An Accurate and Fast Computational Algorithm Based on Hybrid Methods

The numerical solutions of nonlinear equation for a single diode equivalent circuit of a solar cell are introduced. Four numerically algorithms, which include newton's; Predictor-Corrector Hally and Accelerated Predictor-Corrector Hally methods are described and compared in the present work. These algorithms are applied to calculate the voltage; current; power of a solar cell with the different values of load resistance programmed by Matlab language. The results showed that the proposed algorithm is the most efficient compare with other three algorithms.


Introduction
Numerical analysis involves the study and evaluation of methods of calculating required numerical results based on given numerical data, making numerical analysis an important part of the science known as information processing.The numerical data given is the input information, the required results are the output information, and the calculation method is known as the arithmetic system.Numerical treatments for solving nonlinear equations were introduced by several authors in many fields such as engineering and sciences.Iterative algorithms for treating the nonlinear equation of single diode solar cells had been considered.Here we shall consider the numerical solution of a single-diode photovoltaic cell .
In this paper, describes a new algorithm Accelerated Predictor-Corrector Hally method (AHM); so that the nonlinear equation of a solar cell can be solved.It is systematic points: section two characterizing a design of a PV cell (single diode).Section three foundations the zeros finding of Newton Raphson technique.In section four Predictor-Corrector Hally methods has been described.Thus, in section 5 Accelerated Predictor-Corrector Hally method has been demonstrated here; in section six results and discussion are reported while in section seven the conclusions is presented.KCL-Kirchhoff's current law have been applied on Figure 1; a final equation of the PV cell current is extracted according to this equivalent as follows
Subs.Eq. 4 in Eq. 5 yield where I s reverse saturation current= 10 −12 A. In parallel, Based on the first derivative of Eq. 6; V can be determined numerically.

Newton's Method
The following algorithm suggestion for solving Eq. 6 by using NRM INPUT initial approximate solution x 0 = 1,

Accelerated Predictor-Corrector Hally Method (AHM)
To compare the different numerical methods of iterations, algorithm 1 and algorithm 2 has been used against the proposed algorithm 3.In addition; Eq. 6. has been solved to demonstrate the performance of the new proposed algorithm and determine the consistency and stability of results.The results are examined using three iterative algorithms Algorithm 1: Newton Raphson Method (NRM) Algorithm 2: Predictor-Corrector Hally Method (HM) Algorithm 3: Accelerated Predictor-Corrector Hally Method (AHM) Tolerance ε = 10 −9 and σ

Results and Discussion
Four Algorithms are given based on Eqns.7, 9, 12, and 15 is achieved in order to solve the roots of Eq. 6 which is a non-linear equation with predict guess v 0 .To demonstrate the performance of the four methods is used.The approximate solutions produced by the techniques regarded and list the errors obtained by the four methods.Five various examples are utilized by means of equation 6 which are based on the R-values (load resistance) varies from 1 ohm to 5 ohm Figs 2-6 and Tables 1-5.The results indicate AHM need 6 iterations whereas NRM, ANRM and HM need 9, 8 and 8 iterations respectively for reaching the convergence, this prove that AHM is better than the other techniques.

Conclusion
A new Accelerated Predictor-Corrector Hally, Predictor-Corrector Hally, Accelerated Newton's, Newton's algorithms is described and investigated in order to calculate the voltage; current and power of a single-diode equivalent circuit design numerically with a several values of load resistance R.These Several algorithms were applied for illustration and good results were acquired for the determinations of the three electrical parameters of a solar cell.The following steps have been identified: First, the process of computation presented of a new proposed algorithm in the equation of a solar cell approach is simple; the approximate results are easy to obtain by a few computations; so the approach is considerably powerful.Second Good results obtained depend on the selection of the initial value  0 for the three algorithms.Third Good results based on the algorithms used to find the involved model.

Fig. 2 -
Fig. 2 -Number of iterations per PV parameters of transcendental function using four methods.

Fig. 3 -
Fig. 3 -Number of iterations per PV parameters of transcendental function using four methods.

Fig. 4 -
Fig. 4 -Number of iterations per PV parameters of transcendental function using four methods.

Fig. 5 -
Fig. 5 -Number of iterations per PV parameters of transcendental function using four methods.

Fig. 6 -
Fig. 6 -Number of iterations per PV parameters of transcendental function using four methods.