## 1. Introduction

The Energy Union Framework Strategy is aiming to a serious transition from an economy dependent on fossil fuels to one more reliant on renewables [

1] and among the available sources of renewable energy, solar energy is on the most abundant [

2,

3,

4], which could be assertively harnessed, especially in the southern countries of Europe. According to Club of Rome study embracing the circular economy concept could signify up to 70% decrease in carbon emissions by 2030, of five European economies [

5]. By targeting a renewable energy based economy and a circular economy at the same time could be the way achieve the Energy Union Framework Strategy targets [

1]. Although free and available on a planetary scale, the role in the global energy mix is unobtrusive, competing not only with other forms of non-renewable energy, such as gas and coal [

6], but also with its more direct rival—the wind renewable energy source [

7]. Except for a very limited number of countries where proactive and generous income policies were implemented at the beginning of the last decade, there has been a more recent mobilization of European governments in this sector, legislating on specific instruments to stimulate production decentralized and small scale power [

8]. In the last few years, the solar energy has been gaining importance in the worldwide energy evolution tendency due to a constantly increasing efficiency and lifespan, the decrease of the price of PV modules and by being environmentally friendly [

9]. Solar photovoltaic (PV) systems are steadily becoming one of the main three electricity sources in Europe [

10]. The entire installed PV capacity in 2016 reached 303 GWp and in that year Spain and Italy were responsible for 5.4 and 19.3 GWp, respectively [

11].

In a typical PV system several photovoltaic modules are linked in series in order to create a PV string. The aim is to reach a certain voltage and power output. With the intention of accomplishing a greater power, such PV strings can be linked in parallel in order to make a PV array. For the duration of a constant irradiance condition, the power-voltage (P-V curve) characteristics of a PV string show a typical P-V curve peak. Such type of a peak embodies the maximum power of the PV string [

4]. The P-V characteristics of a PV system are nonlinear and are affected by both the ambient temperature and solar irradiance, which in turn reveal distinct MPPs. With the purpose of optimizing the use of PV systems, conventional MPPT algorithms are often used [

12].

In the planning of a photovoltaic power plant the electric power produced is strongly linked to the meteorological conditions (solar radiation and temperature) [

13,

14]. Due to the intermittent nature of solar energy, power forecasting is crucial for a correct interpretation of business profitability and payback time [

15]. In the current market there is a great offer of manufacturers that, of course, have quite different technological production processes. All this leads to two modules with an identical technical sheet, under nominal test conditions, to differ in performance and produce very different results [

16]. The actual operating conditions in both solar radiation and temperature will very rarely coincide with the combination of nominal meteorological variables. Thus, the broader characterization is of utmost importance for studying the differences [

17]. In the end the main goal is to realistically quantify the performance, giving credibility to the estimation process in function of meteorological specificities of each season of the year. Ultimately it is desired that the process has enough resolution to reach the count up to the daily cycle.

The characterization requires the compilation of a large amount of data required for the application of appropriate mathematical model. In the concrete case of the photovoltaic cell the analytical model opens the doors for the detailed description in function of the external variables, which for all effects determine the general forms of the characteristic curves. However, to make modeling effective, it will be the model’s intrinsic parameters which will more or less shape the link to the experimental data [

18]. Molding is particularly critical at three points of operation. First, the predicted forecast of the peak electric power, then the open circuit operating points, that is, the maximum potential difference to be supported by the power electronics in the DC-AC conversation in the cut-off state, and finally the short-circuit, that is the maximum current to be supported by the electric cables in the event of a fault.

The models share in common the same electrical base model. The cell being a photoelectric device is modeled with a DC current source and a junction diode in parallel. From here all models are effectively variations with the introduction of more electrical elements. The elements may be of a series element of resistive nature by recreating the internal losses by Joule, or a parallel resistance simulating the internal leakage current, or a supplementary diode, which is normally associated with the losses by recombination of the carriers in the zone of the depletion layer [

19].

Researchers have been increasingly focusing on MPPT techniques [

20,

21,

22,

23,

24]. Authors in [

25] have proposed a glowworm swarm optimization-based MPPT for PVs exposed to uneven temperature distribution and solar irradiation. A technique based on Radial Movement Optimization (RMO) for detecting the MPPT under partial shading conditions and then compared with the results of the particle swarm optimization (PSO) method is studied in [

19]. Authors in [

26] focus on the analysis of dynamic characteristic for solar arrays in series and MPPT based on optimal initial value incremental conductance strategy under partially shaded conditions. In [

27] the authors optimize the MPPT with a model of a photovoltaic panel with two diodes in which the solution is implemented by Pattern Search Techniques. A PV source that was made by utilizing un-illuminated solar panels and a DC power supply that functions in current source mode is proposed in [

18]. The authors in [

28] address a simple genetic algorithm (GA)—based MPPT method and then compare the experimental and theoretical results with conventional methods. A direct and fully explicit method of extracting the solar cell parameters from the manufacturer datasheet is tested and presented in [

29] and the authors base their method on analytical formulation which includes the use of the Lambert W-function with the aim of turning the series resistor equation explicit. The authors in propose a three-point weight method shared with fuzzy logic for increasing the speed of MPPT [

30] and in this study the simulation was performed in Matlab and was experimentally validated.

The followed methodology was made for the comparison of the models in meteorological conditions as wide as possible. Extreme scenarios of incident solar radiation were simulated. The simulated temperature was considered suitable. The main goal of this study is to simulate and compare the characteristic curves of equivalent circuits of the ideal PV cell and, with one and with two diodes, respectively, namely equivalent circuits with five and seven parameters. The role of every parameter is assessed and compared. The ideal model of the PV cell is given in detail in [

31]. The aim was to find areas of model intervention in which the modeling could lead to identical results. In this study the numerical simulation of the models is used intensively and extensively. The method used to model the equivalent circuits allows an adequate simulation of the photovoltaic array systems by taking into consideration the compromise between accuracy and complexity. By using the Newton–Raphson method the studied models are simulated through the use of Matlab/Simulink. All the simulations were carried out on the basis of a solar cell whose electrical specifications are given in [

32].

The remainder of this paper is organized as follows. In

Section 2 the equivalent circuit with five parameters is presented while in

Section 3 the equivalent circuit with seven parameters is presented. The comparison between the one-diode model and the two-diode model is presented in

Section 4. Finally, the conclusions are addressed in

Section 5.