Numerical approach for extraction of photovoltaic generator single-diode model parameters

ABSTRACT


INTRODUCTION
The photovoltaic (PV) generator is the basic element of the photovoltaic system. This element is made of semiconductor material with a p-n junction which has the property of, when exposed to sunlight, producing direct current electricity proportional to the solar irradiation. The behavior of PV generator has a non-linear current-voltage (I-V) characteristic. Various representations have been employed to describe the currentvoltage (I-V) relationship in solar cells. In practice, the single diode model is the commonly equivalent circuit used to describe the I-V relationship. However, regardless of this model, there are important PV parameters that must be accurate and extracted for the simulation, design, performance, evaluation and control of PV systems. The five main parameters that describe the behavior of PV generator models are the generated photocurrent (Iph), the saturation current (I0), the series resistance (Rs), shunt resistance or parallel resistance (Rp) and the diode ideality factor (n). The electric current produced by a solar cell depends on the intensity of the incident light and its properties.
Several methods have been suggested in the literature to extract the five parameters. We can divide these methods into three categories: analytical, numerical and evolutionary methods [1]- [22]. N. Maouhoub  Numerical approach for extraction of photovoltaic generator single-diode model … (Abdelaaziz Benahmida) 59 [15] has proposed an analytical method to determine the five physical parameters. The key points have been used to calculate the series resistance and the ideality factor. The linear least square technique has been used to determine to parallel resistance, the photo current and the saturation current. F. Ghani et al. [16] provided a numerical method to extract the values of the series and shunt resistors using the Newton-Raphson algorithm and the Lambert W-function. A singularity problem, which is divided by zero, can occur if the initial conditions of the parameters are incorrectly chosen. Moreover, these methods require many assumptions to simplify the problem of extracting the five parameters. C. Zhang et al. [17] proposes a simple and efficient numerical method for extracting all the parameters of a solar cell from a single current-voltage (I-V) curve under constant illumination by reducing the parameters to three. The disadvantage of this approach is that it allows several simplifications by eliminating terms of the analytical equation of current (I-V). Hejri et al. [18] proposed a numerical procedure using three key points from manufacturer's data sheets and by solving a system of four nonlinear equations; this method needs four suitable initial guesses. Recently, Stornelli et al. [19] proposed a new simplified method for the iterative estimation of the five PV module parameters. This method uses some approximations to determine the values of Rs and n.Yadir et al. [20] proposed another numerical method based on key points and solving a system of four nonlinear equations. El Achouby et al [21] proposed numerical method to extract the five physical parameters, operating at standard test conditions (STC). This method is based on the variation of the ideality factor and solving a system of four nonlinear equations. The problem of this iterative method is that needs suitable initial values of four physical parameters.
In the present work, we propose a numerical approach to extract the five physical parameters based on the single diode model. In a first step, we formulate three analytical expressions of the parallel resistance Rp, the photocurrent Iph and the saturation current I0 using the linear least square method. These three expressions depend on the diode ideality factor n and the series resistance Rs. Then, we extract these two parameters numerically by solving a nonlinear system of two transcendent equations at two key points, namely, a short circuit point and maximum power point. The proposed method has the advantage over other methods of employing only two key points of the experimental I-V curve and only two initial guesses and does not use any approximation and any computing of the slope at open-circuit voltage or at short-circuit current. To check the precision of our method, we extract the five physical parameters of the commercial RTC PV cell, the PWP201 PV module and the KC200GT PV module and we compare the theoretical current-voltage curve with the experimental one. We organize our paper into four sections. After an introduction, we present in the second section the one-diode equivalent circuit model and the proposed five-parameter extraction methods. In the third section, we use three photovoltaic generators experimental data to validate our proposed method and to compare with other methods; we evaluate the accuracy of the presented method by calculating the absolute error and the root mean square error. Finally, we close our paper with a conclusion.

PROPOSED METHOD 2.1. Single diode model
In our study, we use the equivalent single diode circuit model to describe the behavior of the photovoltaic generator. This circuit is illustrated in the Figure 1. The five parameters of this model are: the shunt resistance Rp, the saturation current I0, the photocurrent Iph, the ideality factor n and the series resistance Rs. -Ns is the number of cells in series in the PV module (Ns = 1 for one solar cell), -Vth = kBT/q is the thermal voltage, -q is the electronic charge, -kB is the Boltzman's constant -T is the temperature in Kelvin.
This (1) is a nonlinear transcendent equation. We can write the current I = f (V) by introducing the Lambert (2).
VI, m is the measured voltage of the PV generator and N is the number of measured points. In order to formulate the analytical expression of Iph, I0 and Rp, we use the linear least squares method based on minimizing the follow objective (5).
Ii, m is the measured current of the PV generator and Ii is the theoretical current given in the (1). After application of this method, we formulate the follow linear system of three equations [15]: Where Gp = 1 / Rp The solution of the previous system, provides three analytical expressions of Iph (n, Rs), I0 (n, Rs) and Rp (n, Rs) as a function of ideality factor n and series resistance Rs.

2.2.2.Numerical extraction of n and Rs
In order to extract the series resistance Rs and the ideality factor n, we use two transcendent equations. By setting (V, I) = (Vmpp, Impp) using (1), we establish a first transcendent equation linking maximum power point to two physical model parameters: Equations (7) and (8) are two implicit and nonlinear equations with two unknown n and Rs. To determine these two values, we solve numerically this system of nonlinear equations via Newton-Raphson method by applying the nonlinear solver "fsolve" in MATLAB environment. To solve this system, suitable values of initial guesses are required. The main advantage of the presented method is that use only two initial guesses and needs only two key points of the experimental curve and does not use any approximation or slope computing. The flowchart in Figure 2 shows the steps involved in carrying out the proposed numerical method.

RESULTS AND DISCUSSION
In order to validate the proposed method, it is examined for three experimental case studies: the Silicon solar cell RTC France at 33°C (case study 1), Silicon Module PWP201 at 45°C (case study 2), and the KC200GT multi-crystal PV module operating under Standard Test Conditions STC (case study 3). To test the accuracy of our method and to measure the degree of precision of our mathematical model defined above for the extraction of the five parameters, two statistical indicators are selected: the Absolute Error (AE) and the Root Mean Square Error (RMSE) estimator defined as (9)  The Table 1 shows the physical characteristics extracted from the experimental I-V curve of the three photovoltaic generators RTC France at 33°C, PWP201 at 45°C and KC200GT at STC. The experimental I-V curves of these two cases are extracted from [1]. For the KC200GT panel, the experimental I-V data are extracted from the datasheet [23] under standard STC test conditions (irradiation level 1000 W /m2, AM spectrum 1.5 and cell temperature 25°C).  [8], [17], [20] are summarized in Table 2. As can be seen, the approach proposed provides a minimum value of (RMSE). The presented method has the advantage of using only two initial guesses, reducing the research space to the two unknown parameters and does not use any approximation and any computing of the slope at open-circuit voltage or at short-circuit current compared to other methods that use four or five initial guesses. Several of these methods determine the values of the slope at open-circuit voltage or at short-circuit current with many approximations.  Figure 3 shows the experimental characteristic I-V and the theoretical curve using the five estimated parameters, for RTC France solar cell. Theoretical curves are determined using the (2) and using the five extracted parameters. As it can be ascertained, the theoretical curves are in very agreement with the experimental measurement. Figure 4 shows the plot of the absolute error for RTC France solar cell compared with different methods. It is distinctly indicated that the proposed method in this work has a small error. The absolute error does not exceed 0.003 A. The method previously studied and validated in study case 1, it is applied to the PWP201 module. The extracted values are compared to the work of [2], [7], [17], and are abredged in Table 3. As can be seen, the proposed numerical method provides a very low RMSE value compared to other methods.  Figure 5 shows the experimental characteristic I-V and the theoretical curve using the five estimated parameters, for PWP201 module. As we can find out, the theoretical curve is in very good concordance with the experimental measurement extracted from the technical data sheet. Figure 6 shows the absolute error plot for the PWP201 module compared to different methods. It is very clearly indicated that the proposed method in this work has a very low error value compared to other methods. The absolute error does not exceed 0.005A.  Table 4. The extracted values are compared with those of other research authors [15], [18], [19]. In Figure 7, we display the absolute  To study the effect of temperature T and irradiance level G on the I-V characteristic, we suppose that the influence of the temperature and irradiation on n, Rs and Rp are negligible [15]. The other two parameters Iph and I0 are given by (11) and (12). Where Iph, STC is the photo-current in STC conditions, GSTC is the global irradiation under STC conditions (GSTC = 1000 W/m²), TSTC is the temperature under STC conditions (TSTC = 25 °C) and Ki is the temperature coefficient of Iph. The temperature and irradiation dependence of Voc is given by (13).
Ki is the temperature coefficient of Voc and Voc, STC is open-circuit voltage in STC conditions. Figure 8 presents the experimental I-V characteristic and the theoretical curve using the five estimated parameters, for the KC200GT PV modules. As can be viewed, the theoretical curve of the above-mentioned PV module is in very good agreement with the experimental measurement extracted from the data sheet under the conditions of constant temperature T=25°C and at different irradiation levels. In Figure 9, we plot the experimental I-V characteristic and the theoretical curve, for the KC200GT PV module for fixed irradiation 1000W/m² and at different temperatures. It is clear that the theoretical values of the model are in accord with the experimental measurements.

CONCLUSION
This paper proposed a numerical method to extract and evaluate the physical parameters of photovoltaic generators. The equivalent single-diode circuit with five parameters is applied for modeling the electrical behavior of the PV generators. This technique is based on the analytical extraction of the saturation current, the parallel resistance and the photocurrent as a function of the ideality factor and series resistance using the linear least squares method. Then, the estimation of the series resistance and the ideality factor has been found by solving numerically a system of two nonlinear equations. The suggested numerical method reduces the number of the initial values to two and extract the five parameters without approximation. The application of this method for the RTC France solar cell, PWP201 and the KC200GT panels indicated a low error and a good agreement with the experimental data.