Mathematical Problems in Engineering

Volume 2015, Article ID 145258, 13 pages

http://dx.doi.org/10.1155/2015/145258

## A New Six-Parameter Model Based on Chebyshev Polynomials for Solar Cells

^{1}College of New Energy, Bohai University, Jinzhou 121013, China^{2}School of Mathematics and Physics, Bohai University, Jinzhou 121013, China

Received 29 December 2014; Accepted 2 April 2015

Academic Editor: Georgios Veronis

Copyright © 2015 Shu-xian Lun et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This paper presents a new current-voltage (*I-V*) model for solar cells. It has been proved that series resistance of a solar cell is related to temperature. However, the existing five-parameter model ignores the temperature dependence of series resistance and then only accurately predicts the performance of monocrystalline silicon solar cells. Therefore, this paper uses Chebyshev polynomials to describe the relationship between series resistance and temperature. This makes a new parameter called temperature coefficient for series resistance introduced into the single-diode model. Then, a new six-parameter model for solar cells is established in this paper. This new model can improve the accuracy of the traditional single-diode model and reflect the temperature dependence of series resistance. To validate the accuracy of the six-parameter model in this paper, five kinds of silicon solar cells with different technology types, that is, monocrystalline silicon, polycrystalline silicon, thin film silicon, and tripe-junction amorphous silicon, are tested at different irradiance and temperature conditions. Experiment results show that the six-parameter model proposed in this paper is an *I-V* model with moderate computational complexity and high precision.

#### 1. Introduction

Photovoltaic (PV) power generation system directly converts solar energy into electrical energy by using PV arrays. To obtain higher energy efficiency, PV power generation systems need to establish their simulation models to get optimized parameters. PV power generation system is mainly composed of PV arrays, controller [1–3], and inverter [4–6]. PV arrays, the core devices of PV power generation system, usually consist of solar cells in series and/or in parallel. There are two kinds of popular simulation models for solar cells, that is, the single-diode model and the double-diode model [7–10]. Because of having fewer parameters, the single-diode model is simpler than the double-diode model. This makes the single-diode model widely used. Some efforts have been made to improve the accuracy of the single-diode model. According to the number of parameters, there are mainly three kinds of models, that is, the four-parameter model, the five-parameter model, and the seven-parameter model.

The four-parameter model in [11] includes ideality factor, diode reverse saturation current, light-generated current, series resistance. The four-parameter model has the fewest cell parameters, and then its expression is the simplest. However, the predicted accuracy of the four-parameter model is very limited. And the four-parameter model is validated only for monocrystalline silicon modules. On the basis of the four-parameter model, the five-parameter models are obtained by introducing shunt resistance [11–21]. The five-parameter models are more precise than the four-parameter model in [11]. Compromising the number of parameters and the approximate accuracy, the five-parameter models are the most commonly used. The five-parameter models proposed in [11, 19–21] utilize the reciprocals of the slopes at the open-circuit point and short-circuit point to calculate the cell parameters. The two slope values are not usually provided by the manufacturers’ datasheet. And they can be obtained by using enough data pairs of experimental characteristic under the certain condition. This makes it very complicated to obtain the slope values. Therefore, these five-parameter models actually have seven parameters rather than five parameters. The five-parameter model in [12] does not utilize the above-mentioned two slope parameters and can accurately predict the performance of monocrystalline and polycrystalline silicon modules. However, the five-parameter model in [12] exists large modeling errors for amorphous silicon solar cells. This may be due to the facts that series resistance in [12] is assumed to be a constant and the temperature dependence of series resistance is ignored. Here, temperature refers to the temperature of solar cells. The seven-parameter models proposed in [13, 22] add two additional parameters on the basis of the five-parameter model. For example, in [13], the two additional parameters are temperature coefficient for series resistance and the diode reverse saturation current radiation dependence, respectively. Temperature coefficient for series resistance describes the temperature dependence of series resistance. The value of series resistance changes exponentially with the cell temperature. The diode reverse saturation current radiation dependence describes the influence of irradiance on diode reverse saturation current, which is obtained by using the current and the voltage at maximum power point under the irradiance of and the temperature of . The extraction process for this parameter is very complicated. Although the seven-parameter model in [13] provides a more accurate characteristic model, it is not widely used because of the complexity of parameter extraction. Therefore, we propose a new six-parameter model.

In this paper, Chebyshev polynomials are developed to describe the relationship between series resistance and temperature. This makes a new parameter called temperature coefficient for series resistance introduced into the single-diode model. Therefore, the single-diode model has six cell parameters in this paper, that is, ideality factor, light-generated current, diode reverse saturation current, series resistance, shunt resistance, and temperature coefficient for series resistance. The new six-parameter model provides a simpler and more reasonable expression for series resistance and accurately predicts characteristic for silicon solar cells. Five kinds of silicon solar cells with four different technology types, that is, monocrystalline silicon, polycrystalline silicon, thin film silicon and triple-junction amorphous silicon, are tested to validate the six-parameter model in this paper.

#### 2. The Single-Diode Model of Solar Cells

A solar cell is essentially a very large area p-n junction diode. Under illumination, a single-diode model of a solar cell can be described by an equivalent circuit, shown in Figure 1. The equation of a single-diode model is shown as follows:where () is light-generated current, () is diode reverse saturation current, () is shunt resistance, and () is series resistance, is ideality factor which is defined as follows:where is the number of solar cells in series in a PV module, is diode ideality factor, and is thermal voltage which is defined as follows:where is Boltzmann’s constant ( J/K), () is cell temperature, and is electron charge ( C).