1. Introduction
Due to the unlimited use of the Earth’s natural resources, the increase in the concentration of greenhouse gases and the rise in average global temperatures has led to the urgent need for cleaner and renewable energy sources [
1,
2,
3,
4]. In recent years, solar energy has reached installation record values and it is considered one of the most important renewable energy sources. This type of green energy also contributes to the economic development of a more wide group of countries. Although the initial investment might be huge, natural resources are available around the world, being a never-ending source of energy. In the case of solar radiation that reaches the Earth’s surface, electricity is generated through the photoelectric effect/conversion, leading to comparative advantages in cost, technological setup complexity and environmental benefits [
1,
5].
In the last decades, the development of solar cells evolved in three technological generations, mainly due to the use of different materials and architectures [
1,
6,
7]. Solar cells based on crystalline silicon structures are widely used and have high efficiencies. These types of cells are dominating the technology market, since silicon is an abundant element on Earth, with a non-toxic manufacturing process [
1,
7,
8,
9,
10].
As with many other semiconductor devices, the solar cell can be included and simulated in circuit analyses with the aid of an electrical model, which represents its electrical performance and working principle [
1,
8]. Generally, those models are based on continuous I-V functions, which can be obtained from a mathematical fitting of a set of experimental data. This fitting preserves a certain I-V shape, which is considered from the solar cell ideal working principle. The electrical model can be almost ideal, such as the 1M3P (one Model of three Parameters), which only takes into account the p-n junction diode characteristic at dark mode, or can be made more accurate, including resistive losses, such as those depicted in 1M5P (one Model of five Parameters) [
1,
11].
The motivation to proposed this model is based on the graphical connections between consecutive I-V points. When one is analysing the points in any tool as a continuous plot, the tool just connect the scatter points. The aim of this research work is to try to develop a model based on the 1M5P and on that fact: connecting adjacent points (with small voltage difference) must be better than obtaining a certain number of parameters that represents the whole voltage range (as the continuous models do). It is done by discretizing the diode on the 1M5P (or 1MxP in general), and for that reason, the proposed model is named as d1MxP, meaning that it is a discrete (d) model (1M) with x parameters (xP). The discretization is done using the equivalent model of a diode, which is a series of an ideal diode, a resistance and an independent voltage source.
2. Methodology
The solar cell electrical model is an approach to characterise the major physical phenomena and working principles that occur in the device [
1].
The model proposed in this work is based on the discrete analyses of the well-known 1M5P model, illustrated in
Figure 1, but the discretization can also be applied to more complex models, with an increased number o diode brunches, such as the 1M7P (also known as 2D7P) model [
1,
12]. For that reason, the proposed model will be hereafter named d1MxP (d for discrete, with x parameters).
Based on simple circuit analyses and on the p-n junction working principle, it is possible to analytically define the current-voltage (I-V) characteristic of a solar cell through the Equation (
1). Referring again to
Figure 1,
is the photogenerated current,
is the current that passes through the diode,
is the diode reverse current (also known as the diode saturation current) and
is the current that passes through the shunt resistance. In Equation (
1),
is the thermal voltage, which is given by Equation (
2) (usually assumed at room temperature, leading to a value of 26 mV), where
k is the Boltzmann’s constant,
T is the temperature and
q is the electron charge,
n is the diode non-ideality factor,
is the series resistance associated to a voltage loss due to the cell’s connections and
is the shunt/parallel resistance related to a current loss due to current leaks on the cell [
1,
13,
14].
The first difficulty in using this model is that the independent variable, the cell’s output current, depends not only on the cell’s output voltage but also on itself (cell’s output current), leading to an implicit mathematical function. In this scenario, the Newton–Raphson algorithm is applied as depicted in Equation (
3), where
corresponds to the function that translates the current-voltage relation of a solar cell, Equation (
1), and
is its first derivative, with respect to
I (current function’s derivative with respect to the current). For each
V, the output current
I is determined based on the Newton–Raphson algorithm, given by Equation (
3), where
i is the iteration index.
Equations (
4) and (
5) were used to extract the five model parameters:
,
,
,
n and
. This methodology is based on a set of relevant experimental points, namely, the short-circuit current
, and the open-circuit voltage
. To obtain the value for
n based on expression (
5) it is necessary to perform graphical or iterative techniques [
14]. For example, the 1M3P model is obtained when
and
[
1,
14].
and
in Equation (
5) correspond to the I-V coordinates of the point of maximum power, which is the point where the solar cell is delivering the maximum power for a specific load [
1].
The main disadvantage of this classic approach is that the entire voltage range is characterised by the same equation, which is prone to local inaccuracies. Even worse, when the model is deeply analysed and the five parameters are obtained with expressions (
4) and (
5), it is possible to verify that only a few number of points are used for fitting in the whole range. In fact,
is only dependent on a small number of points near the open-circuit voltage (sometimes only two since it is the simplest way to compute the curve slope). The same happens for the
parasitic resistance, but near the short-circuit current path of the I-V experimental data. The photogenerated current is just one single point, at most an average of a few points near the current axis and the
and
n are obtained using points on the curve slope (typically two, as suggested).
The model proposed in this work considers the solar cell performance between two adjacent (experimental) points, which is equivalent to an analytical incremental calculation. In this way, the solar cell diode is decomposed into a set of adjustable parallel diodes, to attain an increased overall precision in the considered solar cell voltage range.
Figure 2 presents an equivalent model of the diode in the solar cell model. The subset model is composed of an ideal diode (an ideal electrical switch, which might be characterised as a diode with null voltage drop at on-state, null on-state resistance, infinite breakdown/reverse voltage and without parasitic capacitive effects), in series with a given resistance and an ideal independent voltage source.
Figure 3 represents the I-V curve obtained from this equivalent model. The ideal diode has its characteristic in red and the curve of the entire serial branch is in light blue. The diode is at on-state (as an ideal closed switch) whenever its anode voltage is above the voltage source V
. With the ideal diode at on-state, the branch collects current from the model current source, leading to less available current at the cell terminals. The slope of the I-V curve with this diode equivalent model is then proportional to
.
Based on this approach, it is possible to extend the concept in order to connect two adjacent points in the I-V curve obtained experimentally, as suggested in
Figure 4. As can be seen, the slope between these two adjacent points is dependent on the resistance of the branch. On the other hand, with a specific sizing methodology, the ideal diode at branch N can become responsible to activate the respective branch, considering the effect of the entire N-1 parallel branches previously activated. The N branch resistance is only associated with the angle to adjust the previous slope to the next one, meaning that the resistance value is based on the previous and on the pretended slope,
m. The resultant calculation process is equivalent to adding weighted resistances in parallel to attain a curve fitting to the experimental data. The mathematical formalism is presented in the system of Equation (
6) and
Figure 5 is the illustration of the obtained equivalent circuit model for N branches.
Shunt and series resistances are not explicitly represented in the model. However, the shunt resistance (current losses) is the one associated with a null-voltage, meaning that the branch is active in the whole range reducing the current flow from the current source to the load (output). On the other hand, the series resistance is not directly computed in the model, but the voltage losses (the meaning of that resistance in 1M5P model) are associated to the slope variation near the solar cell open-circuit characteristic.
Since this model is based on 1MxP/1DxP (1M7P or 1M5P for instance), some assumptions are made. The assumptions are the same to the ones performed on the application of these models. The most important to be highlighted is that these models assume that the I-V slope decreases (becomes more negative) by increasing the output resistive load (or voltage). The exponential behaviour of the diode I-V curve leads to the increase of the slope’s magnitude by increasing the voltage. This assumption comes from the assumptions of 1M5P or 1M7P models [
1,
15,
16,
17]. Experimentally, it is possible to obtain sequences of points that do not follow this rule. In these cases, the model considers these points as outliers. Since the experimental I-V curve data usually have a very low quantity of outliers, it is expected that the proposed model should continue to follow the experimental behaviour.
4. Conclusions
The aim of this research work is to evaluate and validate a more accurate electrical model for characterising solar cells’ performance. Aiming the error minimisation, the proposed electrical model tries to connect every two adjacent points of a I-V curve, in contrast to the classic 1M5P model (or, in general, 1MxP models), which uses a small portion of the data set to characterise the entire voltage range.
The proposed model uses a subset of parallel and finite branches based on the 1M5P diode model, discretizing the diode electrical performance and using its own equivalent model. Thus, its exponential I-V curve is section-wise linear in the model, which is quite useful for characterising the performance of a set of solar cells. All the imperfections in a solar cell might be better characterised using this approach.
The proposed mathematical and electrical model allows us to follow experimental data and the computation of more accurate I-V and P-V curves. Consequently, the estimation of the maximum power point, the Fill Factor, efficiency and other figures of merit can become more accurate and precise, when compared with the 1M5P or 1M7P equivalent models. In this case, it results in an improvement of up to 3.34% in the maximum power, up to 5.70% in its voltage and up to 8.20% in its current. On the other hand, a variation up to 35.98% was registered in the Fill Factor determination.