An efficient Approximate Solution for Non-Linear Solar Cell Equation using Inverse Quadratic Interpolation Method

algorithm


Introduction
Many problems require finding some or all roots of a non-linear equation.In general, an equation containing one variable can be written as f(x) = 0.There are a number of numerical methods to find an approximate value for a given root of the previous equation, the numerical iterative algorithms for example iterative, regular false; Bisection; secant and Newton techniques are used to achieve the approximate numerical solution of these equations [1][2][3][4][5].All these numerical methods need a rough approximate value of a given equation root to enable it to generate sequential initial values of a given equation root to enable it to generate a sequential of better approximate values for that root.There are many techniques improved on the perfection of convergent Newton's method, in order to obtain a superior convergence order than NRM [6][7][8][9][10][11].
This paper is attention with the iterative algorithm to get the voltage's value of the photovoltaic cell V pv in the conditions f(x) = 0, and f ́(x) ≠ 0 where f: R → R be real function.It is systematic as the following steps: section two describes the design of the model chosen (Equation of Non-linear Solar Cell); section three characterizes the two numerical formulas; the proposed method (Inverse Quadratic Interpolation method) have been portrayed; section four results and discussion; finally section five the checked results have been concluded.

Equation of Non-linear Solar Cell
The current and voltage characteristics of the solar cell can be illustrated by the following equations [12][13][14][15][16] Merge Eq. 1 in Eq. 2, yields

Numerical Formulas
Definition 3.1 The Newton's method so called Newton Raphson method, is a root-finding algorithm that uses the first few terms of the Taylor series of a function () approximately a suspected root The algorithm is applied iteratively to obtain [17]  +1 =   − (  )

Definition 3.2
The Inverse quadratic interpolation is a root-finding algorithm, for solving equations of the form  = () = 0.The idea is to use quadratic interpolation to approximate the inverse of .IQIM required three initial values  0 ,  1 ,  2 and realized by the recurrence relation (6) where
Tables 1 to 5 show the results for the proposed method when the load resistance R = 1; and Figure 1, 2, 3, 4 and 5 present the acquired results using the standard and proposed methods.The material below gives the solution of Eq. 3 along with the estimated points solved in MATLAB.The results are found using Newton method.The some discussion on possible improvements.Our allowed tolerance was ε = 10 It would be expected that the IQIM could be as good as or possibly better than the newton method for R = 1, 2, 3, 4 and 5.This is an interesting case because we are only using transcendental function

− 9 . 1 ) 2 ) 4 )
Solutions of Eq. 3 with R = 1, Estimated point for V pv , at a point V: 0.922423135 Estimated point for I pv , at a point I: 0.9224231350 Estimated point for P pv , at a point P: 0.850864439 with number of iteration = 9.Solutions of Eq. 3 with R = 2, Estimated point for V pv , at a point V: 0.917035382 Estimated point for I pv , at a point I: 0.458517691 Estimated point for P pv , at a point P: 0.420476946 with number of iteration n = 9. 3) Solutions of Eq. 3 with R = 3, Estimated point for V pv , at a point V: 0.910403374 Estimated point for I pv , at a point I: 0.303467791 Estimated point for P pv , at a point P: 0.276278101 with number of iteration n = 9.Solutions of Eq. 3 with R = 4, Estimated point for V pv , at a point V: 0.901740602 Estimated point for I pv , at a point I: 0.225435150 Estimated point for P pv , at a point P: 0.203284028 with number of iteration n = 9. 5) Solutions of Eq. 3 with R = 5, Estimated point for V pv , at a point V: 0.889092715 Estimated point for I pv , at a point I: 0.1778185430 Estimated point for P pv , at a point P: 0.158097171 with number of iteration  = 10.