## Introduction

Considering actual energy and environmental context of classical power resources around the world, the photovoltaic (PV) domain provide one of the most efficient ways of producing energy, with real perspectives in the future.

### Modeling PV Cells, Panels and Arrays

Usually, the solar cells are modeled using a specific type of equivalent circuit to deploy the equation of mathematical model. Any PV model is based on diode behavior, which gives to the PV cell its exponential characteristic. In the scientific literature, there are several mathematical models that describe the operation and behavior of the PV generator. These models differ in the procedure of calculation, accuracy and the number of parameters involved in the calculation of current-voltage (I-V) characteristic.

In Simulink^{®} the solar cell can be modeled with three modeling systems. The first possibility of modeling can be done with instruments which can implement any differential equation or algebraic relationship of a highly complex mathematical model. Another possibility is given by Simscape™, which allows direct modeling using physical components of
the electric field (resistors, capacitors, diodes) to implement exactly the same mathematical equation.

A modeling system more complex than those described above is performed using SimElectronics^{®} advanced component library, which contains a block called *Solar Cell*.
The Solar Cell from Matlab^{®} is a solar current source, which includes solar-induced current and temperature dependence.
Solar Cell block is formed from a single solar cell as a resistance *R _{s}* connected in series with a parallel combination of a current source, two exponential diodes and a parallel resistor

*R*.

_{p}The I-V behavior of the solar cells and PV arrays can be described by several different complex mathematical relationships.

### General Equation of Solar-Induced Current (Output Current of Solar Cell)

Where:

*I _{ph}* Solar-induced current

*I*

_{ph}= I_{ph0}·I_{r}/ I_{r}_{0};

*I _{r}* Irradiance (light intensity) that falling on the cell in W/m

^{2};

*I _{ph0}* Measured solar-generated current for the irradiance (light intensity)

*I*

_{r}_{0};

*I _{s}*

_{1}

*, I*

_{s}_{2}Saturation current of the first diode and second diode respectively;

*N _{1}, N_{2}* Quality factor (emission coefficient) of the
first and second diode;

*V _{t}* Thermal voltage

*V*;

_{t}= kT / q*k* Boltzmann constant, *T*: device temperature, *q*: elementary charge on an electron;

*R _{s}, R_{p}* Series and parallel (shunt) resistance;

*V* Voltage at the terminals of solar cell.

### Temperature Dependence

Several
solar cell parameters (the
solar-induced current *I _{ph}*, diodes saturation currents

*I*

_{s}_{1}and

*I*

_{s}_{2}, the series resistance

*R*and the parallel resistance

_{s}*R*) depend on temperature.

_{p}

Where:

*TIPH*1 First temperature coefficient for *I _{ph}*;

*T _{meas}* Extraction temperature parameter.

### 5-Parameters Solar Cell Model

The Solar Cell block from Simulink allows choosing one of two models: a model with 8-parameters and a model with 5-parameters, if for general equation of solar cell model is applied the following simplifying assumptions: the impedance of the parallel resistor is infinite and the saturation current of the second diode is zero. The 5-parameters (single-diode) model provide short-circuit current and open-circuit voltage that the block converts to an equivalent circuit model of the PV cell.

The 5-parameters model is the most used solar cell model. The 5-parameters solar cell model provides a good compromise between accuracy and simplicity. This model is formed from a series resistance *R _{s}* (models bulk and contact resistances) connected in series with a parallel combination of a current source, an exponential diode and a parallel resistance

*R*(models leakage currents, primarily due to defects).

_{p}## PV Array Model

### Simulink Model of PV Array - Plotting the Characteristics of a PV Module

This model represents a PV cell array connected to a variable resistor. This resistor has an input ramp which just varies resistance linearly in closed circuit until it reaches the 30^{th} steps. Control of irradiation is realized by Signal Builder block.

Inside the array subsystem are 6 rows of PV solar cells connected in series, formed by 6 solar cells from SimElectronics library. This structure can be built in any configurations by connecting multiple strings of solar cells in series or in parallel.

The advantage of using of this high level of implementation is to create a simple equivalent circuit, which have much more complex parameters, including the effect of temperature in the device (solar cell) which is very important for behavior of this type of system.

The PV panel model is validated by simulating at a value of irradiance of 1000W/m² and a temperature of 25°C. In the figure are shown the curves of current, voltage and power versus time which are obtained at the output of PV array. When the resistance varies, the current and voltage vary depending on the voltage-time relationship which gives the power curve.

The I-V and P-V characteristic curves of PV array for different levels of irradiation and temperature are given below. If the irradiance decreases, the PV current generated decreases proportionally to that, and variation of no-load (open-circuit) voltage is very small. If the temperature of PV module increases the voltage decreases, the produced current remains practically constant, and the produced electric power of the PV panels is reduced.

The I-V curve represents the standard behavior of both, the PV cells and PV array respectively. In the middle of the I-V curves is the maximum power point (MPP). This point is very critical for this kind of system for maximum power extraction from the PV array. Result that the main objective is to try operating around of this maximum point in order to make the PV cells to work at maximum efficiency.

Since the environmental conditions in which the solar cells are tested are variable, being very difficult to have the same level of solar radiation and temperature, is needed a substantial input for the test system of PV solar arrays.

### Modeling of PV Arrays Using Experimental Test Data

Curve Fitting Toolbox use experimental or test data in the form of typical I-V curves to generate a mathematical model of PV array. The three-dimensional surface fit of I-V characteristic curves represent the experimental behavior of the PV array. This model can be use very quickly as a source for a PV array.

Curve Fitting Tools is a powerful algorithm that affords the achievement of polynomial interpolation, or can run a custom equation, which in this case is very close to the one that is accomplished, because it is known the exponential form of I-V curves (the equation of solar-generated current of PV array have some kind of exponential nature plus a constant term):

Where:

*x, y * Variables on the *x* and *y* axes;

*a, b, m, c, d* Coefficients.

Using a simple interpolation algorithm, like cubic interpolation, is generated a surface whose mathematical equation represents all points on the I-V curves and all points between those curves of PV array.

Steps to follows to create a source for a PV Array:

- Set up figure color map, legend array, figure plotting axis, titles and labels;
- Load data for 5 or more levels of irradiance (W/m
^{2}); - Use the Curve Fitting Toolbox to create a fit for this data;
- Use the auto-generated Matlab function file to create the surface fit;
- Create the irradiance and voltage vectors;
- Use the surface fit object to create the current array;
- Save data;
- Use the array data to parametrize a 2D Lookup Table block: load the Simulink library, create and open a new model, add block to model and parametrize.

Curve Fitting Toolbox is used to create a stable fit because the I-V behavior of PV arrays is static, but the experimental data are for dynamic systems, and for this case is used a predictive model for PV array.

The PV panel is modeled as constant DC source using Photovoltaic I-V Curves 2D Lookup Table block created previously. This block has two inputs: the irradiation input coming from port 1 and have a voltage input, which is like a feedback from the system and in the output is calculated the current value. Therefore, this model generates a current and receives voltage back from the system.

## PV Solar Array Simulator

The PV Solar Array Simulator represents a PV Array connected to a resistive load via a dc-dc step-down (buck) converter with Maximum Power Point Tracking (MPPT) controller implemented with incremental conductance MPPT algorithm, as in the next section of the site.

The variant subsystem of PV Solar Array allows choosing between two 5-parameters PV Array sources (using experimental test data, as shown above, and using fundamental approaches of first principles Simulink, as shown below). These two PV Array models have the same electrical parameters.

This work follows the output power of a PV Array and not the I-V characteristic curves of PV Array in order to establish which of these two PV Array models are more suitable as a source for a PV system.

In case of PV Array model using experimental test data the power obtain at the output of PV Array are significant higher and have smaller oscillations against case of PV Array model realized using fundamental approaches of first principles Simulink, that confirm the choice of PV Array using experimental test data.

At the output of PV Array are obtained different voltages and currents and implicitly different powers as a result of the introduction of dc-dc step-down converter.

In case of direct-coupled PV Array simulations (without dc-dc converter and MPPT controller) at the same condition, the output parameters values of PV Array are close together for both models.

In case of PV System simulations, the variations of the voltage, current and power are caused by incremental conductance algorithm of MPPT controller.

The two PV Array sources relied on models for the PV Solar Array Simulator with step-down dc-dc converter and MPPT control. Given that the PV Solar Array Simulator was simulated for different PV Array sources, and having as the argument the power obtained at the output of PV Panel is decide the superiority of PV Array model using experimental data over the PV Array model using first principles Simulink.

A possible applicability of this obtained results is the achievement of a PV Simulator, that are used to study its operation in different conditions (irradiance, temperature, load profiles).