Advances in Power Electronics

Volume 2015, Article ID 654092, 11 pages

http://dx.doi.org/10.1155/2015/654092

## Comprehensive Analysis and Experimental Validation of an Improved Mathematical Modeling of Photovoltaic Array

^{1}Department of Electrical and Computer Engineering, National University of Singapore, 4 Engineering Drive 3, Singapore 117583^{2}Department of Electrical Engineering and Information Technology, ETH Zurich, Rämistrasse 101, 8092 Zurich, Switzerland^{3}Department of Electrical Engineering, Asian Institute of Technology, Pathumthani 12120, Thailand^{4}School of Computing, National University of Singapore, 13 Computing Drive, Singapore 117417

Received 26 August 2014; Revised 30 November 2014; Accepted 1 December 2014

Academic Editor: Jose Antenor Pomilio

Copyright © 2015 Satarupa Bal 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 proposes a simple, accurate, and easy to model approach for the simulation of photovoltaic (PV) array and also provides a comparative analysis of the same with two other widely used models. It is highly imperative that the maximum power point (MPP) is achieved effectively and thus a simple and robust mathematical model is necessary that poses less mathematical complexity as well as low data storage requirement, in which the maximum power point tracking (MPPT) algorithm can be realized in an effective way. Further, the resemblance of the *P-V* and *I-V* curves as obtained on the basis of experimental data should also be taken into account for theoretical validation. In addition, the study incorporates the root mean square deviation (RMSD) from the experimental data, the fill factor (FF), the efficiency of the model, and the time required for simulation. Two models have been used to investigate the *I-V* and *P-V* characteristics. Perturb and Observe method has been adopted for MPPT. The MPP tracking is realized using field programmable gate array (FPGA) to prove the effectiveness of the proposed approach.
All the systems are modeled and simulated in MATLAB/Simulink environment.

#### 1. Introduction

Electrical energy from photovoltaic is currently regarded as the prerequisite sustainable resource for both stand-alone as well as grid connected applications, since it is abundant and clean, offers zero input fuel cost, and is distributed throughout the earth [1]. In practical cases, photovoltaic modules operate over a highly intermittent nature of temperature and irradiance but the electrical parameters provided in the datasheet are only for the standard test conditions (STC). Moreover, in power generation from PV, optimal utilization of the available solar energy is imperative due to the high costs of PV modules. It is also seen that mathematical models of few individual components of PV system are represented and simulated for better understanding of their performances [2].

This calls for a simple, accurate, and easy to model approach for the simulation of photovoltaic (PV) module to track the maximum power point and to predict PV energy production under varying atmospheric conditions [3]. In order to increase the accuracy, the following can be incorporated, but it leads to the increase in complexity of the modeling [4]:(i)temperature dependence of the diode saturation current,(ii)temperature dependence of the photo current,(iii)inclusion of series resistance: for more accurate shape between the MPP and the open circuit (OC) voltage,(iv)inclusion of shunt resistance in parallel with the diode,(v)variability of diode quality factor,(vi)introduction of two or more parallel diodes.

The accuracy of the simulation of a PV model largely depends on the estimation of the characteristic* I-V* and* P-V* curves. Furthermore, factors such as efficiency, field factor, and simulation time affect the effectiveness of the model. A simplistic and easy to model approach is preferred so as to avoid unwanted complexity due to additional parameters.

So far, among the mathematical models of PV array proposed in the literature, the simplest is the ideal single diode model which involves only three parameters, namely, short circuit current, open circuit voltage, and the diode ideality factor [5]. Improvement has been made, with the simplified single diode model (SSDM) being proposed in [6] which takes the effect of the series resistance () which is the sum of several types of structural resistance of the device into consideration [7–15]. The influence of only becomes dominant when the PV device operates in the voltage source region. Also, it lacks the accuracy when subjected to large temperature variations [16]. Since, the value of is very low, some authors neglect its effect [5, 17–19]. Further improvement has been done by the introduction of the single diode model (SDM) which includes the additional shunt resistance () along with the series resistance [2]. The shunt resistance exists mainly due to the leakage current of the p-n junction. The effect of is dominant when the PV device operates at current source region of operation. However, since the value of is very high many authors [4, 8, 9, 20, 21] neglect it in order to simplify the model. Although it is much more accurate than the previous models, it is not preferred on account of its computational complexity. It is also reported in [16] that the accuracy of this model deteriorates at low irradiance levels. In order to mitigate the inaccuracies offered by the previous models, the two-diode model was proposed in [22]. However, this leads to more model complexity and thus more simulation time due to the involvement of a greater number of parameters. A new mathematical PV model has also been proposed in [23] that includes the advantages of previous models combining the three main considerations, namely, simplicity, ease of modeling, and accuracy. However, it doesnt take into consideration the effect of diode saturation current on temperature which results in model errors at the vicinity of open-circuit voltage and consequently at other regions.

This paper proposes a new, simple, accurate, and easy to model approach for the simulation of PV array and also provides a comparative analysis of the same with the conventional single diode model and the improved ideal single diode model. As the PV systems are generally integrated with specific control algorithms in order to extract the maximum possible power, it is highly imperative that the MPP is achieved effectively and thus it is needed to design a model from which the MPPT algorithm can be realized in an effective way. Some MPPT techniques have been proposed in [1, 3, 4, 10]. However, for simplicity, this paper adopts the Perturb and Observe (P&O) method for MPPT.

The proposed theoretical model is verified and validated with experimental data of commercial PV array. RMSD from the experimental data, maximum efficiency of the design, the fill factor (FF), and the simulation time has also been calculated. In addition, the MPP tracking is realized in digital environment using FPGA kit to prove the effectiveness of the proposed approach. All the systems here are modeled and simulated in MATLAB/Simulink environment. The proposed modeling method can be useful for users who require simple, fast, and accurate models in simulation of PV systems.

#### 2. Mathematical Models for a Photovoltaic Module

The major issue of real-time identification is basically the selection of a proper model. It is therefore necessary to have a proper mathematical model that can represent accurately the current-voltage characteristics of the PV array and which can be solved by analytical methods in a simplified manner [24]. In addition to this to maximize the power extracted from a PV array with the help of MPPT control, the understanding and modeling of PV cell are also important [25].

Assuming the semiconductor diode equation and the Kirchhoff laws, the characteristics for a PV module composed of series connected cells based on single exponential model are expressed as follows [26]: where is the Boltzmann constant ( J/K) and the electron charge ( C). gives the module temperature. The parameter gives the photocurrent, represents the diode saturation current, and gives the output current. and give the series resistance and the shunt resistance. and represent the diode ideality factor and the number of cells connected in series, respectively.

The first term gives the photocurrent and the second part is the ideal dark current that models the emitter and base recombination. All the parameters are mostly calculated through sets of nonlinear equations [27].

##### 2.1. Single Diode Model (SSDM)

The single diode model takes into account both the series resistance as well as the shunt resistance unlike the ideal single diode model or the simplified single diode model as shown in Figure 1. This resistance is the sum of several types of structural resistance of the device. depends mainly on four factors, namely [2],(i)contact resistance of the metal base with p-layer,(ii)resistance of p-layer and n-layer,(iii)contact resistance of the metal grid with the n-layer,(iv)resistance of the grid.