# Mathematical Modelling of a Static Concentrating Photovoltaic: Simulation and Experimental Validation

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## Abstract

**:**

_{opto-e}needs to be included in the mathematical model, which were obtained from experiments. The range of incident angle (±50°) was selected to demonstrate that the RADTIRC is capable of capturing the sun rays within its acceptance angle of ±40°. In each simulation, the I-V and P-V characteristics were produced, and the short circuit current (I

_{sc}), the open-circuit voltage (V

_{oc}), the maximum power (P

_{max}), the fill factor (FF) and the opto-electronic gain (C

_{opto-e}) were determined and recorded. The results from the simulations were validated via experiments. It was found that the simulation model is able to predict the I-V and P-V characteristics of the RADTIRC as well as its angular response, with the highest error recorded for the I

_{sc}, V

_{oc}, P

_{max}, FF and C

_{opto-e}was 2.1229%, 5.3913%, 9.9681%, 4.4231% and 0.0000% respectively when compared with the experiment.

## 1. Introduction

^{2}[1]. Therefore, to allow solar energy to be used for practical applications, solar cells are group together into solar panels. Depending on the application, solar panels can be grouped and arranged into solar arrays, providing a larger surface area, generating significantly more electricity.

_{2}) nanorods embedded nanopillar and argued that this technique can be used to enhance the efficiency of dye-sensitized solar cells (DSSCs). Zhang et al. [5] attempted to optimise the performance of an organic solar cell by studying the mechanisms of exciton diffusion and managed to obtain a power conversion efficiency of 16.35%. Al Kurdi et al. [6] researched on the use of a naphthalene diimide side-chain polymer as an electron-extraction layer to create a perovskite solar cell and achieved a power conversion efficiency of 14%. Angmo et al. [7] reported a roll-to-roll fabrication method which enables the mass production of the perovskite solar and the cell maintains a power conversion efficiency of 15.2%.

^{2}). Wang et al. [31] developed a new metal halide perovskites to enhance the CPV performance and demonstrated that the cell accomplished the highest efficiency of 23.6% under 14 Suns. Huo et al. [32] improved the quantum dots solar cells (QDSC) through the multi-photon absorption (MPA) to produce high densities of excitons, and achieved a 21.29% efficiency when tested with polymer lens CPVs that has a geometrical concentration gain of 4.08. Neo et al. [33] increased the performance of an LSC by introducing a thick zinc chalcogenide and achieved an efficiency of 0.53%. Meinardi et al. [34] on the other hand utilised indirect-bandgap semiconductor nanostructures to attain an LSC with an optical efficiency of 2.85%.

## 2. Methodology

## 3. Mathematical Formulation of the CPV

#### 3.1. RADTIRC as the Reference for the Model

^{2}. The concentrator is designed to achieve a half-acceptance angle of ±40° along its x-axis. The indoor experiments utilised a Class A solar simulator produced by Abet Technologies from the Sun 2000 model, a Keithley 2400 source meter, which are connected to a computer that has been installed with a National Instrument software. The experiments were conducted under STCs to mimic the simulation conditions, and the setup is depicted in the following Figure 2.

_{opto-e}of the RADTIRC to be determine at different angle of incidence. The opto-electronic gain of the RADTIRC at each angle of incidence is presented in Table 1. The opto-electronic gain is a ratio of short circuit current generated from RADTIRC-PV to the one generated from a bare PV cell. Sellami and Mallick [52] define the opto-electronic gain as a product of optical efficiency and the geometrical concentration gain of a concentrator. The optical efficiency value determines the amount of sun rays that travel from the concentrator aperture to its exit aperture [17]. It takes into consideration the rays that are loss due to reflection, scattering and absorption. In the case of dielectric material, it also considers the rays that escape from the side profile of the concentrator. The value of optical efficiency is ≤1 [53]. Freier [54] has demonstrated that the optical efficiency value is also independent of the irradiance value.

_{opto-e}needs to be included in the mathematical model, which was obtained from experiments. Any CPV will focus the solar irradiance from a large area into a smaller area where a solar PV is placed [55]. The relationship between the value of irradiance with the inclusion of CPV, G

_{con}and the opto-electronic gain, C

_{opto-e}, the solar irradiance, G, the irradiance at reference 1000 W/m

^{2}, G

_{ref}and the incidence angle, θ is presented in Equation (1). Note that for a non-CPV, the value of C

_{opto-e}is equal to 1.

#### 3.2. Mathematical Model of a PV Cell

_{ph}, (i.e., a light generated current source). However, in reality, additional parameters must be considered, including the resistivity of the material, the ohmic losses and the shunt resistance’s effect on the cell. These parameters can be modelled as the shunt resistance, R

_{sh}and series resistance, R

_{s}. The equivalent circuit of the PV cell can be represented by a single-diode model (see Figure 3). Several researchers enhanced the single-diode model by including an additional diode into the circuit [56,57,58,59]. The new representation is denoted the two-diode model. This extra diode corresponds to the recombination effects of the charge carriers [60,61]. Despite providing a more precise representation of the PV cell’s outputs, the two-diode model requires much longer computational time compared with the previous single-diode model [60].

_{d}is the diode current. Equation (1) can then be transcribed as [23]:

_{s}is the saturation current for the diode, while n

_{d}corresponds to the diode ideality factor, q is the electron charge (1.602 × 10

^{−19}C), k is the Boltzman constant (1.38 × 10

^{−23}J/K), T is the temperature of the p-n junction (in Kelvin), and V is the voltage across the PV cell.

_{ph}, I

_{rs}and I

_{s}. These values will be determined from the following equations. I

_{ph}is independent of V (or R

_{s}), linearly dependent on the solar irradiance and is also affected by the temperature. This relationship is presented in the following Equation (4) [38].

_{ref}is 25 °C and α is the current temperature coefficient.

_{rs}and the saturation current, I

_{s}are determined using Equations (5) and (6), where the value of the energy bandgap, E

_{g}is provided by the manufacturer. The key parameters for the solar PV cell are tabulated in Table 2. These parameters were provided by the manufacturer of the solar PV cell, Narec (National Renewable Energy Centre).

## 4. Simulation Process

_{ph}, I

_{rs}and I

_{s}.

_{oc}. The corresponding value of power, P is calculated by multiplying the corresponding values of I and V. The I-V and P-V curves are then plotted at the end of the programme. In each simulation, the I-V and P-V characteristics were produced, and the short circuit current (I

_{sc}), the open-circuit voltage (V

_{oc}), the maximum power (P

_{max}), the fill factor (FF) and the opto-electronic gain (C

_{opto-e}) were determined and recorded.

_{s}and t

_{exp}are the simulation and experimental values respectively. Table A1, Table A2, Table A3 and Table A4 in the Appendix demonstrated the comparison of simulation and experimental values (short circuit current, open circuit voltage, maximum power, fill factor, opto-electronic gain and the relative error) for the RADTIRC-PV structure and the bare PV cell.

## 5. Results and Discussions

^{2}and at a temperature of 25 °C. From the simulations, the bare PV cell generated 0.0350 A of short circuit current, 0.5860 V of open-circuit voltage and 0.0165 W of maximum power output. The inclusion of RADTIRC in the structure increases the short circuit voltage to 0.1460 A, the open-circuit voltage to 0.0627 V and the maximum output power to 0.0740 W. The simulated opto–electronic gain, (calculated by dividing the short circuit current generated from the concentrator by the one produced from a bare PV cell), is calculated to be 4.1714.

_{sc}), the open-circuit voltage (V

_{oc}), the maximum power (P

_{max}), the fill factor (FF) and the opto-electronic gain (C

_{opto-e}) were determined and recorded. The I-V and P-V characteristics generated from the simulations were compared with values obtained from the experiment.

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Appendix A

Incident Angle (°) | Simulation | Experiment | Relative Error | ||||||
---|---|---|---|---|---|---|---|---|---|

V_{oc}(V) | I_{sc}(A) | P_{max}(W) | V_{oc}(V) | I_{sc}(A) | P_{max}(W) | V_{oc}(%) | I_{sc}(%) | P_{max}(%) | |

0 | 0.6270 | 0.1460 | 0.0740 | 0.6210 | 0.1460 | 0.0720 | 0.9569 | 0.0000 | 2.7074 |

±5 | 0.6270 | 0.1395 | 0.0706 | 0.6190 | 0.1420 | 0.0706 | 1.2759 | 1.8160 | 0.0168 |

±10 | 0.6250 | 0.1342 | 0.0678 | 0.6095 | 0.1370 | 0.0657 | 2.4800 | 2.1229 | 3.0815 |

±15 | 0.6220 | 0.1219 | 0.0613 | 0.6085 | 0.1240 | 0.0595 | 2.1704 | 1.7529 | 3.0085 |

±20 | 0.6190 | 0.1109 | 0.0556 | 0.6190 | 0.1130 | 0.0554 | 0.0000 | 1.8570 | 0.2555 |

±25 | 0.6160 | 0.0998 | 0.0497 | 0.6010 | 0.1010 | 0.0479 | 2.4351 | 1.1955 | 3.7888 |

±30 | 0.6110 | 0.0843 | 0.0417 | 0.5928 | 0.0851 | 0.0394 | 2.9787 | 0.9538 | 5.5375 |

±35 | 0.6100 | 0.0530 | 0.0255 | 0.5805 | 0.0532 | 0.0238 | 4.8361 | 0.4523 | 6.6623 |

±40 | 0.5960 | 0.0326 | 0.0153 | 0.5662 | 0.0327 | 0.0141 | 5.0000 | 0.3299 | 7.4529 |

±45 | 0.5860 | 0.0235 | 0.0108 | 0.5580 | 0.0235 | 0.0099 | 4.7782 | 0.1969 | 8.5441 |

±50 | 0.5750 | 0.0157 | 0.0070 | 0.5440 | 0.0156 | 0.0063 | 5.3913 | 0.4337 | 9.9681 |

Incident Angle (°) | Simulation | Experiment | Relative Error | ||||||
---|---|---|---|---|---|---|---|---|---|

V_{oc}(V) | I_{sc}(A) | P_{max}(W) | V_{oc}(V) | I_{sc}(A) | P_{max}(W) | V_{oc}(%) | I_{sc}(%) | P_{max}(%) | |

0 | 0.5860 | 0.0350 | 0.0165 | 0.5860 | 0.0350 | 0.0162 | 0.0000 | 0.0000 | 1.4684 |

±5 | 0.5860 | 0.0349 | 0.0164 | 0.5855 | 0.0355 | 0.0165 | 0.0853 | 1.8160 | 0.2868 |

±10 | 0.5860 | 0.0345 | 0.0162 | 0.5810 | 0.0352 | 0.0162 | 0.8532 | 2.1229 | 0.0304 |

±15 | 0.5850 | 0.0338 | 0.0159 | 0.5670 | 0.0344 | 0.0153 | 3.0769 | 1.7529 | 3.7463 |

±20 | 0.5840 | 0.0329 | 0.0154 | 0.5735 | 0.0335 | 0.0152 | 1.7979 | 1.8570 | 1.7812 |

±25 | 0.5830 | 0.0317 | 0.0148 | 0.5800 | 0.0321 | 0.0147 | 0.5146 | 1.1955 | 0.9565 |

±30 | 0.5820 | 0.0303 | 0.0141 | 0.5760 | 0.0306 | 0.0139 | 1.0309 | 0.9538 | 2.0716 |

±35 | 0.5800 | 0.0287 | 0.0133 | 0.5720 | 0.0288 | 0.0129 | 1.3793 | 0.4523 | 3.1050 |

±40 | 0.5790 | 0.0268 | 0.0124 | 0.5720 | 0.0269 | 0.0121 | 1.2090 | 0.3299 | 2.8453 |

±45 | 0.5760 | 0.0247 | 0.0114 | 0.5690 | 0.0247 | 0.0110 | 1.2153 | 0.1969 | 3.3874 |

±50 | 0.5740 | 0.0225 | 0.0103 | 0.5660 | 0.0224 | 0.0099 | 1.3937 | 0.4337 | 3.5023 |

**Table A3.**Comparison of simulation and experimental value for the fill factor of the RADTIRC-PV and the bare PV cell.

Incident Angle (°) | Simulation | Experiment | Relative Error | |||
---|---|---|---|---|---|---|

Bare PV (FF) | RADTIRC-PV (FF) | Bare PV (FF) | RADTIRC-PV (FF) | Bare PV (%) | RADTIRC-PV (%) | |

0 | 80.3597 | 79.1796 | 80.8734 | 79.4441 | 1.4684 | 1.7673 |

±5 | 80.3411 | 79.2020 | 80.7149 | 80.3133 | 1.4179 | 0.4975 |

±10 | 80.2846 | 79.2681 | 80.8361 | 78.6674 | 1.2661 | 2.6828 |

±15 | 80.3282 | 78.3992 | 80.8798 | 78.8056 | 2.4014 | 2.5646 |

±20 | 80.3305 | 78.8794 | 80.9228 | 79.2444 | 1.8064 | 2.0740 |

±25 | 80.2897 | 78.9887 | 80.9121 | 78.8468 | 1.6204 | 2.5525 |

±30 | 80.2020 | 78.6089 | 80.9024 | 78.0245 | 1.9863 | 3.5572 |

±35 | 80.1988 | 78.4407 | 79.0707 | 77.2041 | 2.1922 | 2.3607 |

±40 | 79.9971 | 78.4134 | 78.6697 | 76.3865 | 1.9797 | 2.9023 |

±45 | 80.0004 | 78.3956 | 78.3793 | 75.4280 | 2.0059 | 3.7654 |

±50 | 79.7782 | 78.4123 | 77.6704 | 74.2350 | 1.7121 | 4.4231 |

Incident Angle (°) | Simulation | Experiment | Relative Error (%) |
---|---|---|---|

0 | 4.1714 | 4.1714 | 0.0000 |

±5 | 4.0000 | 4.0000 | 0.0000 |

±10 | 3.8920 | 3.8920 | 0.0000 |

±15 | 3.6047 | 3.6047 | 0.0000 |

±20 | 3.3731 | 3.3731 | 0.0000 |

±25 | 3.1464 | 3.1464 | 0.0000 |

±30 | 2.7810 | 2.7810 | 0.0000 |

±35 | 1.8472 | 1.8472 | 0.0000 |

±40 | 1.2156 | 1.2156 | 0.0000 |

±45 | 0.9514 | 0.9514 | 0.0000 |

±50 | 0.6964 | 0.6964 | 0.0000 |

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**Figure 1.**(

**a**) A small concentrating solar PV window integrating the rotationally asymmetrical dielectric totally internally reflecting concentrators (RADTIRCs); and (

**b**) a single RADTIRC that has an acceptance angle of ±40°.

**Figure 2.**Indoor experimental setup to evaluate the I-V and P-V characteristics of the RADTIRC-PV structure and the bare PV cell.

**Figure 3.**A single-diode equivalent circuit. Adapted from [61].

**Figure 4.**The steps carried out at each angle of incidence to determine the angular response of the RADTIRC from −50° to 50° with an increment 5°. A 0° incident angle means that the sun rays are perpendicular to the RADTIRC.

**Figure 5.**Simulated I-V and P-V characteristics of the RADTIRC-PV structure and the bare PV cell at normal incidence.

**Figure 6.**Comparison of the short circuit currents obtained from simulations and experiments for the RADTIRC-PV structure and a bare PV cell at STCs.

**Figure 7.**Comparison of the open-circuit voltages obtained from simulations and experiments for the RADTIRC-PV structure at STCs.

**Figure 8.**Comparison of the open-circuit voltage obtained from simulations and experiments for the bare PV cells at STCs.

**Figure 9.**Comparison of the maximum powers obtained from simulations and experiments for the RADTIRC-PV structure and a bare PV cell at STCs.

**Figure 10.**Comparison of the fill factor obtained from simulations and experiments for the RADTIRC-PV structure at STCs.

**Figure 11.**Comparison of the fill factor obtained from simulations and experiments for the bare PV cells at STCs.

**Figure 12.**Comparison of the opto-electronic gains obtained from simulations and experiments for the RADTIRC-PV structure and a bare PV cell at STCs.

Angle of Incidence (°) | Opto-Electronic Gain, C_{opto-e} |
---|---|

0 | 4.17143 |

±5 | 4.00000 |

±10 | 3.89205 |

±15 | 3.60465 |

±20 | 3.37313 |

±25 | 3.14642 |

±30 | 2.78105 |

±35 | 1.84722 |

±40 | 1.21561 |

±45 | 0.95142 |

±50 | 0.69643 |

Component | Value |
---|---|

Solar radiation intensity, G_{ref} | 1000 W/m^{2} |

Reference Temperature, T_{ref} | 25 °C |

Square solar cell area, A | 1 cm^{2} |

Short circuit current, I_{sc} (at 0°) | 0.0350 A |

Open-circuit voltage, V_{oc} (at 0°) | 0.586 V |

Ideality factor, n | 1.109 |

Energy bandgap, E_{g} | 1.12 |

Charge of an electron, q | 1.6 × 10^{−19} C |

Boltzman constant, k | 1.38 × 10^{−23} m^{2} kg s^{−2} K^{−1} |

Series resistance, R_{s} | 0.047994 Ω |

Shunt resistance, R_{sh} | 2148.53 Ω |

Short circuit current temperature coefficient, α | 0.00024 A/°C |

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**MDPI and ACS Style**

Muhammad-Sukki, F.; Farooq, H.; Abu-Bakar, S.H.; Ardila-Rey, J.A.; Sellami, N.; Kilpatrick, C.; Muhtazaruddin, M.N.; Bani, N.A.; Zulkipli, M.
Mathematical Modelling of a Static Concentrating Photovoltaic: Simulation and Experimental Validation. *Appl. Sci.* **2021**, *11*, 3894.
https://doi.org/10.3390/app11093894

**AMA Style**

Muhammad-Sukki F, Farooq H, Abu-Bakar SH, Ardila-Rey JA, Sellami N, Kilpatrick C, Muhtazaruddin MN, Bani NA, Zulkipli M.
Mathematical Modelling of a Static Concentrating Photovoltaic: Simulation and Experimental Validation. *Applied Sciences*. 2021; 11(9):3894.
https://doi.org/10.3390/app11093894

**Chicago/Turabian Style**

Muhammad-Sukki, Firdaus, Haroon Farooq, Siti Hawa Abu-Bakar, Jorge Alfredo Ardila-Rey, Nazmi Sellami, Ciaran Kilpatrick, Mohd Nabil Muhtazaruddin, Nurul Aini Bani, and Muhammad Zulkipli.
2021. "Mathematical Modelling of a Static Concentrating Photovoltaic: Simulation and Experimental Validation" *Applied Sciences* 11, no. 9: 3894.
https://doi.org/10.3390/app11093894