#
Equivalent Circuit Model for Cu(In,Ga)Se_{2} Solar Cells Operating at Different Temperatures and Irradiance

^{1}

^{2}

^{*}

^{†}

## Abstract

**:**

_{2}solar cells fabricated by the low-temperature pulsed electron deposition technique. A reduced form approach of the one-diode model has been adopted, leading to an accurate identification of the cell. It was possible to elaborate suitable relations describing the behavior of the parameters as functions of the environmental conditions. This allowed accurately predicting the trends of the parameters from a pair of curves, instead of a whole set of measurements. The developed model describing the dependence on irradiance and temperature was validated by means of a large set of experimental measurements on several Cu(In,Ga)Se

_{2}(CIGS) devices built with the same technological process.

## 1. Introduction

_{2}(CIGS) [1] is the quickest expanding thin-film technology, despite the expensive fabrication costs required for standard multiple-stage cell production routes based on sputtering [2], thermal co-evaporation [3] and the notable NRELthree-stage CIGS deposition process [4]. Nowadays, lab-scale CIGS solar cells with a conversion efficiency ≈23% have been obtained [5]. Low-Temperature Pulsed Electron Deposition (LTPED) is a novel and less expensive alternative growth technology for depositing CIGS on a variety of different substrates including ultra-lightweight metal foils and thermolabile substrates, with no need for post-deposition treatments like selenization or high temperature annealing [6]. Thanks to the combination of low substrate temperatures (≈250 °C) and the unique properties of the single-stage PED ablation process, solar cells with efficiencies up to 17% have been fabricated [7]. The huge expansion experienced by CIGS-based PV technology in this last decade needs suitable tools to estimate and predict the power generation of CIGS-based PV systems. Recent studies on the current-voltage characteristics of CIGS addressed some issues on modeling, such as in [8] or in [9], where analytical models of the J-V characteristics of CIGS-based thin film solar cell was proposed, starting from the study of the junction parameters. On the other hand, few works, to the authors’ knowledge, have taken into account the dependence of the current-voltage characteristic on irradiance and temperature conditions. In particular, in [10], an analysis of the current-voltage curves of a CIGS solar cell from experimental data for different irradiation conditions was proposed. In the cited work, an analysis featuring high irradiance levels (from 1–5 Sun at steps of 0.5 Sun) was taken into account. A specific modified double diode model was proposed, and the dependence of the model parameters on irradiance level was investigated. Unfortunately, the temperature influence on the parameters is omitted both in terms of modeling and in terms of experimental study. The most complete analysis proposed in the literature is probably the one in [11], where an analytical model for CIGS, with dependence on illumination and temperature, is presented. In this paper, starting from a physical model of the junction, a modified version of the one-diode model is presented, where the circuital parameters depend nonlinearly on voltage, irradiance and temperature. This model seems to be extremely complete and, in theory, could be used to predict the power production for a PV system based on CIGS. In terms of usage, it requires a fitting procedure based on current-voltage curves from both dark and illuminated (for example at 1 Sun) conditions, and consequently, it is easily implementable. The main inconvenience is related to the analytical model. The model was developed on CIGS cells featuring moderately high efficiency (as stated by the authors) and requires further analysis for general CIGS cells. These results seem to suggest the necessity for a modified and specific model for CIGS, since the direct adoption of the available model for Si-based devices cannot ensure the required accuracy. Moreover, the study of tailored models for the different technologies is still an open issue [12,13,14,15,16]. Indeed, each PV technology has different characteristics, and only experimental characterization, together with suitable models, allows gaining insight into its behavior. In general, the most widely-used circuital representation for the description of the current voltage relationship of PV cells is the one-diode (or five-parameter) model [17,18], whose lumped circuital parameters can be identified either from datasheet information [19,20] or from experimental data [21]. Clearly, the current-voltage curve depends on environmental conditions, and consequently, even for such a model, the lumped parameters of the circuital model change according to solar radiation and cell temperature [18]. For this reason, it is important to study this dependency for each parameter and develop, for each different technology, a complete model able to take into account this effect and predict the current-voltage curve in any environmental condition. The bottleneck of this kind of approach is the effectiveness of the model extracted, since it is affected by the error in the identification of the circuital parameters: indeed, to identify the model, it is necessary to solve a least squares problem involving noisy experimental data, which is a non-trivial task. The main critical issues of this problem are the size of the solution space and the choice of the initial values of the parameters. These issues have also led to some erroneous considerations about the validity of the one-diode model for different technologies, due to the extraction of non-physical meaning parameters. In some cases, other circuital models employing a higher number of diodes have been adopted [12,14], but this leads to further difficulties in the solution of the identification problem due to a higher number of parameters involved. However, as already shown in [21], the dimension of the search space for the one-diode model can be decreased by using the so-called reduced forms: this approach reduces the number of unknowns of the problem from five to two, making it possible to solve the least squares problem with efficient and simple deterministic methods. Following this approach, significant benefits can be achieved, such as a reduced execution time and better convergence, while preserving accuracy. In addition, the extracted parameters have physical meaning thanks to the tailored boundary, and for this reason, they can be successfully used to model the cell behavior. For example, by observing the trends of the extracted parameters with changing environmental conditions or after prolonged exposure, it is possible to elaborate precise relations for the aforementioned parameters [22].

## 2. Experimental Setup and Measurements

_{3}N

_{4}blocking layer and a 0.5 µm-thick Mo contact. A sodium-reservoir NaF layer was deposited by LTPED at a substrate temperature of 80 °C to a thickness of the order of 8–10 nm. After the temperature of the deposition surface was increased up to 250 °C, the CIGS absorber with a Ga/(In + Ga) ratio = 0.5 was grown by LTPED up to a thickness of about 1.6 µm, monitored in situ by an IR pyrometer. The solar cells were then completed with a 70 nm-thick CdS n-type buffer layer grown by Chemical Bath Deposition (CBD), followed by 50 nm of undoped ZnO and 250 nm of Al:ZnO, both deposited by RF-sputtering. Al contacts (1 µm thick) were finally thermally evaporated on the top surface. The details about the structural and the morphological properties of LTPED CIGS solar cells can be found in previous works [6,7,23]. The representative scheme of the solar cell architecture is shown in Figure 1. From a single fabrication batch, a set of 16 different cells with a total area of 0.25 cm

^{2}was defined by mechanical scribing. Every cell was characterized under Standard Reference Conditions (SRC) (25 °C, AM1.5spectrum and 100 mW/cm

^{2}) using the following experimental setup:

- A class ABB solar simulator equipped with a xenon lamp was used to reproduce the solar spectrum;
- three air mass filters were used to shape the spectrum;
- a system to sense and tune the temperature of the cells made by a thermocouple, a Peltier cell and a temperature controller was used to set and check cell temperature;
- a curve-tracer was used to extract the current from the cells by setting a voltage in the range [−0.2, 1] V with a step of 0.01 V;
- a calibrated test cell measured the irradiance;
- all the acquisition and measurements were controlled by a LabVIEW interface.

^{2}and increasing the temperature from 25–55 °C with a step of 5 °C. The second set was obtained by setting the temperature at 25 °C and varying the irradiance S in the range [30, 130] mW/m

^{2}with a step of 20 mW/cm

^{2}. The dataset of each curve is made up of 120 samples; the measured curves are shown in Figure 2 and Figure 3.

^{2}and 25 °C) vs. SRC (100 mW/cm

^{2}and 25 °C) measurement is reported. This measurement allows assessing the difference between voltage-dependent collection and the shunt effect and is important for a physical interpretation of the device operating behavior. Indeed, for a complete physical investigation of the device behavior, larger ranges in temperatures and irradiances should be considered, especially for better understanding of the transport phenomena inside the solar cells. For further details, valid results can be found in [24,25].

## 3. Identification of the One-Diode Model from Measurements

- Initial determination of the model parameters by solving the reduced-form problem (i.e., extraction of n and ${R}_{s}$). The problem is solved by means of the least squares method on I-V curves in the voltage range (0–${V}_{oc}$). The algorithm iteratively adjusts [n, ${R}_{s}$] by minimizing the squared error between computed and measured current samples.

## 4. Model for Irradiance and Temperature Dependence

#### 4.1. Temperature Set

^{−9}. Figure 8 shows the comparison between the measured current-voltage characteristics taken from Figure 2 and the simulated values for all available samples. Figure 9 plots the error between measured and simulated samples. It is easy to note that the error is almost always below 1 × 10

^{−4}A.

#### 4.2. Irradiance Set

^{2}):

^{−4}; thus, even in this case, the method is able to identify the cell accurately.

## 5. Validation of the Proposed Model on a Further Measurement Set

^{2}–130 mW/cm

^{2}) and temperature range (from 25–50 °C) was made in order to consider typical environmental conditions for operative solar cells. The characterization process followed the same approach described previously: a couple of current-voltage curves were used to identify the model parameters, and the error in estimation for all the operating conditions was evaluated. For all the devices used in this validation, the results were satisfactory. Indeed, the mean squared error between simulated and measured values for the whole set of measurements was always below 1 × 10

^{−8}and exactly between 1.8 × 10

^{−9}and 7.7 × 10

^{−9}. In the two Figure 12 and Figure 13, we report the comparison, between measured and simulated values for all the available current-voltage curves, at different irradiance and temperatures of a single device. The device belongs to the validation set (i.e., was not previously used during the development of the model). This is done for the sake of simplicity, but analogous results were achieved for all the validation tests.

## 6. Conclusions

_{2}solar cells fabricated by the LTPED technique, the lumped circuital parameters have been found. In order to obtain an accurate identification, a reduced form approach of the one-diode model has been adopted. It was possible to elaborate a comprehensive model for irradiance and temperature dependence, i.e., suitable relations describing the behavior of the parameters as functions of the environmental conditions. The set of introduced formulas was effective and allowed the prediction of the current-voltage characteristics for different environmental conditions outside SRC. An important strength of the proposed model is that the identification of the involved parameters can be achieved by using a few measured curves. Consequently, this can be considered a further step towards the development of a tailored PV simulator for CIGS-based modules. As a final consideration, it can be seen that the the high ideality factor obtained from the model identification suggests recombination phenomena. To account for this phenomena, a more complex model, the two-diode model, could be used. Investigating the temperature and irradiance dependence of the two-diode model is a very challenging open problem, for which we believe this work can be a good starting point.

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**Schematic representation of the architecture of the Cu(In,Ga)Se

_{2}(CIGS)-based cell under study. CBD, Chemical Bath Deposition; LTPED, Low-Temperature Pulsed Electron Deposition.

**Figure 4.**Dark (0 mW/cm

^{2}and 25 °C) vs. SRC (100 mW/cm

^{2}and 25 °C) I-V characteristics of the CIGS cells.

**Figure 5.**Irradiance [30, 130] mW/m

^{2}dependence of the short-circuit current (

**left**) and open-circuit voltage (

**right**).

**Figure 6.**Temperature [25, 55] °C dependence of the short-circuit current (

**left**) and open-circuit voltage (

**right**).

**Figure 8.**Comparison between measured and simulated values (obtained by the proposed set of formulas) for all the available current voltage curves at different temperatures. Irradiance was fixed at 100 mW/cm

^{2}, and the temperature range was [25, 55] °C.

**Figure 9.**Absolute error in logarithmic scale between the measured and simulated value for all datasets with different temperatures. Irradiance was fixed at 100 mW/cm

^{2}, and the temperature range was [25, 55] °C.

**Figure 10.**Comparison between measured and simulated values (obtained by proposed set of formulas) for all the available current voltage curves at different irradiance. Temperature was fixed at 25 °C, and the irradiance range was [30, 130] mW/m

^{2}.

**Figure 11.**Absolute error in logarithmic scale between measured and simulated values for all datasets with different irradiance. Temperature was fixed at 25 °C, and the irradiance range was [30, 130] mW/m

^{2}.

**Figure 12.**Comparison between measured and simulated values (obtained by the proposed set of formulas) for all the available current voltage curves at different irradiance levels [30, 130] mW/m

^{2}and a temperature of 25°C.

**Figure 13.**Comparison between measured and simulated values (obtained by the proposed set of formulas) for all the available current voltage curves at different temperatures [25, 55] °C for an irradiance value of 100 mW/cm

^{2}.

**Figure 14.**Comparison between measured and simulated values (obtained by the proposed set of formulas) for all the available current voltage curves at conditions outside SRC for both temperature and irradiance.

**Table 1.**Photovoltaic parameters of CIGS cells measured under Standard Reference Conditions (SRC) ((*) measured under dark conditions).

Parameter | Measured Value |
---|---|

${J}_{irr}$ | 27.4 ± 1.3 mA/cm^{2} |

${V}_{oc}$ | 0.686 ± 0.010 V |

FF | 0.67 ± 0.02 |

$\eta $ | 12.6 ± 0.82% |

${R}_{sh}$ | 135 kΩcm^{2} (*) |

${R}_{s}$ | 12 Ωcm^{2} (*) |

50 mW/cm^{2} | 90 mW/cm^{2} | 110 mW/cm^{2} | |
---|---|---|---|

${I}_{irr}$ (A) | 0.003328 | 0.006292 | 0.007516 |

${I}_{o}$ (A) | 4.913 × 10^{−10} | 1.958 × 10^{−9} | 3.316 × 10^{−9} |

${R}_{sh}$ (Ω) | 1.802 × 10^{3} | 1.077 × 10^{3} | 1.030 × 10^{3} |

${R}_{s}$ (Ω) | 3.611 | 3.153 | 3.342 |

n | 1.618 | 1.777 | 1.847 |

25 °C | 40 °C | 55 °C | |
---|---|---|---|

${I}_{irr}$ (A) | 0.006840 | 0.006848 | 0.006899 |

${I}_{o}$ (A) | 1.018 × 10^{−9} | 9.904 × 10^{−9} | 1.771 × 10^{−7} |

${R}_{sh}$ (Ω) | 925 | 1243 | 1807 |

${R}_{s}$ (Ω) | 3.877 | 2.606 | 2.882 |

n | 1.729 | 1.740 | 1.887 |

Case | MSE |
---|---|

Case 1 (30 °C–45 °C) | 3.0651 × 10^{−9} |

Case 2 (35 °C–55 °C) | 3.1132 × 10^{−9} |

Case 3 (25 °C–40 °C–55 °C) | 2.8143 × 10^{−9} |

Case 4 (complete dataset) | 2.7682 × 10^{−9} |

Case | MSE |
---|---|

Case 1 (30 mW/cm^{2}–110 mW/cm^{2}) | 3.8654 × 10^{−9} |

Case 2 (complete dataset) | 3.4023 × 10^{−9} |

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## Share and Cite

**MDPI and ACS Style**

Bronzoni, M.; Colace, L.; De Iacovo, A.; Laudani, A.; Lozito, G.M.; Lucaferri, V.; Radicioni, M.; Rampino, S. Equivalent Circuit Model for Cu(In,Ga)Se_{2} Solar Cells Operating at Different Temperatures and Irradiance. *Electronics* **2018**, *7*, 324.
https://doi.org/10.3390/electronics7110324

**AMA Style**

Bronzoni M, Colace L, De Iacovo A, Laudani A, Lozito GM, Lucaferri V, Radicioni M, Rampino S. Equivalent Circuit Model for Cu(In,Ga)Se_{2} Solar Cells Operating at Different Temperatures and Irradiance. *Electronics*. 2018; 7(11):324.
https://doi.org/10.3390/electronics7110324

**Chicago/Turabian Style**

Bronzoni, Matteo, Lorenzo Colace, Andrea De Iacovo, Antonino Laudani, Gabriele Maria Lozito, Valentina Lucaferri, Martina Radicioni, and Stefano Rampino. 2018. "Equivalent Circuit Model for Cu(In,Ga)Se_{2} Solar Cells Operating at Different Temperatures and Irradiance" *Electronics* 7, no. 11: 324.
https://doi.org/10.3390/electronics7110324