# Metaheuristic Algorithm for Photovoltaic Parameters: Comparative Study and Prediction with a Firefly Algorithm

^{1}

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^{*}

## Abstract

**:**

## Featured Application

**The parameter identiﬁcation of solar cell and photovoltaic module are used for evaluation, control and optimization of photovoltaic systems**.

## Abstract

^{2}. The second, is a flexible hydrogenated amorphous silicon a-Si:H solar cell single diode. The third is a commercial photovoltaic module (Photowatt-PWP 201) in which 36 polycrystalline silicon cells are connected in series, single diode, at 25 °C and 1000 W/m

^{2}from experimental current-voltage. The proposed constrained objective function is adapted to minimize the absolute errors between experimental and predicted values of voltage and current in two zones. Finally, for performance validation, the parameters obtained through the Firefly algorithm are compared with recent research papers reporting metaheuristic optimization algorithms and analytical methods. The presented results confirm the validity and reliability of the Firefly algorithm in extracting the optimal parameters of the photovoltaic solar cell.

## 1. Introduction

- the first is the so-called local attraction, since the light intensity decreases with distance (the attractions of fireflies can be local or global and depend on the absorbing coefficient);
- the second is related to the subdivision of fireflies and their regrouping into subgroups because a neighboring attraction is stronger than a long-distance attraction, and each subgroup will swarm around a local mode, making the firefly algorithm suitable for multimodal global optimization problems [66].

^{2}and at a temperature of 300 K [54]; (iii) a Photowatt-PWP 201 photovoltaic module which 36 polycrystalline silicon cells are connected in series and the data is measured at an irradiance of 1000 W/m

^{2}, and a temperature of 25 °C [29]. To verify the performance of the proposed approach and the quality of the obtained results, statistical analyses are carried out to measure the accuracy of the calculated parameters and model suitability. The results obtained are compared with recent techniques such as the Biogeography-Based Optimization algorithm with Mutation strategies (BBO-M) [68], Levenberg-Marquardt algorithm combined with Simulated Annealing (LMSA) [47], Artificial Bee Swarm Optimization algorithm [48], Artificial Bee Colony optimization (ABC) [49], hybrid Nelder-Mead and Modified Particle Swarm Optimization (NM-MPSO) [50], Repaired Adaptive Differential Evolution (RADE) [59], Chaotic Asexual Reproduction Optimization (CARO) [69] for solar cell single and double diodes. For organic flexible hydrogenated amorphous silicon, a-Si:H solar cell will be compared with the Quasi-Newton (Q-N) method and Self-Organizing Migrating Algorithm (SOMA) [54]. The optimal parameters of Photowatt-PWP 201 are compared with the Newton-Raphson [29] Pattern Search (PS) [55], Genetic algorithm (GA) [56] and Simulated Annealing algorithm (SA) [58]. The obtained results are in accordance with experimental data, there is good agreement for most of the extracted parameters and the proposed algorithm outperformed the compared techniques.

## 2. Presentation and Modelling of the Solar Cell

_{ph}is the photocurrent, I

_{SD}

_{1}and I

_{SD}

_{2}are the saturation currents, ${a}_{1}$ and ${a}_{2}$ are the diffusion and recombination diode quality factors; R

_{s}and R

_{p}are the resistances in series and parallel, respectively, ${V}_{T}$ is the thermal voltage (which will be defined in the followings), and:

_{i}represents the thickness of the i-layer, the effective µτ-product (µτ)

_{eff}represents average mobility-lifetime product for election and hole, and quantifies the quality of the active layer in terms of recombination of photo-generated carriers. The thermal voltage is ${V}_{T}=KT/q$ where K is Boltzmann’s constant, T is the cell absolute temperature in Kelvin and q is the electronic charge, a is the diode quality factor.

_{rec}is the current sink and it represents the recombination current in the i-layer of a P-I-N; the current through the diode represents the diffusion process of charge carriers and the last term represents the shunt leakage current ${I}_{p}$ and is modelled as a space charge limited current [77,78].

## 3. Problem Formulation

- for a single diode: $X=$ ${x}_{1}=\left[{I}_{ph}\text{\hspace{0.17em}}{I}_{SD}\text{\hspace{0.17em}}a\text{\hspace{0.17em}}{R}_{S}\text{\hspace{0.17em}}{R}_{P}\right]$;
- for a double diode: $X=$ ${x}_{2}=\left[{I}_{ph}\text{\hspace{0.17em}}{I}_{SD1}\text{\hspace{0.17em}}{I}_{SD2}\text{\hspace{0.17em}}{a}_{1}\text{\hspace{0.17em}}{a}_{2}\text{\hspace{0.17em}}{R}_{S}\text{\hspace{0.17em}}{R}_{P}\right]$;
- for a flexible solar cell: $X=$ ${x}_{3}=\left[{I}_{ph}\text{\hspace{0.17em}}{d}_{i}\text{\hspace{0.17em}}\mu \tau \text{\hspace{0.17em}}{V}_{bi}\text{\hspace{0.17em}}{R}_{s}\text{\hspace{0.17em}}{I}_{0}\text{\hspace{0.17em}}a\text{\hspace{0.17em}}{R}_{sh}\right]$.

_{1}, x

_{2}and x

_{3}. The Equations (1) and (3) is rewritten in the following homogeneous equations.

## 4. Firefly Optimization Algorithm

- No sex distinctions, i.e., fireflies are attracted to other fireflies regardless of their sex.
- The degree of the attractiveness of a firefly is proportional to its brightness, thus for any two flashing fireflies, the less bright one will move towards the brighter one; the more brightness, the less the distance between two fireflies. If there is no brighter firefly, it will move randomly.
- The brightness of a firefly is determined by the value of the objective function.

#### 4.1. Attractiveness

#### 4.2. Distance and movement

## 5. Results, Discussions and Comparison

- Test scenario 1: Apply to commercial solar cell for both single diode and double model under standard irradiance level with relevant example comparisons to other methods.
- Test scenario 2: Apply to a flexible hydrogenated amorphous silicon a-Si:H photovoltaic cell using single diode module.
- Test scenario 3: Apply to a commercial photovoltaic array using the single diode model, with 36 solar cells connected in series.

#### 5.1. Case 1: Single and Double Diode Model (RTC France Company)

^{2}at 33 °C. The extracted parameters are compared with those found by: Biogeography-Based Optimization algorithm with Mutation strategies (BBO-M) [68], Levenberg-Marquardt algorithm combined with Simulated Annealing (LMSA) [47], Artificial Bee Swarm Optimization algorithm [48], Artificial Bee Colony optimization (ABC) [49], hybrid Nelder-Mead and Modified Particle Swarm Optimization (NM-MPSO) [50], Repaired Adaptive Differential Evolution (RADE) [59], Chaotic Asexual Reproduction Optimization (CARO) [69], and the results for each model are reported in Table 3 and Table 4.

#### 5.2. Case 2: Organic Flexible Hydrogenated Amorphous Silicon a-Si:H Solar Cell

^{2}and at a temperature of 300 K. The experimental data are used from [48]; only the open circuit voltage V

_{o}

_{c}and short circuit current I

_{sc}are obtained. Moreover, the optimal parameters are compared with several other techniques based on the same experimental data. The extracted optimal parameters by Firefly algorithm have been reported in Table 9, compared with the Quasi-Newton method and Self-Organizing Migrating Algorithm. Since it is difficult to extract the flexible amorphous silicon solar cell circuit model parameters and the research is still comparatively rare, the Quasi-Newton (Q-N) method and Self-Organizing Migrating Algorithm (SOMA) [48] have been chosen for comparison because in [29,48] they were demonstrated to provide good results for parameter extractions.

_{3}found by the Firefly algorithm, the extracted values of ${I}_{ph}$, ${d}_{i}$, $\mu \tau $, ${V}_{bi}$, ${R}_{s}$, ${I}_{0}$, $a$ and ${R}_{sh}$ are put into Equation (3), then the current-voltage and power-voltage characteristics of this model is reconstructed with 16 pairs of current-voltage. The current-voltage and power-voltage characteristics resulting from the extracted parameters by Firefly algorithm along with experimental data have been illustrated in Figure 11. The Figures show the reconstructed model is in good agreement with the experimental data.

#### 5.3. Case 3: Commercial Silicon Photovoltaic Module Photowatt-PWP 201

^{2}. In this case, 26-pair current-voltage measured values are the same as [29], which are derived from 36 polycrystalline silicon cells which are connected in series. The extracted optimal parameters values for the photovoltaic module by Firefly algorithm have been reported in Table 12. Moreover, the optimal parameters are compared with several other techniques: Newton-Raphson [29] Pattern Search (PS) [55], Genetic Algorithm (GA) [56] and Simulated Annealing algorithm (SA) [58] based on the same experimental data. The purpose of comparison is to validate the accuracy of the Firefly algorithm in the parameter extraction process with a short time of convergence.

## 6. Conclusions

## Author Contributions

## Conflicts of Interest

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**Figure 1.**Equivalent circuit solar cell model: (

**a**) single and double diode, (

**b**) flexible hydrogenated amorphous silicon a-Si:H.

**Figure 3.**A conceptual view of the firefly algorithm relationships, including locations $x$, distance $r$, brightness $I\left(r\right)$, and attractiveness $\beta \left(r\right)$.

**Figure 5.**Individual absolute error (IAE) plots for single and double diode for Mono-crystalline silicon solar cell, RTC France Company.

**Figure 6.**Relative Error (RE) plots for single and double diode for Mono-crystalline silicon solar cell, RTC France Company.

**Figure 7.**Experimental current-voltage data compared with estimated data of the ono-crystalline silicon solar cell single diode, RTC France Company.

**Figure 8.**Experimental power-voltage data compared with estimated data of the mono-crystalline silicon solar cell single double diode, RTC France Company.

**Figure 9.**Compared experimental current-voltage and power-voltage of the mono-crystalline single diode silicon solar cell, RTC France Company.

**Figure 10.**Individual absolute error compared to, (

**a**) I-V and (

**b**) P-V for each current measurement by different algorithms.

**Figure 11.**Comparison between, (

**a**) I-V and (

**b**) P-V characteristics resulting from the experimental data, Q-N, Soma and FA.

**Figure 12.**Comparison between, (

**a**) IAE and (

**b**) RE using the extracted parameters by FA and Newton-Raphson, PS, GA and SA for photovoltaic module Photowatt-PWP 201.

**Figure 13.**Comparison of (

**a**) I-V and (

**b**) P-V curve between experimentally recorded data for photovoltaic module Photowatt-PWP 201 and the estimated results by FA.

Optimization Method | Reference |
---|---|

Least squares and Newton-Raphson method | [29] |

Iterative curve fitting | [30] |

Lambert W-functions | [20,31,32,33,34,35] |

Integral-based linear least square identification method | [36,37] |

Linear interpolation/extrapolation | [38] |

Chebyshev polynomials | [39] |

Taylor’s series expansion | [40] |

Padé approximants | [41] |

Symbolic function | [42] |

Analytical mathematical method | [43,44,45] |

Simple methods based on measured points | [46] |

Metaheuristic Methods | Reference |
---|---|

Levenberg-Marquardt algorithm combined with Simulated Annealing | [47] |

Artificial Bee Swarm | [48] |

Artificial Bee Colony | [49] |

Hybrid Nelder-Mead and Modified Particle Swarm | [50] |

Firefly Algorithm | [51,52,53] |

Self-Organizing Migrating Algorithm | [54] |

Pattern Search | [55] |

Genetic Algorithm | [56,57] |

Simulated Annealing algorithm | [58] |

Repaired Adaptive Differential Evolution | [59] |

Particle Swarm Optimization | [60] |

Bird Mating Optimization approach | [61] |

**Table 3.**Comparison of various parameter identification techniques for single diode model (RTC France Company). FA: Firefly Algorithm; BBO-M: Biogeography-Based Optimization with Mutation strategies; RADE: Repaired Adaptive Differential Evolution; LMSA: Levenberg-Marquardt algorithm combined with Simulated Annealing; CARO: Chaotic Asexual Reproduction Optimization; ABC: Artificial Bee Colony optimization; NM-MPSO: hybrid Nelder-Mead and Modified Particle Swarm Optimization.

Approaches | Parameter | ||||
---|---|---|---|---|---|

${\mathit{I}}_{\mathit{p}\mathit{h}}\left(\mathbf{A}\right)$ | ${\mathit{I}}_{0}\left(\mathsf{\mu}\mathbf{A}\right)$ | $\mathit{a}$ | ${\mathit{R}}_{\mathit{s}}\text{\hspace{0.17em}}\left(\mathbf{\Omega}\right)$ | ${\mathit{R}}_{\mathit{p}}\text{\hspace{0.17em}}\left(\mathbf{\Omega}\right)$ | |

FA | 0.76069712 | 0.4324411 | 1.45245666 | 0.03341059 | 53.40180803 |

BBO-M | 0.76078 | 0.31874 | 1.47984 | 0.03642 | 53.36227 |

RADE | 0.760776 | 0.323021 | 1.481184 | 0.036377 | 53.718526 |

LMSA | 0.76078 | 0.31849 | 1.47976 | 0.03643 | 53.32644 |

CARO | 0.76079 | 0.31724 | 1.48168 | 0.03644 | 53.0893 |

ABC | 0.7608 | 0.3251 | 1.4817 | 0.0364 | 53.6433 |

NM-MPSO | 0.76078 | 0.32306 | 1.48120 | 0.03638 | 53.7222 |

**Table 4.**Comparison of various parameter identification techniques for a double diode model (RTC France Company).

Approaches | Parameter | ||||||
---|---|---|---|---|---|---|---|

${\mathit{I}}_{\mathit{p}\mathit{h}}\left(\mathbf{A}\right)$ | ${\mathit{I}}_{01}\left(\mathsf{\mu}\mathbf{A}\right)$ | ${\mathit{I}}_{02}\left(\mathsf{\mu}\mathbf{A}\right)$ | ${\mathit{a}}_{1}$ | ${\mathit{a}}_{2}$ | ${\mathit{R}}_{\mathit{s}}\text{\hspace{0.17em}}\left(\mathbf{\Omega}\right)$ | ${\mathit{R}}_{\mathit{p}}\text{\hspace{0.17em}}\left(\mathbf{\Omega}\right)$ | |

FA | 0.760820 | 0.591126 | 0.245384 | 1.0246 | 1.3644 | 0.036639 | 55.049 |

RADE | 0.760781 | 0.225974 | 0.749347 | 1.451017 | 2.0000 | 0.036740 | 55.485443 |

CARO | 0.76075 | 0.29315 | 0.09098 | 1.47338 | 1.77321 | 0.03641 | 54.3967 |

ABSO | 0.76078 | 0.26713 | 0.38191 | 1.46512 | 1.98152 | 0.03657 | 54.6219 |

ABC | 0.7608 | 0.0407 | 0.2874 | 1.4495 | 1.4885 | 0.0364 | 53.7804 |

NM-MPSO | 0.76078 | 0.22476 | 0.75524 | 1.45054 | 1.99998 | 0.03675 | 55.5296 |

Item | ${\mathit{V}}_{\mathit{E}\mathit{x}\mathit{p}}\left(\mathbf{V}\right)$ | ${\mathit{I}}_{\mathit{E}\mathit{x}\mathit{p}}\left(\mathbf{A}\right)$ | ${\mathit{I}}_{\mathbf{Calculated}}\left(\mathbf{A}\right)$ | FA (A) | Individual Absolute Error (IAE) | ||
---|---|---|---|---|---|---|---|

RADE | BBO-M | NM-MPSO | |||||

1 | −0.2057 | 0.7640 | 0.76407143 | 7.1420 × 10^{−5} | 9.5590 × 10^{−5} | 6.0000 × 10^{−6} | 8.7000 × 10^{−5} |

2 | −0.1291 | 0.7620 | 0.76263790 | 6.3789 × 10^{−4} | 6.6611 × 10^{−4} | 6.0400 × 10^{−4} | 6.6200 × 10^{−4} |

3 | −0.0588 | 0.7605 | 0.76132213 | 8.2213 × 10^{−4} | 8.5473 × 10^{−4} | 8.1700 × 10^{−4} | 8.5400 × 10^{−4} |

4 | 0.0057 | 0.7605 | 0.76015347 | 3.4652 × 10^{−4} | 3.5034 × 10^{−4} | 3.6400 × 10^{−4} | 3.4600 × 10^{−4} |

5 | 0.0646 | 0.7600 | 0.75905434 | 9.4565 × 10^{−4} | 9.4298 × 10^{−4} | 9.4600 × 10^{−4} | 9.4500 × 10^{−4} |

6 | 0.1185 | 0.7590 | 0.75804099 | 9.5900 × 10^{−4} | 9.5528 × 10^{−4} | 9.4300 × 10^{−4} | 9.5700 × 10^{−4} |

7 | 0.1678 | 0.7570 | 0.75702642 | 2.6419 × 10^{−5} | 9.5100 × 10^{−5} | 1.2000 × 10^{−4} | 9.1000 × 10^{−5} |

8 | 0.2132 | 0.7570 | 0.75614154 | 8.5846 × 10^{−4} | 8.4950 × 10^{−4} | 8.1700 × 10^{−4} | 8.5800 × 10^{−4} |

9 | 0.2545 | 0.7555 | 0.75509107 | 4.0892 × 10^{−4} | 4.1823 × 10^{−4} | 3.6100 × 10^{−4} | 4.1300 × 10^{−4} |

10 | 0.2924 | 0.7540 | 0.75367808 | 3.2191 × 10^{−4} | 3.2967 × 10^{−4} | 2.7600 × 10^{−4} | 3.3600 × 10^{−4} |

11 | 0.3269 | 0.7505 | 0.75111180 | 6.1180 × 10^{−4} | 8.9542 × 10^{−4} | 9.5300 × 10^{−4} | 8.8800 × 10^{−4} |

12 | 0.3585 | 0.7465 | 0.74691657 | 4.1656 × 10^{−4} | 8.5737 × 10^{−4} | 9.1400 × 10^{−4} | 8.4800 × 10^{−4} |

13 | 0.3873 | 0.7385 | 0.73945849 | 9.5848 × 10^{−4} | 1.6042 × 10^{−3} | 1.6680 × 10^{−3} | 1.5960 × 10^{−3} |

14 | 0.4137 | 0.7280 | 0.72757692 | 4.2308 × 10^{−4} | 5.9912 × 10^{−4} | 5.8300 × 10^{−4} | 6.0400 × 10^{−4} |

15 | 0.4373 | 0.7065 | 0.70650197 | 1.9700 × 10^{−6} | 4.4631 × 10^{−4} | 4.8500 × 10^{−4} | 4.5200 × 10^{−4} |

16 | 0.4590 | 0.6755 | 0.67551809 | 1.8089 × 10^{−5} | 1.9600 × 10^{−4} | 2.3000 × 10^{−4} | 2.0600 × 10^{−4} |

17 | 0.4784 | 0.6320 | 0.63102588 | 9.7411 × 10^{−4} | 1.1090 × 10^{−3} | 1.2710 × 10^{−3} | 1.1170 × 10^{−3} |

18 | 0.4960 | 0.5730 | 0.57300627 | 6.2700 × 10^{−6} | 9.1027 × 10^{−4} | 1.1120 × 10^{−3} | 9.2000 × 10^{−4} |

19 | 0.5119 | 0.4990 | 0.49898281 | 1.7190 × 10^{−5} | 4.9902 × 10^{−4} | 5.6300 × 10^{−4} | 4.9000 × 10^{−4} |

20 | 0.5265 | 0.4130 | 0.41270839 | 2.9160 × 10^{−4} | 4.9030 × 10^{−4} | 6.1200 × 10^{−4} | 4.9200 × 10^{−4} |

21 | 0.5398 | 0.3165 | 0.31629674 | 2.0325 × 10^{−4} | 7.1532 × 10^{−4} | 9.8500 × 10^{−4} | 7.1800 × 10^{−4} |

22 | 0.5521 | 0.2120 | 0.21218495 | 1.8495 × 10^{−4} | 1.0468 × 10^{−4} | 1.4200 × 10^{−4} | 1.0200 × 10^{−4} |

23 | 0.5633 | 0.1035 | 0.10350897 | 8.9700 × 10^{−6} | 7.8397 × 10^{−4} | 1.2540 × 10^{−3} | 7.7900 × 10^{−4} |

24 | 0.5736 | −0.0100 | −0.01025607 | 2.5607 × 10^{−4} | 7.5437 × 10^{−4} | 1.2680 × 10^{−3} | 7.5100 × 10^{−4} |

25 | 0.5833 | −0.1230 | −0.12309841 | 9.8410 × 10^{−5} | 1.3775 × 10^{−3} | 2.5370 × 10^{−3} | 1.3810 × 10^{−3} |

26 | 0.5900 | −0.2100 | −0.21005316 | 5.3159 × 10^{−5} | 8.0320 × 10^{−4} | 1.4690 × 10^{−3} | 8.0700 × 10^{−4} |

Item | ${\mathit{V}}_{\mathit{E}\mathit{x}\mathit{p}}\left(\mathbf{V}\right)$ | ${\mathit{I}}_{\mathit{E}\mathit{x}\mathit{p}}\left(\mathbf{A}\right)$ | ${\mathit{I}}_{\mathbf{Calculated}}\left(\mathbf{A}\right)$ | Individual Absolute Error (IAE) | ||
---|---|---|---|---|---|---|

FA | RADE | NM-MPSO | ||||

1 | −0.2057 | 0.7640 | 0.76404800 | 4.7990 × 10^{−5} | 9.2680 × 10^{−5} | 2.3000 × 10^{−5} |

2 | −0.1291 | 0.7620 | 0.76265838 | 6.5837 × 10^{−4} | 6.5394 × 10^{−4} | 5.9800 × 10^{−4} |

3 | −0.0588 | 0.7605 | 0.76138191 | 8.8191 × 10^{−4} | 8.5755 × 10^{−4} | 8.3200 × 10^{−4} |

4 | 0.0057 | 0.7605 | 0.76020876 | 2.9123 × 10^{−4} | 3.3747 × 10^{−4} | 3.3000 × 10^{−4} |

5 | 0.0646 | 0.7600 | 0.75912329 | 8.7671 × 10^{−4} | 9.4000 × 10^{−4} | 8.9500 × 10^{−4} |

6 | 0.1185 | 0.7590 | 0.75806245 | 9.3754 × 10^{−4} | 9.4935 × 10^{−4} | 8.8000 × 10^{−4} |

7 | 0.1678 | 0.7570 | 0.75700411 | 4.1100 × 10^{−6} | 9.6350 × 10^{−5} | 1.8700 × 10^{−4} |

8 | 0.2132 | 0.7570 | 0.75750201 | 5.0201 × 10^{−4} | 8.5535 × 10^{−4} | 7.5700 × 10^{−4} |

9 | 0.2545 | 0.7555 | 0.75557754 | 7.7540 × 10^{−5} | 4.1885 × 10^{−4} | 3.2300 × 10^{−4} |

10 | 0.2924 | 0.7540 | 0.75409595 | 9.5950 × 10^{−5} | 3.3126 × 10^{−4} | 2.7700 × 10^{−4} |

11 | 0.3269 | 0.7505 | 0.75031932 | 1.8060 × 10^{−4} | 8.9511 × 10^{−4} | 8.9600 × 10^{−4} |

12 | 0.3585 | 0.7465 | 0.74651818 | 1.8185 × 10^{−5} | 8.4939 × 10^{−4} | 7.9800 × 10^{−4} |

13 | 0.3873 | 0.7385 | 0.73873379 | 2.3370 × 10^{−4} | 1.6021 × 10^{−3} | 1.4950 × 10^{−3} |

14 | 0.4137 | 0.7280 | 0.72816539 | 1.6540 × 10^{−4} | 6.1216 × 10^{−4} | 7.2900 × 10^{−4} |

15 | 0.4373 | 0.7065 | 0.70628557 | 2.1442 × 10^{−4} | 4.5162 × 10^{−4} | 3.4400 × 10^{−4} |

16 | 0.4590 | 0.6755 | 0.67594242 | 4.4242 × 10^{−4} | 1.9888 × 10^{−4} | 2.5900 × 10^{−4} |

17 | 0.4784 | 0.6320 | 0.63286049 | 8.6045 × 10^{−4} | 1.1123 × 10^{−3} | 1.0990 × 10^{−3} |

18 | 0.4960 | 0.5730 | 0.57381689 | 8.1689 × 10^{−4} | 9.2523 × 10^{−4} | 8.4500 × 10^{−4} |

19 | 0.5119 | 0.4990 | 0.49879214 | 2.0785 × 10^{−4} | 4.9417 × 10^{−4} | 5.8600 × 10^{−4} |

20 | 0.5265 | 0.4130 | 0.41276355 | 2.3644 × 10^{−4} | 4.9125 × 10^{−4} | 5.7100 × 10^{−4} |

21 | 0.5398 | 0.3165 | 0.31674212 | 2.4212 × 10^{−4} | 7.1918 × 10^{−4} | 7.5300 × 10^{−4} |

22 | 0.5521 | 0.2120 | 0.21202519 | 2.5196 × 10^{−5} | 1.0831 × 10^{−4} | 8.8000 × 10^{−5} |

23 | 0.5633 | 0.1035 | 0.10350359 | 3.5935 × 10^{−6} | 7.7968 × 10^{−4} | 8.2700 × 10^{−4} |

24 | 0.5736 | −01000 | −0.01049021 | 4.9021 × 10^{−4} | 7.5539 × 10^{−4} | 7.1100 × 10^{−4} |

25 | 0.5833 | −0.1230 | −0.12300588 | 5.8808 × 10^{−6} | 1.3767 × 10^{−3} | 1.3880 × 10^{−3} |

26 | 0.5900 | −0.2100 | −0.21005362 | 5.3621 × 10^{−5} | 8.0501 × 10^{−4} | 8.6500 × 10^{−4} |

Item | FA | BBO-M | RADE | LMSA | CARO | ABC | NM-MPSO |
---|---|---|---|---|---|---|---|

Total IAE | 9.92230 × 10^{−3} | 21.3000 × 10^{−3} | 17.7036 × 10^{−3} | 21.5104 × 10^{−3} | 18.1550 × 10^{−3} | 20.5000 × 10^{−3} | 17.700 × 10^{−3} |

RMSE | 5.138165 × 10^{−4} | 9.8634 × 10^{−4} | 9.8602 × 10^{−4} | 9.8640 × 10^{−4} | 9.86650 × 10^{−4} | 9.86200 × 10^{−4} | 9.8602 × 10^{−4} |

SSE | 5.723673 × 10^{−6} | 2.52997 × 10^{−5} | 1.5625 × 10^{−5} | 2.5297 × 10^{−5} | 1.65385 × 10^{−5} | 25.7000 × 10^{−6} | 15.6295 × 10^{−6} |

MAE | 3.81630 × 10^{−4} | 8.1923 × 10^{−4} | 6.8090 × 10^{−4} | 8.2732 × 10^{−4} | 6.98260 × 10^{−4} | 7.8846 × 10^{−4} | 6.8077 × 10^{−4} |

Item | FA | RADE | CARO | ABSO | ABC | NM-MPSO |
---|---|---|---|---|---|---|

Total IAE | 8.570300 × 10^{−3} | 17.7093 × 10^{−3} | 69.330 × 10^{−3} | 17.768 × 10^{−3} | 20.3929 × 10^{−3} | 17.356 × 10^{−3} |

RMSE | 4.548499 × 10^{−6} | 9.82480 × 10^{−4} | 9.8260 × 10^{−4} | 9.8344 × 10^{−4} | 9.8610 × 10^{−4} | 9.8250 × 10^{−4} |

SSE | 5.379100 × 10^{−6} | 15.6338 × 10^{−6} | 16.9587 × 10^{−6} | 15.3457 × 10^{−6} | 25.600 × 10^{−6} | 14.9455 × 10^{−6} |

MAE | 3.2963 × 10^{−4} | 17.7093 × 10^{−3} | 69.330 × 10^{−3} | 17.768 × 10^{−3} | 20.3929 × 10^{−3} | 6.6754 × 10^{−4} |

Algorithm | ${\mathit{I}}_{\mathit{p}\mathit{h}}\left(\mathsf{\mu}\mathbf{A}\right)$ | $\mathit{d}\left(\mathbf{m}\right)$ | $\mathsf{\mu}{\mathit{\tau}}_{\mathit{e}\mathit{f}\mathit{f}}\left(\frac{\mathbf{c}{\mathbf{m}}^{2}}{\mathbf{V}}\right)$ | ${\mathit{V}}_{\mathit{b}\mathit{i}}\left(\mathbf{V}\right)$ | ${\mathit{R}}_{\mathit{s}}\left(\mathsf{\Omega}\right)$ | ${\mathit{I}}_{0}\left(\mathbf{A}\right)$ | $\mathit{a}$ | ${\mathit{R}}_{\mathit{s}\mathit{h}}\left(\mathsf{\Omega}\right)$ |
---|---|---|---|---|---|---|---|---|

FA | 0.3167 | 5.8065 × 10^{−8} | 3.3306 × 10^{−5} | 0.9895 | 0.4242 | 3.0691 × 10^{−14} | 2 | 13.4978 |

Q-N | 0.3043 | 5.8065 × 10^{−8} | 4.8812 × 10^{−5} | 0.9759 | 0.4242 | 3.0691 × 10^{−14} | 1.9998 | 11.9138 |

SOMA | 0.3181 | 4.9743 × 10^{−8} | 3.3277 × 10^{−5} | 0.9963 | 0.4706 | 3.0783 × 10^{−14} | 1.9931 | 13.9288 |

Experiment Current | FA | Q-N | SOMA | |||
---|---|---|---|---|---|---|

Current (A) | IAE | Current (A) | IAE | Current (A) | IAE | |

0 | 7.3656 × 10^{−4} | 7.3656 × 10^{−4} | 0.0041 | 0.0041 | 8.6804 × 10^{−4} | 8.6804 × 10^{−4} |

0.0158 | 0.0152 | 6.0 × 10^{−4} | 0.0100 | 0.0058 | 0.0131 | 0.0027 |

0.0302 | 0.0361 | 0.0059 | 0.0305 | 0.0003 | 0.0334 | 0.0032 |

0.0619 | 0.0653 | 0.0034 | 0.0591 | 0.0028 | 0.0623 | 0.0004 |

0.0868 | 0.0744 | 0.0124 | 0.0680 | 0.0188 | 0.0715 | 0.0153 |

0.1142 | 0.1023 | 0.0119 | 0.0955 | 0.0187 | 0.1004 | 0.0138 |

0.1604 | 0.1623 | 0.0019 | 0.1549 | 0.0055 | 0.1679 | 0.0075 |

0.3044 | 0.3002 | 0.0042 | 0.2835 | 0.0209 | 0.3018 | 0.0026 |

Statistical Errors | FA | Q-N | SOMA |
---|---|---|---|

Standard deviation (SD) | 4.925 × 10^{−3} | 8.46 × 10^{−3} | 7.86 × 10^{−3} |

Root mean square error (RMSE) | 6.1634 × 10^{−3} | 12.3924 × 10^{−3} | 7.9529 × 10^{−3} |

Residual sum of squares (SSE) | 3.6384 × 10^{−4} | 1.2286 × 10^{−3} | 5.0604 × 10^{−4} |

Mean bias error (MBE) | 6.62401 × 10^{−3} | 1.2424 × 10^{−2} | 7.4912 × 10^{−3} |

**Table 12.**Optimal parameter values identified by FA for Photowatt-PWP 201 polycrystalline photovoltaic module single diode compared with other methods.

Item | FA | Newton-Raphson | PS | GA | SA | NM-MPSO |
---|---|---|---|---|---|---|

${I}_{ph}\left(A\right)$ | 1.0306 | 1.0318 | 1.0313 | 1.0441 | 1.0331 | 1.0305 |

${I}_{0}\text{\hspace{0.17em}}\left(\mu A\right)$ | 3.4802 | 3.2875 | 3.1756 | 3.4360 | 3.6642 | 3.6817 |

$a$ | 48.6551 | 48.4500 | 48.2889 | 48.5862 | 48.8211 | 48.8598 |

${R}_{s}\text{\hspace{0.17em}}\left(\mathsf{\Omega}\right)$ | 1.2014 | 1.2057 | 1.2053 | 1.1968 | 1.1989 | 1.1944 |

${R}_{sh}\text{\hspace{0.17em}}\left(\mathsf{\Omega}\right)$ | 971.1396 | 555.5556 | 714.2857 | 555.5556 | 833.3333 | 983.9970 |

**Table 13.**Measured and calculated current of photovoltaic module Photowatt-PWP 201 at 25 different working conditions compared with SA and PS.

Item | ${\mathit{V}}_{\mathit{E}\mathit{x}\mathit{p}}\left(\mathbf{V}\right)$ | ${\mathit{I}}_{\mathit{E}\mathit{x}\mathit{p}}\left(\mathbf{A}\right)$ | ${\mathit{I}}_{\mathbf{Calculated}}\left(\mathbf{A}\right)$ | Individual Absolute Error | ||
---|---|---|---|---|---|---|

FA | SA | PS | ||||

1 | 0.1248 | 1.0315 | 1.02919209 | 2.30790 × 10^{−3} | 6.0000 × 10^{−5} | 2.2000 × 10^{−3} |

2 | 1.8093 | 1.0300 | 1.02743525 | 2.56480 × 10^{−3} | 6.4000 × 10^{−4} | 3.7800 × 10^{−3} |

3 | 3.3511 | 1.0260 | 1.02577555 | 2.24450 × 10^{−4} | 1.4100 × 10^{−3} | 2.6500 × 10^{−3} |

4 | 4.7622 | 1.0220 | 1.02412139 | 2.12140 × 10^{−3} | 3.4900 × 10^{−3} | 1.4100 × 10^{−3} |

5 | 6.0538 | 1.0180 | 1.02228609 | 4.28610 × 10^{−3} | 5.4100 × 10^{−3} | 2.4000 × 10^{−4} |

6 | 7.2364 | 1.0155 | 1.01990640 | 4.40640 × 10^{−3} | 5.2900 × 10^{−3} | 1.0100 × 10^{−3} |

7 | 8.3189 | 1.0140 | 1.01632679 | 2.32680 × 10^{−3} | 2.9600 × 10^{−3} | 3.8800 × 10^{−3} |

8 | 9.3097 | 1.0100 | 1.01045436 | 4.54360 × 10^{−4} | 830.00 × 10^{−6} | 6.4200 × 10^{−3} |

9 | 10.2163 | 1.0035 | 1.00062757 | 2.87240 × 10^{−3} | 2.8200 × 10^{−3} | 10.320 × 10^{−3} |

10 | 11.0449 | 0.9880 | 0.98458550 | 3.41450 × 10^{−3} | 3.7000 × 10^{−3} | 11.260 × 10^{−3} |

11 | 11.8018 | 0.9630 | 0.95960866 | 3.39130 × 10^{−3} | 4.0300 × 10^{−3} | 11.450 × 10^{−3} |

12 | 12.4929 | 0.9255 | 0.92293341 | 2.56660 × 10^{−3} | 3.5000 × 10^{−3} | 10.590 × 10^{−3} |

13 | 13.1231 | 0.8725 | 0.87243997 | 6.00000 × 10^{−5} | 1.0000 × 10^{−3} | 7.5600 × 10^{−3} |

14 | 13.6983 | 0.8075 | 0.80712359 | 3.76410 × 10^{−4} | 1.5200 × 10^{−3} | 7.4200 × 10^{−3} |

15 | 14.2221 | 0.7265 | 0.72772952 | 1.22950 × 10^{−3} | 4.4000 × 10^{−4} | 4.7100 × 10^{−3} |

16 | 14.6995 | 0.6345 | 0.63619518 | 1.69520 × 10^{−3} | 1.2200 × 10^{−3} | 3.0900 × 10^{−3} |

17 | 15.1346 | 0.5345 | 0.53538376 | 8.83760 × 10^{−4} | 3.6000 × 10^{−4} | 3.0700 × 10^{−3} |

18 | 15.5311 | 0.4275 | 0.42846560 | 9.65600 × 10^{−4} | 8.0000 × 10^{−4} | 1.7300 × 10^{−3} |

19 | 15.8929 | 0.3185 | 0.31828380 | 2.16190 × 10^{−4} | 7.4000 × 10^{−4} | 2.3400 × 10^{−3} |

20 | 16.2229 | 0.2085 | 0.20744219 | 1.05780 × 10^{−3} | 1.8900 × 10^{−3} | 2.5500 × 10^{−3} |

21 | 16.5241 | 0.1010 | 0.09791334 | 3.08670 × 10^{−3} | 5.3400 × 10^{−3} | 5.0500 × 10^{−3} |

22 | 16.7987 | −0.008 | −0.00863233 | 6.32300 × 10^{−4} | 5.9000 × 10^{−4} | 6.7000 × 10^{−4} |

23 | 17.0499 | −0.111 | −0.11145028 | 4.50280 × 10^{−4} | 6.0000 × 10^{−5} | 2.2800 × 10^{−3} |

24 | 17.2793 | −0.209 | −0.20961535 | 6.15350 × 10^{−4} | 0000000000 | 3.1900 × 10^{−3} |

25 | 17.4885 | −0.303 | −0.30253352 | 4.66470 × 10^{−4} | 2.6200 × 10^{−3} | 6.7500 × 10^{−3} |

Item | FA | Newton-Raphson | PS | GA | SA |
---|---|---|---|---|---|

Total IAE | 42.6725 × 10^{−3} | 56.8800 × 10^{−3} | 115.610 × 10^{−3} | 153.479 × 10^{−3} | 50.710 × 10^{−3} |

RMSE | 2.1540 × 10^{−3} | 780.500 × 10^{−3} | 11.8000 × 10^{−3} | 6.9828 × 10^{−3} | 2.700 × 10^{−3} |

SSE | 1.1600 × 10^{−4} | 2.3249 × 10^{−4} | 8.1725 × 10^{−4} | 1.2190 × 10^{−3} | 1.7703 × 10^{−4} |

MAE | 1.7069 × 10^{−3} | 2.2752 × 10^{−3} | 4.6244 × 10^{−3} | 6.1392 × 10^{−3} | 2.0284 × 10^{−3} |

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Louzazni, M.; Khouya, A.; Amechnoue, K.; Gandelli, A.; Mussetta, M.; Crăciunescu, A.
Metaheuristic Algorithm for Photovoltaic Parameters: Comparative Study and Prediction with a Firefly Algorithm. *Appl. Sci.* **2018**, *8*, 339.
https://doi.org/10.3390/app8030339

**AMA Style**

Louzazni M, Khouya A, Amechnoue K, Gandelli A, Mussetta M, Crăciunescu A.
Metaheuristic Algorithm for Photovoltaic Parameters: Comparative Study and Prediction with a Firefly Algorithm. *Applied Sciences*. 2018; 8(3):339.
https://doi.org/10.3390/app8030339

**Chicago/Turabian Style**

Louzazni, Mohamed, Ahmed Khouya, Khalid Amechnoue, Alessandro Gandelli, Marco Mussetta, and Aurelian Crăciunescu.
2018. "Metaheuristic Algorithm for Photovoltaic Parameters: Comparative Study and Prediction with a Firefly Algorithm" *Applied Sciences* 8, no. 3: 339.
https://doi.org/10.3390/app8030339