞ Þ ðN MÞV oc V NIRs I2 ¼ c ð2Þ I ph MðRsh þ Rs Þ In this equation, Rs and Rsh denote the average series and shunt resistance of a solar cell inside the PV module, respectively, and c is a constant. Term Iph Iph(shade)/Iph is included to express the different slopes in the I–V graph according to the variable ratio of the partially shaded PV module.
where Vc denotes the potential difference at an identical photocurrent point between the non-shaded (irradiance of 1 kW/m2) and low irradiation conditions. qðV 1 =N þ IRs Þ ðV 1 =N þ IRs Þ I ¼ I ph I 0 exp 1 nkT Rsh qðV 1 =N þ IRs Þ ffi I ph I 0 exp ð4Þ nkT qðV 2 =N þ IRs Þ ðV 2 =N þ IRs Þ 1 I ¼ I phðshadeÞ I 0 exp nkT Rsh qðV 2 =N þ IRs Þ ffi I phðshadeÞ I 0 exp ð5Þ nkT The current equations could be approximately expressed as Eqs. (4) and (5) for the conditions stated above. The current due to large shunt resistance can be ignored, as discussed previously. The respective voltages at identical photocurrent points of the solar cell under different shading ratios are represented by V1 and V2, respectively. The voltage difference (Vc) can be expressed as Vc = V1 V2 and it is rearranged as shown in the following equation: Vc ¼
nkT I ph I ln q I phðshadeÞ I
ð6Þ
Substitution of Eq. (6) into Eq. (3) gives Eq. (7); by securing an absolute value, the equation can be expanded to include voltages higher than the open circuit voltage. I phðshadeÞ I exp qðV =N þ IRs Þ 1 I 3 ¼ I ph I 0 nkT I ph I ðV =N þ IRs Þ ð7Þ Rsh The sudden variation in the photocurrent near the open circuit voltage due to the application of a potential difference between different shading ratios of solar cells in the one diode model is expressed by Eq. (7).
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2.4. Proposed model The total current obtained from the sum of I1, I2, and I3 can be written as shown in the following equation: h i I I qðV =NþIRs Þ ðV =NþIRs Þ 1 I ¼ I ph I 0 phðshadeÞ exp nkT Rsh I ph I h ih ib ðI I Þ ðNMÞV oc ðVþNIRs Þ MV b þa ph IphðshadeÞ V b ðNMÞV oc þVþNIRs MðRs þRsh Þ ph h i ðI ph I phðshadeÞ Þ ðNMÞV oc ðVþNIRs Þ þc I ph MðRsh þRs Þ ð8Þ A proposed equivalent circuit, as shown in Fig. 1, can be represented by the one diode model, which includes Vc, R(M), Vm(N, M), and S(N, M), as defined by the following equations. RðMÞ ¼ MðRs þ Rsh Þ
ð9Þ
V m ðN ; MÞ ¼ ðN MÞV oc
SðN ; MÞ ¼ a
ð10Þ b
ðI ph I phðshadeÞ Þ MV b I ph V b ðN MÞV oc þ V þ NIRs
þc
ðI ph I phðshadeÞ Þ I ph ð11Þ
3. Experiment and simulation The procedures of measurements and simulations in this study are showed to validate the proposed equation and the output of a partially reverse biased PV module, as shown in Fig. 2. Fig. 3 shows a PV module fabricated with identical thirty polycrystalline solar cells that have 16.8% efficiency and contains individual external connections between the solar cells. It would be possible to reduce the measurement data error by changing the number of measuring solar cells due to the usage of one PV module. The tape of paper material is used to intentionally protect light exposure to the designed parts of PV module as shown in Fig. 4. The I–V curves of the whole PV module are measured by using Nisshinbo PV module simulator
217
having Xenon light source with a maximum power accuracy of ±5%. Fig. 2 shows block diagram of the experimental method under the diverse conditions. First, the I–V curves of PV module under the different shaded areas for one solar cell in PV module conditions are measured to confirm the variable outputs of the whole PV module according to avalanche effect of solar cell. Next, three different numbers, 22, 26 and 30, of solar cells are selected to obtain their module performances. All the measurements used one fixed solar cell having half shaded area. The outputs of the whole PV module are also measured by including 1, 2 and 3 solar cells with shading ratio of 33% for each cell. On the other hand, different values of resistors (0.5, 0.25 and 0 X) are connected in series to the external electrode of the module to trace the avalanche effect. Last, one and 30 solar cells, shaded by one third of each, are used in the module (5 6 cells) respectively to find the effect of potential differences near open circuit voltage of the module. The proposed Eq. (8) was solved by using MATLAB and it was validated by comparing simulation data with experiment results. The values shown in Table 1 were used to simulate Eq. (8). 4. Results and discussion The simulation results obtained with Eq. (8) were compared to the experimental data shown in Fig. 5 by increasing the shading ratio of 1/3 for one non-shaded solar cell in the PV module. The simulation results were obtained using a different photo current Iph in Eq. (8). It was proved that the output of a PV module with different shading can be predicted by the proposed Eq. (8). The maximum root mean square error (RMSE) between the simulated and measured currents for the PV module with a completely shaded solar cell is approximately 2.37%. Fig. 6 shows the output of module with a half shaded solar cell that was fabricated with 30, 26, and 22 individual solar cells. The short circuit current of the PV module increases with the number of total solar cells. This is attributed to the voltage reaching the breakdown point of the shaded solar cells. However, although the number of total
Fig. 1. Proposed equivalent circuit.
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Fig. 2. The block diagram for measurement and simulation method.
Fig. 3. PV module structure with an individual external plug between two adjacent solar cells.
Fig. 4. PV module including partially shaded solar cell.
solar cells in the module is the same, the increase in the number of shaded solar cells brings about reducing the
short circuit current of module, as illustrated in Fig. 7. The current obtained when more than two solar cells are
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Table 1 Simulation parameters for PV module. Parameter
Value
Reverse saturation current Ideality factor Temperature of PV module Boltzmann’s constant Series resistance Shunt resistance Open circuit voltage Photocurrent (irradiance of 1 kW/m2) a b c Breakdown voltage
1.9 108 (A) 1 300 (K) 1.39 1023 (J/K) 6.0 103 (X) 10 (X) 0.6 (V) 8.33 (A) 3.5 103 3 3.0 103 20.5 (V)
Fig. 7. I–V curves of PV modules with different numbers of shaded solar cells.
Fig. 5. I–V curves of a PV module with a variety of shading ratios of one solar cell.
Fig. 6. I–V curves of PV modules with different numbers of solar cells.
shaded is almost identical to the current of module with two shaded solar cells, because the generated voltage of the non-shaded module is lower than the breakdown voltage (51 V) of two shaded solar cells. Eq. (8) is used to validate these module conditions. The simulation results
of the proposed equation closely match the measured output under the various module conditions such as the number of shaded solar cells and the number of total solar cells in the PV module. The different outputs of the PV module attributed to the variation in the series and shunt resistance are shown in Fig. 8. In this experiment, the output of the PV module was measured using the different values of the series resistance by adding different resistances to the module. The measured short circuit current increases with a decrease in the series resistance, and it is illustrated in Fig. 8. The simulated results are obtained by using Eq. (8) according to only the variable series resistance (Rs) under the assumption that each solar cell has an identical shunt resistance. The simulation results obtained with Eq. (8) were compared with the measured data of the PV module with additional series resistance variations. The results show that the proposed model could predict the reduced avalanche effect by increasing the series resistance. The output of the PV module having shading ratio of 1/3 for one solar cell is shown in Fig. 9. Simulation results of the module under non-shaded and overall low irradiation conditions could also be obtained by using the general one diode model in Eq. (7), without the supplementary current parameter (Vc). The experimental results obtained by the module with one shaded solar cell are distinctly different from the calculated results of a general one diode model under overall low irradiation conditions. Specifically, the current of the PV module with the shaded solar cell decreased rapidly near the open circuit voltage, and the current calculated by using the general one diode model for overall low irradiated module was less than that of the partially shaded module. The modified one diode model as Eq. (8) could express sharp decrease in the current for partially shaded PV module near open circuit voltage. The simulation results for PV module with one shaded solar cell were obtained using Eq. (8) that included the potential difference between a partially shaded and a fully
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by the proposed one diode model considering a potential difference between the partially shaded PV module and the overall low irradiated PV module. This model was validated by comparing the simulation results with the measured data acquired from a partially shaded PV module. Acknowledgments This work was supported by the New & Renewable Energy of the Korea Institute of Energy Technology Evaluation and Planning (KETEP) Grant funded by the Korea government Ministry of Knowledge Economy (No. 2009T100100579) (No. 20113010010010-11-1-000). Fig. 8. I–V curves of PV module with variable series resistances.
Fig. 9. I–V curves of PV module under low irradiation and partially shaded conditions.
low irradiated PV module. The relative mean error (RME) of the short circuit current and the maximum power point (MPP) of the PV module with a shaded solar cell are 8.4% and 1.33%, respectively. These results prove that the effect of the potential difference, as expressed in Eq. (8), provides good agreement with the data obtained experimentally. 5. Conclusion The various outputs of a PV module with a partially reverse biased solar cell were determined using the shading method. A newly proposed equation for the reverse bias effect in a PV module contains three factors accurately to predict the output characteristics of partial shaded PV module. First, the new equation could reflect the module conditions such as the number of shaded and non-shaded solar cells. Second, the proposed model could express the output of the PV module with variable resistances. Third, the rapidly declining current of the partially shaded PV module near an open circuit voltage could be expressed
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