Modeling and parameters extraction of photovoltaic cell and modules using the genetic algorithms with lambert W-function as objective function

Abstract

In this paper, a method based on genetic algorithms is proposed for improving the accuracy of solar cell parameters extracted using novel technique. We propose a computational based binary-coded genetic algorithm (GA) to extract the parameters ( $I_{0}$ , $I_{{ph}}$ , $n$ , $R_{{s}}$ and $R_{{sh}}$ ) for a single diode model of solar cell from its current-voltage (I–V) characteristic. The algorithm was implemented using Matlab as a programming tool and validated by applying it to the I–V curve synthesized from the literature using reported values. The characterization, current-voltage data used was generated by simulating a one-diode solar cell model of specified parameters. The new approach is based on formulating I–V equation of solar cell, with Lambert function, the parameter extraction as a search and optimization problem. Compared with other optimization techniques in literatures, the approach proposed for the determination of parameters are in good agreement.

Keywords

Photovoltaic solar cell parameter extraction genetic algorithm optimization Lambert function

1. Introduction

The utilization of renewable energy has experienced steady increase over the past years due to limited supply of fossil fuels and air pollution. Solar energy is one of the most promising renewable energy sources due to readily available, easy installation, low maintenance, and almost zero pollution [1, 2, 3]. With the increasing utilization of solar energy, researche on photovoltaic (PV) generation system have received significant attention in the past few years [4, 5, 6, 7, 8]. Accurate extraction and optimization of solar cells and solar panel parameters are very important in improving the device quality during fabrication and in device modeling and simulation [9, 10, 11]. To simulate and characterize a photovoltaic cell, and therefore the various parameters that affect these characteristics, several methods have been proposed for the identification of these parameters, to simulate its behavior, and optimize different characteristics.

The sensitivities of the parameters are directly related to the current-voltage characteristic of the solar cell; five parameters were substituted into equation described in the following paragraph to draw an I–V curve. The shape of the I–V curve in the voltage source region is depressed horizontally with a gradual increase in the value of resistance series from zero. As shunt resistance decreases, the current diverted through the shunt resistor that increases for a given level of junction voltage. The ideality factor and the reverse saturation current have a large influence on the open circuit voltage of the solar cell.

In practice, two main equivalent circuit models have been widely used: single and double diode models [12, 13, 14]. A relative of the photovoltaic generator empirical model, which is currently the most widely used due to the quality its results: is the model with a one diode. Generally, in the literature, there are three categories such as analytic, iterative and evolutionary computational methods [15]; analytic parameter extraction methods, iterative parameter extraction methods and evolutionary parameter extraction methods.

Analytical methods are common approaches in estimating the parameters by mathematical equations. Similar analytical solution methods [16, 17, 18], a new analytical solution technique, using the so called “Co-content function” which is based on Lambert function [19], Taylor series expansion is used to approximate the analytical solution of single diode equation with series resistance [20, 21], Lambert W-function is used but the mathematical formulation is complex and the method is computationally expensive. All this deterministic techniques impose several model restrictions such as convexity and differentiability in order to be correctly applied [22].

Iterative Parameter Extraction Methods, a number of iterative methods are available in the past few years. We quote the method to calculate solar cell parameters; Newton algorithm genetic method [23], a comparative study of three different methods, namely curve-fitting method, iterative 5-point method, and analytical 5-point method, for extracting solar cell parameters is presented in reference [1].

Evolutionary Parameter Extraction Methods, to resolve issues with analytical and iterative methods, Evolutionary computational algorithms which use artificial intelligence technique has received considerable attention for solar cell parameter extraction recently. These evolutionary algorithms are global optimization techniques. Evolutionary algorithms use iterative progress, such as growth or development in a population. This population is then selected in a guided random search using parallel processing to achieve the desired end. Such methods include GA (genetic algorithms) [22, 23, 24, 25], PSO (particle swarm optimization) [26, 27], SA (simulated annealing) [28], HS (harmony search) [29], BFA (Bacterial Foraging Algorithm) [30], TBLO (teaching-learning based optimization) [31] and BMO (bird matting optimization) [32]. Although heuristic methods present a higher probability of obtaining a global solution in comparison with deterministic ones, they have important limits [33]. In case of GA and PSO, they maintain a trend that concentrates toward local optima, since their elitist mechanism forces premature convergence [34, 35]. Such a behavior becomes worse when the optimization algorithm faces multi-modal functions [36, 37], pattern search (PS) [38], differential evolution (DE) [39], and artificial neural network (ANN) [40, 41] and Parameter identification of solar cells using artificial bee colony optimization (BCO) [42].

This paper proposes an efficient technique for estimating the solar cell parameters as it introduces a new objective function to this estimation problem, based non-linear equation of Lambert. The objective consisted to extract, parameters from experimental data using genetic algorithms (GAs) via an explicit sum equation of the photovoltaic cell expressed in terms of Lambert W, which represents a considerable growth due to their accuracy, reliability and robustness [43, 44, 45, 46, 47]. We decided to work with a solar cell and a solar panel, to make our choice be based on the parameters in existing literature, in order to properly evaluate our proposed method. Section 2 discusses solar cell single diode modeling and its equivalent circuit and Mathematical formulation with Lambert function. A description of the Genetic algorithm approach is provided in Section 3. Section 4 presents testing and simulation results. The paper is then concluded in Section 5.

2. Photovoltaic modeling – single diode model

In general, most semiconductor solar cells are based on single p-n junctions. Description of cell behavior can therefore be given in terms of a model based on the diode equation [12]. There are many possible mathematical relationships of varying complexity which can be used to describe PV current-voltage behavior. In this model, the solar cell is modeled as a current source connected in parallel with a diode. So a shunt resistance and a series resistance component are added to the model. The model of the solar cells for which the superposition principle is applicable can be represented by the equivalent circuit in Fig. 1 and expressed as Eq. (1) which includes light-generating current source ( $I_{L}$ ), series resistance ( $R_{s}$ ) and shunt resistance ( $R_{P}$ ) of diode with the diode equation shown by Eq. (1) [48].

Figure 1.

Simplified conceptual model, with one diode, for a photovoltaic solar cell.

Substituting these into the first equation produces the characteristic equation of a solar cell, which relates solar cell parameters to the output current and voltage:

$\displaystyle I=I_{l}-I_{0}\left[\textit{Exp}\left(\frac{{q}(V+IR_{s})}{% \textit{nKT}}\right)-{1}\right]{}-\frac{V+IR_{{s}}}{{{R}}_{{p}}}$ (1)

$I_{0}$ is the reverse saturation current, $n$ the quality factor, $k$ the Boltzmann’s constant, $T$ the absolute temperature, $q$ the electron charge.

We noticed that the implicit nonlinear equation transcendent above Eq. (1) cannot be solved explicitly in general $I$ or $V$ using common elementary functions. To solve the equation and extract its parameters, there are explicit analytical solutions based on the use of the Lambert function, a special function which is not expressible in terms of two basic analytic functions. This approach has also been used recently to study the bright solar cells [43, 44].

In this context, we exploited the offering of circuit (Fig. 1) and solving Eq. (2) in terms of the Lambert function. Thus, the solutions of each variable $I$ and $V$ as an explicit function of the other parameters and model. The five parameter model has been shown to offer improved accuracy over the more basic four parameter model which neglects the effects of shunt resistance [49] while avoiding the additional complexity of the double diode model [50].

The implicit nature of Eq. (2) and its computationally burdensome evaluation has however resulted in the growing use of an alternative expression based on the Lambert $W$ -function [51, 52, 53]. By using the Lambert $W$ -function the current can be expressed as an explicit function of voltage as shown by Eq. (3). The result is a more convenient and computationally efficient calculation of current.

Solution of equation, which is transcendental in nature using Lambert $W$ -function, is as follows [54, 55, 56, 57]:

$\displaystyle I=-\frac{V}{R_{s}+R_{p}}{}-\frac{\text{Lambert W}\left[\begin{% array}[]{c}\frac{{{I}}_{0}{{R}}_{{S}}{{R}}_{{p}}}{{n}{{V}}_{{th}}({{R}}_{{s}}{% +}{{R}}_{{p}})}\textit{Exp}\\ \left(\frac{{{R}}_{{p}}\left({{I}}_{{ph}}{{R}}_{{s}}{+}{{I}}_{0}{{R}}_{{S}}{+V% }\right)}{{n}{{V}}_{{th}}\left({{R}}_{{s}}{+}{{R}}_{{p}}\right)}\right)\\ \end{array}\right]}{{{R}}_{{s}}}{}+\frac{\left({{I}}_{{ph}}{+}{{I}}_{0}\right)% {{R}}_{{p}}}{{{R}}_{{s}}{+}{{R}}_{{p}}}$ (2)

Lambert W, defined by: $W(x)e^{W(x)}=x$ .

3. Genetic algorithm

The genetic algorithms are methods used to solve constrained and unconstrained optimization problems. They have been used to solve optimization problems in engineering and the sciences [58, 59], and are considered global methods for the purpose of optimization.

Three main rules are used by genetic algorithm at each step to form the next generation from the present population, and these rules are: selection rules to select the individuals (parents) that is considered the source for the next generation, crossover rules that combine two parents to produce children for the next generation, and mutation rules that randomly apply changes to individual parents to produce children [60].

Coding for a solution, termed a chromosome in GA literature, is usually described as a string of symbols from ${\{}$ 0, 1 ${\}}$ . These components of the chromosome are then labeled as genes. The number of bits that must be used to describe the parameters is problem dependent. L et each solution in the population of m such solutions $x_{i}$ , $i=1,2,\ldots,m$ , be a string of symbols $\left\{0.1\right\}$ of length $l$ . Typically, the initial population of m solutions is selected completely at random, with each bit of each solution having a 50% chance of taking the value 0 [61].

GA uses proportional selection, the population of the next generation is determined by $n$ independent random experiments; the probability that individual $x_{i}$ is selected from the tuple $\left\{x_{1},x_{2},\ldots,x_{n}\right\}$ to be a member of the next generation at each experiment is given by:

$\displaystyle P\left(x_{i},\text{ is selected}\right)=\frac{f\left(x_{i}\right% )}{\sum^{n}_{j=1}{f\left(x_{i}\right)}}$ (3)

This process is also called roulette wheel parent selection and may be viewed as a roulette wheel where each member of the population is represented by a slice that is directly proportional to the member’s fitness. A selection step is then a spin of the wheel, which in the long run tends to eliminate the least fit population members [61].

Crossover is an important random operator in GA and the function of the crossover operator is to generate new or ‘child’ chromosomes from two ‘parent’ chromosomes by combining the information extracted from the parents. The method of crossover used in GA is the one-point crossover. Typically, the probability for crossover ranges from 0.6 to 0.95 [61].

Mutation is another important component in GA, though it is usually conceived as a background operator. It operates independently on each individual by probabilistically perturbing each bit string [61].

By adjusting the curve to the measured data, the solar cells parameters extraction involves minimizing the cost function $f$ with respect to the five parameters $\left(I_{ph},I_{S},R_{S},R_{P},n\right)$ :

$\displaystyle\textit{fitness}=\sum^{N}_{i=1}{{\left(I_{m}-I_{e}(V_{i},\theta% \right))}^{1/2}}$ (4)

Where, $N$ is the number of experimental data. In Eq. (4), for the single diode model, ( $I_{m}$ , $V_{m}$ ) are respectively the measured current and voltage, $I_{e}$ is a calculated value by substitution $V_{m}$ in the Eq. (3).

Equation (3) is explicit in $I$ ; one way of simplifying the computation of $I\left(V_{i},\theta\right)$ is to substitute $I_{i}$ and $V_{i}$ in Eq. (2). Hence, we obtain Eq. (3). The Eq. (3) is used to calculate the currents in a discrete way for each iteration of the voltage Vi, we have 40 voltages (iteration).

$\displaystyle\quad{I}\left({{V}}_{{i}},{\theta}\right)=-\frac{{{V}}_{{i}}}{{{R% }}_{{s}}+{{R}}_{{p}}}$ $\displaystyle{}-\frac{\text{Lambert W}\left[\begin{array}[]{c}\frac{{{I}}_{0}{% {R}}_{{S}}{{R}}_{{p}}}{{n}{{V}}_{{th}}({{R}}_{{s}}+{{R}}_{{p}})}\textit{Exp}\\ \left(\frac{{{R}}_{{p}}\left({{I}}_{{ph}}{{R}}_{{s}}+{{I}}_{0}{{R}}_{{S}}+{{V}% }_{{i}}\right)}{{n}{{V}}_{{th}}\left({{R}}_{{s}}+{{R}}_{{p}}\right)}\right)\\ \end{array}\right]}{{{R}}_{{s}}}$ (5) $\displaystyle{}+\frac{\left({{I}}_{{ph}}+{{I}}_{0}\right){{R}}_{{p}}}{{{R}}_{{% s}}+{{R}}_{{p}}}$

Vector of floating numbers: $\theta=(I_{ph},I_{S},R_{S},R_{P},\linebreak n)$ .

In Fig. 2, we give the flow chart of the GAs. The chromosome here is the vector h containing the five parameters $I_{s}$ , $I_{ph}$ , $n$ , $R_{s}$ , and $R_{p}$ . The initial population ( $N_{\text{POP}}$ ) of chromosomes is a matrix given by Eq. (7) [62].

$\displaystyle\textit{Initial Pop}=\left(h_{i}-I_{0}\right).\textit{random}% \left[N_{\textit{ipop}},N_{\textit{par}}\right]{}+I_{0}$ (6)

Where $N_{\textit{ipop}}$ is the initial number of chromosomes in Initial Pop, $N_{\textit{par}}$ is the number of parameters in the chromosome ( $N_{\textit{par}}=$ 10 in our case), $I_{o}$ and $h_{i}$ are respectively the lowest and the highest values of parameters $I_{s}$ , $I_{ph}$ , $R_{s}$ , $R_{sh}$ and $n$ .

A canonical genetic algorithm can be implemented as shown in Fig. 2. The organization follows the steps (Fig. 2):

The measured data are located, the voltages and currents are calculated by bias of Eq. (1) the parameter values are estimated. A first generation of chromosomes is generated.

A feature ( $I$ , $V$ ) of each individual in the current generation is calculated.

The fitness values of all individuals of the current generation are calculated using Eq. (3).

Evaluation of the number of iterations, if $N$ is not equal to the number $N_{\textit{Max}}$ , then.

We create a new generation reserving the best individuals of the current generation.

The growth of the new generation continues with the operation croissement. In this case based on the roulette operation.

The new generation is performed using the Gaussian mutation operation.

Figure 2.

Flow diagram for the GA crossing-mates algorithm.

4. Results and discussion

The proposed method for the parameters identification of the single and single diode models is coded and executed in the Matlab environment. The proposed technique is used in this section to estimate the solar cell model parameters. Practical measured I–V data of a solar cell and solar module are considered for testing [63]. The objective function is present by Eq. (4) is minimized in order to be to reach an optimal set of parameters that reflects the solar cell characteristics. A value of zero for the objective function would yield an optimal solution. Experimental data of a silicon solar cell are taken from the bibliography. Voltage and current data measurements were taken using a 57 mm diameter commercial (R.T.C France) silicon solar cell under 1 sun (1000 W/m ${}^{2}$ ) at 33 ${{}^{\circ}}$ C. In Section 2, voltage and current measurements were taken using a solar module in which 36 polycrystalline silicon cells (60 W) are connected in series under 1 sun (1000 W/m ${}^{2}$ ) at 45 ${{}^{\circ}}$ C.

In order to test the quality of the fit to the experimental data, statistical analysis of the results is performed. The root mean squared error (RMSE), the mean bias error (MBE) and the mean absolute error (MAE) are the fundamental measures of accuracy [63, 64].

The RMSE, MBE and MAE are given by:

$\displaystyle\textit{RMSE}=\sqrt{{1}/{N}\sum^{N}_{i=1}{{e_{i}}^{2}}}$ (7) $\displaystyle\textit{MBE}=\sum^{N}_{i=1}{e_{i}}$ (8) $\displaystyle\textit{MAE}={1}/{N}\sum^{N}_{i=1}{\left|e_{i}\right|}$ (9)

Where, $e$ : is the relative error, i.e.

$\displaystyle e_{i}=\frac{I_{\textit{mesured}}-I_{\textit{cacluled}}}{I_{% \textit{mesured}}}$

$N$ : is the number of measurements.

In case one, solar cell data given in [23] is employed to extract the cell parameters using the single model. The extracted parameters for the single diode model are shown in Table 1 along with the root mean squared error (RMSE).

Table 1

Extracted parameters of solar cell by the current method and the previous work

	PS		SA		GA		This work
$I_{ph}(A)$	0.	7617	0.	7620	0.	7619	0.	7608
$I_{0}(\mu A)$	0.	9980	0.	4798	0.	8087	0.	1459
$a$	1.	6000	1.	5172	1.	5751	1.	4022
$R_{s}(\Omega)$	0.	0313	0.	0345	0.	0354	0.	0404
$R_{sh}(k\Omega)$	0.	0156	43.	1034	42.	3729	0.	0199
RMSE (%)	0.	0149	0.	0190	0.	01908	0.	00501
MBE (%)	/		/		/		$-$ 0.	00802
MAE (%)	/		/		/		0.	00802

The results of proposed method and those found by genetic algorithm-GA, with an ordinary I–V equation as an objective function [12], pattern search-PS [13], and simulated annealing-SA [28], for parameters identification of the solar cell single diode model are summarized in Table 1. It can be seen that proposed method yields better results than GA, PS, and SA. Table 1 lists the relative error of each measurement along with the MAE. The optimal parameters are put into the single diode model and genetic algorithm method is used to reconstruct the I–V characteristic.

Figures 3 and 4 shows the comparison between the experimental data and the results produced by the single diode model using the optimal parameters. It can be seen that the I–V characteristic obtained by the identified model using genetic algorithm based Lambert function is in good accordance with the experimental one meaning the high quality of the identification process.

Figure 3.

Comparison between the solar cell I–V characteristics resulted from the experimental data and the single diode model.

Figure 4.

Comparison between the photovoltaic module I–V characteristics resulted from the experimental data and the single diode model.

A comparison between the obtained results indicates that the accuracy of the genetic algorithm with Lambert function of single diode model is slightly more than that genetic algorithm using classical equation of the single diode model, because the RMSE value for the genetic algorithm with Lambert function is 0.00501.

In case of photovoltaic module, PV module data from Table 2 is used to extract the PV model parameters given in Eq. (3). The single diode model is used in this case.

Table 2

Extracted parameters of photovoltaic module by the current method and the previous work

	Ref. [65]	Ref. [66]	Ref. [67]	Ref. [67]	This work
$I_{ph}(A)$	3.8119	3.82006	3.812	3.812	3.	8119
$I_{0}(\mu A)$	0.1859	0.2446	0.1866	0.1802	0.	1845
$a$	1.360	0.9001	1.358	1.277	1.	34
$R_{s}(\Omega)$	0.180	0.3862	0.178	0.171	0.	1774
$R_{sh}(k\Omega)$	360	146.2	3376.71	3585.69	334.	8961
RMSE (%)	/	/	/	/	0.	00261
MBE (%)	/	/	/	/	$-$ 0.	00505
MAE (%)	/	/	/	/	0.	005388

The extracted parameters for the PV module are shown in Table 2 along with comparable results reported from previous studies. Similar curve fitting procedure, as in case of solar cell. Examining Table 2 reveals that similar pattern is noted. Comparing the proposed method outcomes with other reported results, extraction of parameters via the proposed method (Genetic Algorithms with Lambert W-function as objective function) clearly outperformed other competing methods in terms of RMSE.

Figure 5.

Results of the whiteness test for the solar cell case.

Figure 6.

Results of the whiteness test for the PV module case.

The objective of the whiteness test is to ensure that a selected model adequately describes a given set of data. The whiteness test can be achieved by the following two steps:

Examination of the estimated residual graph (exploratory analysis); and

Calculation of the residual autocorrelation function (RACF) at different time lags (confirmatory analysis) [28].

The RACF can be calculated as:

$\displaystyle\textit{RAFC}_{k}=\frac{\sum^{N}_{t=k+1}{e_{t}e_{t-k}}}{\sum^{N}_% {t=1}{{e_{t}}^{2}}}$ (10)

Where, $k$ : is the time lag, $e_{t}$ is the estimated residual at time $t$ calculated as

$\displaystyle e_{t}=I_{\textit{measured}}-I_{\textit{calculated}}$

The $\textit{RACF}\in\left[{-}{1,+1}\right]$ . If a given value (other than the first one) is significantly different from zero, it will fall outside a confidence interval level. Whiteness test results shown in Figs 5 and 6 confirm the higher quality of results obtained. TIME LAG: A time interval between two related voltage points (iteration).

5. Conclusion

In this paper, the use of genetic algorithm to accurately estimate the parameter of solar cells has been presented. The proposed technique provides a good balance between exploration and exploitation. In the simulation and experiment, novel technique is applied to extract the parameters of different solar cell models and photovoltaic module. Comparison results are in favor of novel approach, which outperforms other metaheuristic algorithms, such as GA classical, SA, PS. The method has been successfully applied to a silicon solar cell, and module. The results obtained are in good agreement with those published previously.

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