# Photovoltaic Cell and Module I-V Characteristic Approximation Using Bezier Curves
## 1 Introduction  
The forecast of the total photovoltaic (PV) installations, offered by Bloomberg New Energy Finance (BNEF) predicts an optimistic growth at 111 GW in 2018, rising to 121 GW in 2019, along with a polysilicon factory growth boom and module prices drop to US$ 0.30/W [1, 2]. This robust growth explains the high interest in PV research, modeling, and simulation-along with design and development of PV equipment. The electrical characteristics of the PV cell and PV modules have been of interest for several decades, and different models have been proposed. [PERSON] and [PERSON] [3] were among the first to propose a solution for PV cell parameter extraction. [PERSON] [4] uses a double exponential model to extract the PV cell parameters using the experimental I-V curve. [PERSON] et al. [5] developed an algorithm to find the parameters defining the I-V characteristic for the single diode model of a PV cell, using the Newton-Raphson method and imposing a minimum error threshold for the maximum power point. [PERSON] and [PERSON] [6] introduced a simplified two-diode model for the PV module, while [PERSON] et al. [7] used the Lambert W-Function for finding the solar panel equivalent circuit parameters and also proposed an LTSpice model according to their findings. Temperature influence has been studied by [PERSON] et al. [8]. [PERSON] and [PERSON] [9] use differential evolution to find the PV modules parameters. [PERSON] et al. provided extensive studies of the most widely used models both in single diode case [10] and for the double-diode version [11], introducing a criterion for rating the accuracy and usability of the analyzed models. In our previous paper [12], we proposed a complete SPICE modelincluding all the parameters variation and selfheating. In a recent work, [PERSON] et al. [13] claim a good accuracy for their electrical model for a PV module and they also discuss energy and exergy efficiency as a reliable substitute for the fill factor. All the aforementioned works use an electrical model to describe the behavior of the circuit and rely on a specific circuit to generate the I-V characteristic of the PV cell or module. This paper introduces a new approach. This time, the cell or module are not involved at the electrical level, being defined by just the specific points \(P_{sc},\ P_{mp},\ P_{oc}\) and by the parallel and series resistances \(R_{sh0}\) and \(R_{s0}\), specified at \(I_{sc}\) and \(V_{oc}\), respectively. Careful inspection of the typical I-V characteristic of the PV module or PV cell (Figure 1a,b) shows a similar pattern in all curves. Our aim was to find a way to model it using smooth curves and the datasheet information currently available. Bernstein polynomials have been studied since the beginning of the 20 th century and they form the foundation for Bezier curves [14]. The core applications for graphics came first in 1959 when the French mathematician [PERSON] developed an algorithm able to evaluate a family of specific curves at Citroen. In 1962 the French engineer [PERSON] also used them to design automobile bodies at Renault and afterwards they achieved wide acceptance [15]. Bezier curves are largely used in computer graphics [16,17] and also in the time domain for smoothing the trajectory of the robotic arms, for an accurate gluing or welding path or for trajectory generation [18]. Further development for shape representation was proposed by [PERSON] et al. [19]. The current proposal analyses the use of Bezier curves [20] in order to accurately represent the I-V characteristic of a PV cell or module. Although Bezier curves are commonly used as 2D curves (which are usually not functions) [21], our approach uses such curves to approximate an I-V characteristic (a function). A complete mathematical solution is provided, separately validated for a PV cell and a PV module and the error is analyzed. The results are also studied for different temperature and irradiances and finally compared with the ones offered by the least squares fitting method.