Ece060422.qxd The Journal of Engineering Research Vol.4, No.1 (2007) 75-81 1. Introduction Photovoltaic (PV) systems have been used worldwide for the last three decades. Their early applications were mainly concentrated in remote areas and in uses were other types of energy are either very expensive or not fea- sible. However, with the reduction in their fabrication costs, PV systems have seen a tremendous increase in dif- ferent applications. The maximum efficiency of utiliza- tion of solar panels is obtained at maximum power oper- ating point, which is a function of panel physical charac- teristics and fabrication parameters, solar irradiation, and operating temperature. To design a PV-based system, a model of the panel to be used in simulation prior to imple- mentation is often required. Some PV panel equivalent circuit approximations based on a single-diode model _____________________________________ *Corresponding author’s e-mail: hadj@squ.edu.om have been studied (Ouennoughi, Z and Cheggar, M., 1999; Lee, J.I., Brini, J. and Dimitriadis, C.A., 1998; Araujo, G.L., Sanchez, E. and Marti, M., 1982; Gottschalg, R. et al. 1999; Kaminski, A. et al. 1997). Whereas, the simple models presented give acceptable results only for single crystalline cells. However, for poly- crystalline cells, which are cost effective, the models pre- sented are not accurate enough and hence the extended model of two-diode gives better results (Araujo, G.L., Sanchez, E. and Marti, M., 1982). In addition to this, a precise determination of the internal physical parameters of cells and panels is not always possible and the different errors introduced on these parameters during parameter extraction process induce large errors in the models of solar panels. In this paper, analytical model for equivalent circuit parameters of solar panels and arrays are derived from basic cell models used to build them. The effects of equiv- alent series and shunt resistances on the panel and array Analytical Modelling and Simulation of Photovoltaic Panels and Arrays H. Bourdoucen* and A. Gastli Electrical and Computer Engineering Department, College of Engineering, Sultan Qaboos University, P.O. Box 33, P.C. 123, Al-Khod, Muscat, Oman Received 22 April 2006; accepted 11 September, 2006 Abstract: In this paper, an analytical model for PV panels and arrays based on extracted physical parameters of solar cells is developed. The proposed model has the advantage of simplifying mathematical modelling for different configurations of cells and panels without losing efficiency of PV system operation. The effects of external parameters, mainly temperature and solar irradiance have been considered in the modelling. Due to their critical effects on the operation of the panel, effects of series and shunt resistances were also studied. The developed analytical model has been easily implemented, simulated and validated using both Spice and Matlab packages for different series and parallel configurations of cells and panels. The results obtained with these two programs are in total agreement, which make the proposed model very useful for researchers and designers for quick and accurate sizing of PV panels and arrays. Keywords: Photovoltaic cell, Solar cell, PV system, PV panel, Analytical modelling á«Fƒ° àdƒØdG ±ƒØ°üdGh ìGƒdÓd á«∏«∏ëàdG áLòªædGh IÉcÉÙG »∏£°ü≤dG ∫OÉY ,ø°ShOQƒH êÉM áá°°UUÓÓÿÿGG§°ùÑe »°VÉjQ êPƒ‰ Ω~≤j ¬fG ìÎ≤ŸG êPƒªædG ÉjGõe øeh .á«°ùª°ûdG ÉjÓî∏d á«©«Ñ£dG ⁄É©ŸG ≈∏Y É«æÑe á«Fƒ° àdƒØdG ±ƒØ°üdGh ìGƒdÓd É«∏«∏– ÉLPƒ‰ åëÑdG Gòg Ω~≤j : á°SGQO ” ɪc .»°ùª°ûdG ´É©°T’Gh IQGô◊G á°UÉNh á«LQÉÿG πeGƒ©dG ÒKÉJ QÉÑàY’G ‘ êPƒªædG òNÉjh .»Fƒ° àdƒØdG Ωɶæ∏d 𫨰ûàdG IAÉØc ~≤a ¿h~H áØ∏àıG ìGƒ∏dGh ÉjÓÿG ∫ɵ°T’ ∫ɵ°TG IÉcÉëà ∂dPh èFÉàædG ábO øe ≥≤ëàdGh ádƒ¡°ùH ìÎ≤ŸG êPƒªædG ≥«Ñ£J ”h .á«Fƒ° àdƒØdG ìGƒd’G πªY ≈∏Y áLôM äGÒKÉJ øe É¡d ÉŸ …RGƒàdGh ‹GƒàdG äÉehÉ≤e ÒKÉJ G~L G~«Øe ìÎ≤ŸG êPƒªædG π©éj ɇ ÉeÉ“ á≤aGƒàe Ú›ÉfÈdG èFÉàf âfÉch .ÜÓJÉeh ¢ù«Ñ°S èeGôH ΩõM ΩG~îà°SÉH …RGƒàdGh ‹GƒàdG ≈∏Y á∏°üàe ìGƒd’Gh ÉjÓÿG øe áØ∏àfl .á«Fƒ° àdƒØdG ±ƒØ°üdGh ìGƒdÓd ≥«bOh ™jô°S º«ª°üJ ≈∏Y ∫ƒ°üë∏d Úªª°üŸGh ÚãMÉÑ∏d áá««MMÉÉààØØŸŸGG ääGGOOôôØØŸŸGGGG.á«∏«∏ëàdG áLòªædG ,á«Fƒ° àdƒØdG ìGƒd’G ,á«Fƒ° àdƒØdG º¶ædG ,á«°ùª°ûdG ÉjÓÿG : 76 The Journal of Engineering Research Vol.4, No.1 (2007) 75-81 characteristics are studied as their effects are important for the operation of PV systems. Spice and Matlab are used for model validation and sensitivity evaluation of the series and shunt resistances. The approach presented is valid for single-diode model as well as for two-diode model provided the specific parameters for a given model are considered, as it will be shown in subsequent section of the paper. 2. Some Models for Solar Cells and Panels 2.1 Analytical Model for Panel and Array Solar panels and arrays may be described in terms of a set of electric and optical parameters that represent solar cell properties. This is in addition to the number of cells and panels connected in series and/or in parallel. Most of solar cell models available in the literature represent the solar cell by a single diode in parallel with a current source as shown in 1-a (Ouennoughi, Z. and Cheggar, M., 1999; Lee, J.I., Brini, J. and Dimitriadis, C.A., 1998). However, in order to have a more generalized and accu- rate model, a two-diode equivalent circuit model shown in 1-b has been developed (Araujo, G.L., Sanchez, E. and Marti, M., 1982; Gottschalg, R. et al. 1999; Kaminski, A. et al. 1997). This model consists of an ideal current source, which represents the optical irradiation connected in parallel with two different diodes D1 and D2 and a shunt resistance RSH. All these elements are connected to a series resistance RS. Note that diode D1 models the generated photocurrent in the space charge region, which dominates the total current at low diode voltages whereas diode D2 models the recombination photocurrent outside the space charge region. This photocurrent is more dominant at high diode voltages. The shunt and series resistances are two important parameters in the operation of solar cells. This is because the shunt resistance affects mainly the panel power output and the series resistance affects the efficiency as well as the fill factor (Lee, J.I. et al.; McMahon, T.J. et al.). Therefore, the accuracy in determining these two parame- ters and the knowledge of the different errors involved in their determination is a key point for a better simulation of the panel and array characteristics. Note also that the absolute values of RSH are very important in cells qualifi- cation testing, module performance testing and failure analysis (McMahon, et al.). With reference to Fig. 1-b, for a generalized panel equivalent circuit, the I-V characteristics of solar cells can be expressed in terms of physical and electrical parame- ters as, (1) where, Iph is the total photogenerated current, IL the load current, ID1 and ID2 the equivalent diode currents and ISH the net current through the shunt resistances RSH. For a single solar cell or a single panel that can be also consid- ered as a cell, the currents ID1, ID2 and ISH may be expressed by the following equations (Gottschalg, R. et al. 1999). (2) (3) (4) (5) In the above equations, RS and RSH are the series and shunt resistances respectively, ISD1 and ISD2 are diffusion and saturation currents respectively, n1 and n2 are the dif- fusion and recombination diode ideality factors, k is the Boltzman's constant, q is the electronic charge, T is tem- perature in Kelvin, C0 and C1 are empirical constants mod- eling the temperature and the irradiation dependence, and G is the irradiation in W/m2. Note that in case of a single-diode model shown in Fig. 1-a ID2 must be removed from Eq. (1). A typical connection configuration of cells and panels that form a PV array used for power system applications to feed a resistive load is shown in Fig. 2. For this typical array configuration with m horizontal units and n vertical units, the cells or panels connected in series are numbered as P/Ci1 to P/Cim for a row i, where i varies from 1 to n, whereas, the panels or cells connected in parallel placed in a given column j are numbered as P/C1j to P/Cnj where j varies from 1 to m. Note that P/C represents a cell or panel unit. RSH ID I VISH Load D L LRSIph a) Single-diode model. RSHID1 ID2 I VISH Load D1 D2 L L Iph RS b) Two-Diode model Figure 1. Solar cell models SHI2DI1DIphILI −−−= ( ) ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ −⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ += 1RIV kTn q expII sLL 1 1SD1D ( ) ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ −⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ += 1RIV kTn q expII sLL 2 2SD2D SH sLL SH R RIV I + = ( ) GTCCI 10ph ×+= a) Single-diode model b) Tw-diode model 77 The Journal of Engineering Research Vol.4, No.1 (2007) 75-81 As indicated above and has been reported in the litera- ture, the extracted values of both the series resistance RS and the shunt resistance RSH have significant errors (Jervase, J., Bourdoucen, H. and Al-Lawati, 2001). It is therefore important to take this in consideration and study the effects of these two parameters on the panel and array characteristics. Based on the models developed elsewhere (Araujo, G.L. et al.; Gottschalg, R. et al.; Kaminski, A. et al.) and by taking into account typical changes of RS and RSH from one cell to another, an improved and useful ana- lytical panel model is proposed. The expressions to be used for the model together with Eq. (1) are given in sub- sequent section. Typical figures for errors on the RS and RSH have been suggested and their effects are studied by simulation using Matlab/Simulink and Spice software packages. For demonstrating the validity of the analytical model developed, panels having different number of cells and different sizes have been considered. However, typi- cal results will be shown for panel of 72 cells (6x12) with six cells connected in series and twelve in parallel. Assuming the basic cell parameters similar except for RS and RSH, and making appropriate change of variables on saturation currents and ideality factors, then expres- sions of total currents, ideality factors, equivalent resist- ances RS and RSH can then be formulated using analytical expressions. This is supported by practical considerations where values of these parameters are affected by the way external connections of cells and panels are done. Based on models shown in Fig. 3 for a set of two ideal cells D1 and D2 connected in series and in parallel, one can write a set of equations for each type of configuration: These are parallel, series and combination of parallel-series connec- tions. Parallel connection: with reference to model shown in Fig. 3a, one can write I1 = IS1 [exp (vPq / (n1kT)) - 1] for D1 and D2. Assuming D1 and D2 having equivalent character- istics, the parallel equivalent circuit composed of the two diodes can be represented by current source IT = 2I1 and diode D as shown on the figure. The I-V expression of this circuit can be expressed as: (6) For m cells connected in parallel, saturation current of the equivalent diode is to be multiplied by m and hence the total current becomes: (7) Similar analysis can be done for finding equivalent series and shunt resistances RS and RSH. For the series resistance of a parallel connection of two cells, the current through each resistance is half the total current IT , while the voltage drop stays unchanged. Thus, equivalent series resistance RSE of two cells in parallel circuit equals half the single-cell series resistance RS. Hence, for m cells connected in parallel RSE = Rs / m. The same approach applies for shunt resistances and hence, the equivalent shunt resistance for m cells is RSHE = RSH / m. Series connection: with reference to circuit models of Fig. 3-b, one can write: (8) for D1 and D2. The voltage vs for a single cell can be expressed as: (9) Since the two cells connected in series are assumed to have equivalent characteristics, the voltage across the equivalent cell D is 2vs. One needs to multiply the ideali- ty factor n1 in Eq. (8) and Eq. (9) by 2. For the case of n equivalent cells connected in series, the value of n1 is to be multiplied by n. Thus, the equivalent voltage and cur- rent for a panel of n cells can be expressed as (10) (11) Similar analysis can also be done for finding equiva- lent series and shunt resistances RS and RSH. The equiva- lent series resistance of the two cells equals two times the single cell series resistance. Hence, for n cells connected in series the total series resistance RSE = nRS. For shunt resistance, the same approach applies, and the equivalent shunt resistance RSH for n cells is RSHE = nRSH. Note that the above approach has also been used to deduce the equivalent panel parameters for a two-cell model. Combined series-parallel connections: In case of panel or array with a combined series and parallel connections of many solar cells, a change of variables can be done on the physical parameters of a single cell described by Eqs. (1-5) to determine the model of the panel. Hence, expres- sions of the equivalent parameters can be derived as fol- lows. P/C1 1 P/C1 2 P/C1 3 1 mP/C P/C 2 1 P/C2 2 P/C2 3 2 mP/C P/Cn 1 P/Cn 2 P/Cn 3 P/C Resitive Load IL n m Figure 2. Typical connection of panels and cells forming a PV array that feeds a resistive load. [ ]1))kTn/(qvexp(mImII 1P1S1T −== [ ]1))kTn/(qvexp(II 1S1S1 −= ) I II ln( q kT nv 1S 1S1 1s + = ) I II ln( q kT n.nv 1S 1S1 1T + = [ ]1))kTnn/(qvexp(II 1T1ST −= [ ]1))kTn/(qvexp(I2I 1P1ST −= P/C11 P/C12 P/C13 P/C1m P/C21 P/C22 P/C23 P/C2m P/Cn1 P/Cn2 P/Cn3 P/Cnm 78 The Journal of Engineering Research Vol.4, No.1 (2007) 75-81 (12) Where, n1,2E and ISD1,2E are respectively the ideality factors and saturation currents for the panel equivalent two diode circuit. Thus, Eqs. (2-4) can be extended to panels and arrays and expressed as follows, (13) (14) (15) Note that since RS and RSH vary also with optical irra- diance and ambient temperature, the equivalent array series and shunt resistances RSE and RSHE will also vary with these two parameters. These are considered in simu- lations as shown in the following section. A formulation of this dependency for one single cell can be found in (Veissid, N., De-Andrade, A.M., 1991). To validate the proposed analytical model of panels and arrays to be used for designing PV systems, simulations were done using Matlab and Spice. The following sections describe how the above models were implemented using these two packages. 2.2 Matlab/Simulink Simulation A Matlab/Simulink program was used for to validate the developed solar panel model. Figure 4 shows the solar panel model as it is implemented with Simulink based on previous equations. The inputs to the model are the cell temperature T in oK and the solar irradiation G in W/m2. The output is a voltage that drives a resistive load RL. Note that the solar panel current can be varied by changing the load resistance and both voltage and current at the output of the solar panel can be tracked and meas- ured. The model of the panel was based on the implemen- tation of Eqs. (1), (5) and (12-15) with the assumption that all cells' parameters are interrelated according to expres- sions (12). 2.3 Spice Simulation Circuit models for cells' and panels' configurations shown in Fig. 2 have been implemented by generating models using the model editor in Spice. This has been done for basic solar cells having actual parameters as well as for panels and arrays modeled using the developed ana- lytical approach. The results obtained were compared with the ones obtained using Matlab/Simulink for similar con- figurations. These are presented in a subsequent section. It is worth noting that the analytical model presented above is very useful for simulation and modeling of PV- based systems. It simplifies the mathematical computa- tions and design of solar panels without altering physical parameters that are very well known for solar cells. Hence, the parameters derived even though purely mathe- matical give accurate physical behavior of the PV panel and array. 3. Results Typical parameters of solar cells used to evaluate the developed analytical model are given in (Gottschalg, R. et al. 1999). These are C0=2.19x10-3 A.m2/W, C1=0 A.m2/W/K, G=1000W/m2, T=55oC, n1=0.99, n2=1.9, ISD1=2.4x10-9 A, ISD2=5.5x10-5 A. The series and shunt resistances are kept variable to see their effect on the out- put power of the simulated arrays. However, their values for the basic cell units are RSH = 200 Ω and RS = 2.5x10-2 Ω. I1 vP D IT D2 vP D1 vP I1 a) vS D1 I1 IT D2 I1 vS D 2vS b) Figure 3. Parallel (a) and series (b) connections of cells, and corresponding equivalent circuits used to build analytical model of panels and arrays ∑ = − ∑ = −∑ = − ∑ = − ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ =⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ = n 1j 1m 1i 1 ijP SHE n 1j 1m 1i 1 ijS SE RR,RR 2E21E1 n.nn,n.nn == 2SDI.mE2SDI,1SDI.mE1SDI == ( ) ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ −⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ += 1RIV kTn q expII SELL E1 E1SD2D ( ) ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ −⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ += 1RIV kTn q expII SELL E2 E2SD2D SHE SELL sh R RIV I + = 79 The Journal of Engineering Research Vol.4, No.1 (2007) 75-81 Variations of the output power as a function of the load voltage, for different values of cell and panel equivalent series resistance RS and shunt resistance RSH for the panel of 72 cell configuration are shown in Figs. 5 and 6 respec- tively. The percentage changes in the values of RS (rela- tive to typical value of Rs=2.5x10-2 Ω) used to obtain these characteristics are 50, 100, 150, and 200%. The val- ues of shunt resistance RSH in percentage relative to typi- cal value of RSH = 200 Ω used to obtain these character- istics are also 50, 100, 150, and 200%. Note the significant effect of series resistance fluctua- tions on the output power (refer to Fig. 5). However, the shunt resistance has practically an insignificant effect. Theses significant fluctuations of the output power versus the output voltage occur at low load resistances as they are more affected by RS rather than RSH which is dominated by the two diodes connect to it in parallel. On the other hand, the main parameters that affect the output current and voltage of the modeled panel have been considered. These are the solar irradiation G of the site and the temperature T of the panel. To illustrate their effects, the circuit model of Fig. 4 was simulated with dif- ferent temperatures and solar irradiations using a 72 cell panel. Figure 7 shows the simulation results. Notice that, if the temperature of the panel decreases, the output volt- age increases. For instance, when the temperature of the panel is 20oC and the irradiation is 1000 W/m2 the open circuit voltage (Voc) is about 6.3 V. However, if the tem- perature is decreased to 0oC and the irradiation is kept constant, the open circuit voltage (Voc) becomes equal to about 6.6 V. Notice also that reducing the irradiation will result in decreasing the output voltage of the panel. Note also that, if the irradiation is decreased from 1000 W/m2 to 500 W/m2, the open circuit voltage is decreased from about 6.3 V to 5.8 V at a constant temperature of 20oC. (a) Solar panel simulation block diagram (b) Details of solar panel given in a) above 2 m 1 V+ + - v + - v Iph ISD RSE RSHE u[1]-Is1*(exp(q/n1/K/u[2]*u[3])-1)-Is2*(exp(q/n2/K/u[2]*u[3])-1) ISD Np*(C0+C1*u[2])*u[1] Iph + i- + i- signal 2 T 1 G 27+273.15 T G T V+ m Solar Panel R L 1000 G 74.77 18.69 Figure 4. Solar panel model implemented with Matlab/Simulink Output Voltage, ( ) 0 2.0 4.0 6.0 8.0 O ut pu t P ow er , ( W ) 0 20 40 60 Figure 5. Output power as a function of load voltage for different values of panel equivalent series resistance RS (upper curve correspo- nding to lower % change in Rs and lowest curve to highest change). The values of RS in % of typical value 2.5x10-2 Ω are: 50, 100, 150 and 200 Output Voltage, (V) 80 The Journal of Engineering Research Vol.4, No.1 (2007) 75-81 The results obtained using Spice and Matlab were in very good agreement. Simulation results shown in Figs. 8- a) and 8-b) for IL-VL and PL-VL characteristics confirm this fact. 4. Conclusions An analytical model for PV panels and arrays based on extracted physical parameters of solar cells has been pre- sented in this paper. The proposed approach has the advantage of simplifying mathematical modeling of dif- ferent cells' and panels' configurations without losing nec- essary accuracy of system operation. The effects of tem- perature and solar irradiance have been considered in the modeling. The developed analytical model has been sim- ulated and validated using both Matlab and Spice pack- ages for different cells and panels connected in series and parallel. This makes the proposed model very useful for researchers and systems designers as it allows a quick and accurate sizing of PV panels and arrays. Figure 6. Output power as a function of load voltage for different values of panel equivalent shunt resistsance RSH. The values of RSH in % of typical value 200 Ω are: 50, 100, 150, and 200 Figure 7. Simulated solar panel I-V characteristics for different temperatures and solar irrad- itions Output Voltage (b) P-V characteristics Figure 8. Comparison of I-V and P-V characteristics obtained by Matlab for T = 25oC and G = 1000 W/m2 (72 cells panel) Output Voltage (V) (a) I-V characteristics A sensitivity analysis of the output power with changing RSE and RSHE for a panel of 72 cells has been also carried out. Errors on RSE and RSHE of up t o ±40 % of the nominal values of RSE and RSHE which are respectively 0.05 Ω and 400 Ω . The results of simulations have shown that deviations of output power due to 40% change in RSE and RSHE were less that 7% and 1%, respectively. Output Voltage, (V) Output Voltage, (V) 81 The Journal of Engineering Research Vol.4, No.1 (2007) 75-81 References Araújo, G. L., Sanchez, E. and M. Marti, 1982, "Determination of the Two-Exponential Solar Cell Equation Parameters from Empirical Data," Solar Cells, 5, pp. 199-204. Gottschalg, R., Rommel, M., Infield, D.G. and. Kearney, M.J., 1999, "The Influence of the Measurement Environment on the Accuracy of the Extraction of the Physical Parameters of Solar Cells," Journal Meas. Sci. Technol., 10. pp. 797-804. Jervase, J., Bourdoucen, H. and Al-Lawati, A., 2001, "Solar Cell Parameter Extraction using Genetic Algorithms," Meas. Sci. Technol., 12, pp. 1922-1925. Kaminski, A., Marchand, J. J., Fave, A. and Laugier, A., 1997, "New Method of Parameters Extraction from Dark I-V Curve," 26th PVSC, pp. 203-205. Lee, J. I., Brini, J. and Dimitriadis, C. A., 1998, "Simple Parameter Extraction Method for Non-Ideal Schottky Barrier Diodes," Electronics Letters, 34(12), pp. 1268-1269. McMahon, T. J., Bosso, T. S. and Rummel, S. R., 1996, "Cell Shunt Resistance and Photovoltaic Module Performance," 25th PVSC, pp. 1291-1294. Ouennoughi, Z. and Cheggar, M., 1999, "A Simpler Method for Extracting Solar Cell Parameters using the Conductance Method," Solid-State Electronics, 43, pp. 1985-1988. Veissid, N. and De-Andrade, A. M., 1991, "The I-V Silicon Solar Cell Characteristics Parameters Temperature Dependence, An Experimental Study using the Standard Deviation Method," 10th PSEC, Portugal, pp. 43-47.