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According to the theory of semiconductors, an ideal PV cell can be considered as a current source which is in parallel with one or two diodes depending on the manufacturing technology and semi-conducting materials , . However, in case of representing practical behavior of a PV cell, observations show that the model should include two additional elements: parallel resistance and series resistance . Based on this model, the fundamental equation describing the current-voltage characteristics of a cell is given as:
_ Iph = Photo generated current
_ Is= Saturation current of diode
_ Rs= Cell series resistance
_ Rp= Cell shunt resistance
_ A= Diode quality factor
_ T=Cell operating temperature, Kelvin
_ q= Electronic charge, (1.6x10-19 C)
_ k= Boltzmann's constant, (1.38 x 10-23 J/K)
According to (1), the five parameters Iph, Is, Rs, Rp and A which are regarded hereafter as model parameters, should be known in order to construct the PV model. Model parameters are not usually provided by manufacturers. They can be either determined from the data provided in the datasheet or obtained through experiments. Nevertheless, model parameters do not have constant values as they all vary depending on temperature and solar irradiation levels. In other words, each set of temperature and solar irradiance corresponds to a unique set of model parameters. Therefore, in order to model PV characteristics under various environmental conditions, the effects of solar irradiance and temperature on PV parameters should be taken into account. One approach that has been widely proposed in the literature is to mathematically formulate the relationship of model parameters with temperature and solar irradiance , , . Thus far, several equations have been developed. The most popular and accepted one is the equation derived for the photo -generated current, Iph, which relates it to both temperature and solar irradiance:
where, Iph(n) is the photo-generated current at the standard condition (25_C and 1000 W=m2). T and T0 are the cell actual and standard temperatures (K) respectively. Similarly, E and En are the actual and standard irradiance levels (W=m2) on the PV surface.
Unlike the photo-generated current, the diode saturation current, Is, has been reported to have a non-linear dependence on temperature. Despite its strong dependence on temperature, it is not considerably affected by irradiance changes:
where, Is(0) is the saturation current at the standard temperature and Eg is the band gap energy of semiconductor. Another electrical parameter that can be formulated is Voc:
Voc = Voc(n) + k(T - T0) (4)
In comparison to above equations, there has been less success in presenting consistent equations relating Rs and Rp to the environmental parameters. In  the following equations are presented:
where, k1 - k4 are constants and Rp(n) is the nominal parallel resistance at the standard condition. In many modeling techniques […], Rs and Rp are assumed to be constant under all environmental conditions, i.e. being equal to their nominal values. Such assumption simplifies the modeling technique at the cost of the accuracy. On the other hand, an accurate model is needed to analyze the PV characteristics under partially shaded condition in which PV array is modeled under both non-shaded and shaded conditions. Therefore, a comprehensive modeling of PV parameters is required to represent PV characteristics under various weather conditions.
In order to be worthwhile, the comprehensive modeling of PV should include another crucial stage which is experimental validation. Since the proposed technique in this work has been initially developed based on experimental study, it was able to generate satisfactory results in agreement with experimental results.
Modeling of PV cell based on experimental study
Formulation of model parameters based on environmental parameters
In order to model PV parameters based on environmental parameters, temperature and irradiance level, the above-mentioned equations need to be verified through experimental study. The experiments have been carried out on a single PV module subject to various temperature and uniform solar irradiation levels. By using an algorithm which is described in section III.B, the model parameters corresponding to experimental data have been determined. The model parameters together with temperature and irradiance values have been applied to the equations and ultimately the accuracy of each equation has been investigated. Table I provides eight sets of experimental data including the model parameters determined for each set. Based on the available experimental data, equations (2) and (4) have been validated and the parameters associated to each equation are determined. However, equations (3), (5) and (6) failed to satisfy the experimental conditions. Instead, the equations relating Voc, Impp and Vmpp to temperature and solar irradiance were proved to be effective. Impp and Vmpp are formulated in the same form as Isc and Voc, respectively. In other words, (2) can be applied to Iph, Isc and Impp while (4) can be applied to Voc and Vmpp.
Table 1: Experimental data for various temperature and solar irradiance conditions
Equation (2) can be rewritten in the short form of:
In order to find the coefficient values k0 and k1, two different sets of experimental data Iph, E and T are needed to be substituted in (7). From the available experimental data presented in Table I, all possible combinations of two sets have been attempted to ensure the effectiveness of equation (7) under various conditions. The resulted k1
has the mean value of -0.00247. Using this value for k1 in (7), k0 is determined by solving the equation for each set of data. Table II shows the results for k0. According to Table II, k0 varies in a small range and could hence be approximated by its mean value which is 21.648. The complete form of equation can be written as:
Iph = 21.648 x E(1 - 0.00247 x T) (8)
The similar procedure can be followed for Isc and Impp equations. The results for k0 corresponding to each equation are also shown in Table II. Incorporating the determined coefficients into (7) leads to:
Isc = 4.2348 x E(1 - 0.000015 x T) (9)
Impp = 7.104851 x E(1 - 0.0015 x T) (10)
The equation used for Voc and Vmpp is the simplified form of (4), expressed as:
Voc = Voc(n) + k(T - T0) = k0 + k1T (11)
Like aforementioned equations for currents, the coefficients
k0 and k1 can be determined through substitution of two sets of experimental data Voc and T into (11). This method has been attempted for all possible combinations of two out of eight sets. The resulted mean value of k1 equals -0.09019 which, upon substitution in (11), leads to a single value of k0 for each set of data. The results are shown in Table II. According to Table II, the mean value of k0 equals 93.1442 and hence (11) can be written as:
Voc = 93.1442 - 0.09019 x T (12)
The same procedure can be followed for Vmpp. Consequently, equation (13) can be written for Vmpp:
Vmpp = 118.9274 - 0.2153 x T (13)
Having developed mathematical formulas (9) - (13), the PV electrical parameters can now be calculated for any temperature and solar irradiance levels. The model parameters are now needed to be determined based on these electrical parameters. This can be easily fulfilled through developing an algorithm, capable of determining model parameters from electrical ones; voltages and currents. The algorithm has been used earlier to determine the model parameters associated with each set of experimental data.
Table 2. Coefficient k0 based on experimental study
k0 - ISC
k0 - IMPP
k0 - VMPP
k0 - VOC
Determination of PV model parameters
In this section, the PV model parameters; Iph, Is, Rs, Rp and A are determined based on electrical parameters; Isc, Impp, Vmpp and Voc. The objective is to find the model parameters such that the resulted I-V and P-V curves accurately match the experimental curves, specifically at three key points: short-circuit, maximum power and open circuit. In order to satisfy this criterion, the model parameters are obtained by solving the fundamental equation (1), for the key points. The electrical parameters for a particular weather condition are substituted in these equations. The values for Iph, Is, Rs and Rp are then determined through an iterative procedure. As A is an empirical value which expresses the quality of diode used in PV, an initial value can be chosen in order to obtain other model parameters. Depending on PV device structure and material, A usually varies in the range of 1 to 1.5. In this paper, A is assumed to be equal 1. This value can be modified later if fine tune of the model to the experimental study is needed.
In order to determine the other four parameters, four equations are required. In addition to the equations derived for short circuit, maximum power and open circuit conditions, the forth equation governs that the models maximum power occurs at the same voltage as the measured Vmpp.
The equations for the three key points can be written as:
The forth equation can be derived using the fact that the derivative of power with respect to voltage should be zero at the maximum power point, Vmpp and Impp:
The derivative of power can be written as:
According to (18), the derivative of current with respect to the voltage should be found first:
Solving the above equation for dI/dV and substituting the value in (18) results in:
Substituting Vmpp and Impp in (20) leads to the forth equation:
The iterative procedure is illustrated in Fig.1. As shown, by using (15), Impp(cal) is calculated and compared to its experimental value. Depending on the difference, Rs is slowly incremented or decremented. Based on the new value of Rs and above equations, Rp, Iph and Is can be calculated. These values are substituted in (15) in the next iteration and the same procedure repeats. The iteration continues until the difference between Impp(cal) and Impp reaches zero and all the values become stable. Unlike the algorithm proposed in  in which Rs is incremented starting from zero, in this paper an initial value for Rs has been assumed based on the equation proposed in :
Assuming an initial value for Rs reduces the number of iterations needed to reach the final answer. It also has a great impact on the convergence of the algorithm. The other three parameters; Rp, Iph and Is are set to initial values :
Depending on the software in which the algorithm is implemented, the number of iterations may be required as well. In Matlab, the algorithm reaches the stable final answer with less than 100 iterations, whereas in PSCAD, there is no need to define a number and it automatically reaches the stable answers within less than 0.1 second.