# An Improved Cheetah Optimizer for Accurate and Reliable Estimation of Unknown Parameters in Photovoltaic Cell and Module Models

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## Abstract

**:**

## 1. Introduction

#### 1.1. Motivation and Incitement

#### 1.2. Literature Review

#### 1.3. Contribution and Paper Organization

- A simplified and improved version of the CO is introduced for parameter estimation of PV models.
- The search phase is controlled according to the leader’s position, and its step length is adjusted following the sorted population. Hence, the proposed search strategy facilitates both global and local search capabilities.
- In the attack phase, the interaction factor is adjusted based on the prey’s position, while the turning factor is determined based on a random value. Using the proposed attack operator, the algorithm is expected to perform better during global searches and achieve faster convergence.
- An extensive study is conducted and compared with other well-established metaheuristic algorithms to evaluate the ICO’s effectiveness.

## 2. PV Modeling and Problem Formulation

#### 2.1. The Model of a Solar Cell

#### 2.1.1. SDM

#### 2.1.2. DDM

#### 2.2. PVMM

#### 2.3. Problem Formulation

## 3. Proposed Optimization Algorithm

#### 3.1. Overview of the CO Algorithm

#### 3.1.1. Searching Strategy

#### 3.1.2. Sitting-and-Waiting Strategy

#### 3.1.3. Attacking Strategy

#### 3.1.4. Strategy Selection Mechanism

Algorithm 1. The CO Algorithm |

#### 3.2. Improved Cheetah Optimizer (ICO) Algorithm

#### 3.2.1. Searching Strategy

#### 3.2.2. Attacking Strategy

Algorithm 2. The ICO Algorithm |

## 4. Experimental Results

#### 4.1. Population Size Analysis

^{−4}with n = 80. In addition, the proposed algorithm showed significant robustness with all initial populations, except for the population of 10. The CPU times and Friedman test results through 30 runs are represented in the last three columns of Table 3, and their average values for the three models are shown in Figure 4. Based on these results, it can be seen that the proposed algorithm with n = 80 had the best relative performance in the three models, with an average sum rank of 80 and a CPU time of 39.5 s. The population sizes of 40 and 50 ranked second and third among all examined population sizes, respectively.

#### 4.2. Results of Parameter Extraction Based on the SDM

^{−4}was obtained from the CO and ICO. For n = 40, SEDE and WSO (and for n = 80, WSO) gave the second-best solutions. It should be noted that the lower values of the RMSE indicate a higher accuracy of the estimation of the model parameters. The curves of the current and power in terms of voltage are illustrated in Figure 6a,b to verify the accuracy of the algorithm. Additionally, the values of IAEI and IAEP are drawn in Figure 6c,d over the voltage ranges. In all cases, the individual absolute error of current (IAEI) was less than 2.52 × 10

^{−3}, and the individual absolute error of power (AIEP) was less than 1.375 × 10

^{−3}, indicating that the CO and ICO were highly accurate in estimating the SDM parameters.

#### 4.3. Results of Parameter Extraction Based on the DDM

^{−4}for n = 40, followed by the ICO with an RMSE of 9.824860991382 × 10

^{−4}. Additionally, for n = 80, these optimizers obtained the best results out of the 12 algorithms. Conversely, SSA and GWO produced the worst results. Figure 7a,b illustrates the I–V and P–V curves, respectively, using the measured and estimated data for the DDM model. The corresponding IAEI and IAEP are illustrated in Figure 7c,d, respectively, indicating that the CO and ICO were incredibly accurate in estimating the DDM parameters.

#### 4.4. PVMM-Based Photo Watt-PWP 201

#### 4.5. Comparison of Statistical Results

^{−17}showed the best performance among the competitive algorithms. PGJAYA and WSO showed the second- and third-best accuracies, respectively. According to the Friedman test, the CO showed the best performance, and the ICO had the second-best performance among the 12 algorithms. Aside from that, when n = 80, the ICO and CO showed the best accuracy, and WSO showed the second-best accuracy. However, in terms of reliability, the ICO and CO with SD values of 5.21 × 10

^{−17}and 1.02 × 10

^{−16}were the best and second-best among the competitive algorithms, respectively. Based on the Wilcoxon signed rank test, there was no significant difference between the ICO and CO, while they obtained significantly superior results compared with the other competitive algorithms.

^{−17}), and the second- and third-best values were obtained by the ICO and CO (4.998 × 10

^{−17}and 1.105 × 10

^{−16}, respectively). According to Friedman’s test, the ICO provided the best performance, and the CO provides the second-best performance when n = 40. For n = 80, this ranking was shifted. The final ranking of the comparative algorithms for identifying the unknown parameters of the SDM, DDM, and PVMM is shown in Figure 9. For these models, the best sum rank result among the 12 algorithms with n = 40 was obtained by the ICO, followed by the CO and SEDE. While n = 80, the CO, ICO, and PGJAYA exhibited the first, second, and third sum rank results in the three models, respectively.

#### 4.6. Computational Time

#### 4.7. Convergence Characteristics

#### 4.8. Exploration and Exploitation Analysis

#### 4.9. Results for STM6-40/36 PV Module

^{2}and a temperature of 51 °C were used to measure 20 pairs of current and voltage values [44]. Table 10 presents the optimal parameters and statistical results of the proposed algorithm and recent well-established methods for the SDM and DDM of STM6-40/36. Aside from the algorithms used in the previous sections, the African vultures optimizer (AVO) [45], tuna swarm optimizer (TSO) [46], and artificial hummingbird technique (AHT) [43] were also compared based on the results reported in [43]. The results indicate that the CO and ICO performed better and were more reliable than the other algorithms for determining the optimal solutions. Also, SEDE and PGJAYA provided good performance. In conclusion, this algorithm offers the most reliable and efficient method for obtaining the most efficient results for various solar modules.

#### 4.10. Comparison with the State-of-the-Art Methods

^{−17}. After that, the CO and APLO were placed in the following ranks. For the DDM, hARS-PS showed the best performance in terms of standard deviation with a value of 1.45 × 10

^{−7}, followed by MSSA and the ICO, which were second and third best, respectively. For the PVMM, Rcr-JADE provided the best stability in comparison with the other algorithms. The ICO could also achieve the second-best SD value of 3.19 × 10

^{−17}, followed by the CO with the SD of 4.37 × 10

^{−17}. The results indicate that the ICO provided the most accurate and reliable algorithm for identifying solar PV model parameters. Furthermore, the original CO algorithm also exhibited reasonable performance compared with other hybrid and improved algorithms.

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 5.**Convergence curves of ICO with different population sizes in solving (

**a**) SDM, (

**b**) DDM, and (

**c**) PVMM.

**Figure 6.**Estimated and measured data of the RTC France silicon solar cell based on the SDM with the ICO: (

**a**) I–V, (

**b**) P–V, (

**c**) IAEI, and (

**d**) IAEP.

**Figure 7.**Estimated and measured data of the RTC France silicon solar cell based on DDM with the ICO: (

**a**) I–V, (

**b**) P–V, (

**c**) IAEI, and (

**d**) IAEP.

**Figure 8.**Estimated and measured data yielded by the ICO for the PV module model based on Photo Watt-PWP 201: (

**a**) I–V, (

**b**) P–V, (

**c**) IAEI, and (

**d**) IAEP.

**Figure 9.**Final ranking of applied algorithms for three models based on the Friedman test: (

**a**) n = 40 and (

**b**) n = 80.

**Figure 12.**Convergence curves of comparative algorithms with n = 40 for three models: (

**a**) SDM, (

**b**) DDM, and (

**c**) PVMM.

**Figure 13.**Exploration-exploration and diversity of comparative algorithms with SDM; (

**a**) exploration, (

**b**) exploitation, and (

**c**) diversity measurement.

Algorithm | PV Type | PV Model | Disadvantage | Advantage |
---|---|---|---|---|

PGJAYA [6] | RTC France Si cell and PhotoWatt-PWP201 | SDM, DDM, PVMM | Insufficient reliability | Acceptable accuracy |

DE [10] | SM55 module | SDM | The parameters need to be adjusted and insufficient capability for exploitation | Accurate performance under a variety of operating conditions |

Possessing good exploration capabilities | ||||

PSO [12] | Not specified | SDM, DDM | Stuck in local minima and convergence at the beginning | High level of accuracy in the solution |

Ease of implementation | ||||

Robustness | ||||

SEDE [14] | RTC France Si cell and PhotoWatt-PWP201 | SDM, DDM, PVMM | High computation time | High accuracy and robustness |

WSO [13] | RTC France silicon solar cell, Photo watt-PWP 201, and STM6-40/36 PV modules | SDM, DDM, PVMM | Insufficient robustness | New optimization algorithm for parameter extraction of PV cells and modules and low CPU time |

SSA [15] | TITAN-12-50 | DDM | Caught within local minimums, and convergence occurs early in the process | Low computational time |

IJAYA [16] | RTC France Si cell | SDM, DDM | Caught by local minima and inaccurate solution | A simpler and more efficient algorithm |

Convergence and robustness are high | ||||

Rao [17] | RTC France Si cell and PhotoWatt-PWP201 | SDM, DDM | Stuck in local minima, and commercial modules have not been tested | Ease of implementation |

The ability to explore well | ||||

MABC [18] | RTC France Si cell | SDM, DDM | Excessive computation time | High accuracy and robustness |

Parameters need to be adjusted frequently and achieving convergence early | Insensitive to noise | |||

IMFO [19] | Q6-1380 solar cell and CS6P-240P module | SDM, DDM | It takes a long time to compute, and commercial modules have not been tested | Convergence speed is high, and it is simpler |

SFLA [20] | KC200GT and MSX-60 | SDM | Not accurate | Fast convergence |

A lot of control parameters | ||||

TPTLBO [21] | RTC France Si cell | SDM, DDM | High computational costs and uncertainty about the solution | Ease of implementation |

Fewer control parameters | ||||

Fast convergence | ||||

ICWO [22] | KC200GT | SDM, DDM | Inability to explore, caught within local minimums, and convergence occurs early in the process | Easily implemented and a lower cost of computation |

Capacity for fair exploitation | ||||

SCA [23] | KC200GT | SDM | KC200GT module only tested and a local minimum trap | Easy to implement and simple to use, with a fair degree of accuracy |

GWO-CS [24] | KC200GT | SDM | The convergence speed is very slow | A robust design |

Reduced possibility of local optima trapping | ||||

The accuracy of the solution is high | ||||

COA [25] | RTC France Si cell, PhotoWatt-PWP201, KC200GT, ST40, and SM55 | SDM, DDM | Insufficient ability to exploit and convergence at an early stage | The quality of the solution is high and high convergence speed |

MPA [26] | KC200GT and MSX-60 | DDM | Convergence at an early stage | A high degree of accuracy in the solution |

Stuck in local minima | Excel exploratory skills | |||

AGA [27] | RTC France Si cell | SDM | Caught in the trap of local minima and a lack of local search capability | A reasonable degree of accuracy and identifying promising search areas to find solutions |

IEO [28] | RTC France Si cell, PhotoWatt-PWP201, ST40, and SM55 | SDM, DDM | Long computation times | High level of accuracy |

A good ability to explore and exploit | ||||

SMA [29] | RTC France Si cell and PhotoWatt-PWP201 | SDM, DDM | It takes a long time to compute | A high degree of accuracy and a good ability to explore and exploit |

OAHHO [30] | RTC France Si cell, PhotoWatt-PWP201, PVM 752 GaAs, ST40, and SM55 | SDM, DDM | Not specified | Rapid convergence rates |

Avoiding local optimum situations | ||||

High-quality solutions |

Model | ${\mathit{I}}_{\mathit{p}}\left(\mathbf{A}\right)$ | $\mathit{I},{\mathit{I}}_{1},{\mathit{I}}_{2}\left(\mathsf{\mu}\mathbf{A}\right)$ | $\mathit{u},{\mathit{u}}_{1},{\mathit{u}}_{2}$ | ${\mathit{R}}_{\mathit{s}}(\Omega )$ | ${\mathit{R}}_{\mathit{s}\mathit{h}}(\Omega )$ | |||||
---|---|---|---|---|---|---|---|---|---|---|

Min | Max | Min | Max | Min | Max | Min | Max | Min | Max | |

SDM | 0 | 1 | 0 | 1 | 1 | 2 | 0 | 0.5 | 0 | 100 |

DDM | 0 | 1 | 0 | 1 | 1 | 2 | 0 | 0.5 | 0 | 100 |

PV module | 0 | 2 | 0 | 50 | 1 | 50 | 0 | 2 | 0 | 2000 |

Model | n | Min | Mean | Max | SD | CPU Time (s) | Mean Rank in the Freidman Test | Sum Rank in the Freidman Test |
---|---|---|---|---|---|---|---|---|

SD | 10 | 9.860219 × 10^{−4} | 1.006557 × 10^{−3} | 1.268819 × 10^{−3} | 5.47 × 10^{−5} | 48.59 | 6.0 | 180 |

20 | 9.860219 × 10^{−4} | 9.860219 × 10^{−4} | 9.860219 × 10^{−4} | 8.17 × 10^{−17} | 48.02 | 3.6 | 106.5 | |

40 | 9.860219 × 10^{−4} | 9.860219 × 10^{−4} | 9.860219 × 10^{−4} | 6.30 × 10^{−17} | 37.42 | 2.6 | 76.5 | |

50 | 9.860219 × 10^{−4} | 9.860219 × 10^{−4} | 9.860219 × 10^{−4} | 3.24 × 10^{−17} | 40.23 | 2.5 | 74.5 | |

80 | 9.860219 × 10^{−4} | 9.860219 × 10^{−4} | 9.860219 × 10^{−4} | 4.85 × 10^{−17} | 41.11 | 2.9 | 87 | |

100 | 9.860219 × 10^{−4} | 9.860219 × 10^{−4} | 9.860219 × 10^{−4} | 4.66 × 10^{−17} | 50.89 | 3.5 | 105.5 | |

DD | 10 | 9.849747 × 10^{−4} | 1.046876 × 10^{−3} | 1.199696 × 10^{−3} | 5.90 × 10^{−5} | 52.52 | 5.6 | 169 |

20 | 9.832470 × 10^{−4} | 9.909079 × 10^{−4} | 1.016172 × 10^{−3} | 8.43 × 10^{−6} | 55.21 | 3.9 | 116 | |

40 | 9.824888 × 10^{−4} | 9.869534 × 10^{−4} | 1.002805 × 10^{−3} | 4.35 × 10^{−6} | 53.12 | 2.9 | 87 | |

50 | 9.825601 × 10^{−4} | 9.861925 × 10^{−4} | 9.948859 × 10^{−4} | 2.43 × 10^{−6} | 50.93 | 3.0 | 90 | |

80 | 9.824849 × 10^{−4} | 9.860955 × 10^{−4} | 9.895027 × 10^{−4} | 1.43 × 10^{−6} | 38.32 | 2.6 | 78 | |

100 | 9.836909 × 10^{−4} | 9.878873 × 10^{−4} | 1.014303 × 10^{−3} | 6.14 × 10^{−6} | 36.53 | 3.0 | 90 | |

PVM | 10 | 2.425075 × 10^{−3} | 2.435752 × 10^{−3} | 2.498069 × 10^{−3} | 1.99 × 10^{−5} | 48.36 | 6.0 | 180 |

20 | 2.425075 × 10^{−3} | 2.425075 × 10^{−3} | 2.425075 × 10^{−3} | 2.14 × 10^{−16} | 44.27 | 4.1 | 124 | |

40 | 2.425075 × 10^{−3} | 2.425075 × 10^{−3} | 2.425075 × 10^{−3} | 1.24 × 10^{−16} | 46.07 | 2.7 | 81 | |

50 | 2.425075 × 10^{−3} | 2.425075 × 10^{−3} | 2.425075 × 10^{−3} | 3.65 × 10^{−17} | 44.76 | 2.8 | 84.5 | |

80 | 2.425075 × 10^{−3} | 2.425075 × 10^{−3} | 2.425075 × 10^{−3} | 2.65 × 10^{−17} | 39.16 | 2.5 | 75 | |

100 | 2.425075 × 10^{−3} | 2.425075 × 10^{−3} | 2.425075 × 10^{−3} | 3.17 × 10^{−17} | 39.84 | 2.9 | 85.5 |

n | Algorithm | ${\mathit{I}}_{\mathit{p}}\left(\mathbf{A}\right)$ | $\mathit{I}\left(\mathbf{A}\right)$ | ${\mathit{R}}_{\mathit{s}\mathit{h}}(\Omega )$ | ${\mathit{R}}_{\mathit{s}}(\Omega )$ | $\mathit{u}$ | RMSE |
---|---|---|---|---|---|---|---|

40 | ICO | 0.761 | 3.23 × 10^{−7} | 53.719 | 0.0364 | 1.4812 | 9.860218778914 × 10^{−4} |

CO | 0.761 | 3.23 × 10^{−7} | 53.719 | 0.0364 | 1.4812 | 9.860218778914 × 10^{−4} | |

DE | 0.763 | 3.18 × 10^{−6} | 100.000 | 0.0243 | 1.7547 | 5.274028415510 × 10^{−3} | |

PSO | 0.761 | 2.67 × 10^{−7} | 48.768 | 0.0371 | 1.4623 | 1.049908843005 × 10^{−3} | |

GA | 0.764 | 2.63 × 10^{−6} | 70.532 | 0.0257 | 1.7285 | 5.028715197625 × 10^{−3} | |

TLBO | 0.761 | 3.77 × 10^{−7} | 63.546 | 0.0358 | 1.4967 | 1.061394487359 × 10^{−3} | |

SEDE | 0.761 | 3.23 × 10^{−7} | 53.719 | 0.0364 | 1.4812 | 9.860218778915 × 10^{−4} | |

JAYA | 0.761 | 6.08 × 10^{−7} | 70.138 | 0.0337 | 1.5478 | 1.596303286167 × 10^{−3} | |

PGJAYA | 0.761 | 3.23 × 10^{−7} | 53.713 | 0.0364 | 1.4812 | 9.860219332331 × 10^{−4} | |

WSO | 0.761 | 3.23 × 10^{−7} | 53.719 | 0.0364 | 1.4812 | 9.860218778915 × 10^{−4} | |

GWO | 0.838 | 0.0000000 | 1.139 | 0.0000 | 2.0000 | 2.228699161204 × 10^{−1} | |

SSA | 0.835 | 0.0000000 | 1.162 | 0.0000 | 1.0000 | 2.228762271791 × 10^{−1} | |

80 | ICO | 0.761 | 3.23 × 10^{−7} | 53.719 | 0.0364 | 1.4812 | 9.860218778914 × 10^{−4} |

CO | 0.761 | 3.23 × 10^{−7} | 53.719 | 0.0364 | 1.4812 | 9.860218778914 × 10^{−4} | |

DE | 0.763 | 1.54 × 10^{−6} | 99.600 | 0.0296 | 1.6569 | 3.541687987531 × 10^{−3} | |

PSO | 0.761 | 3.54 × 10^{−7} | 56.556 | 0.0360 | 1.4903 | 1.001530647734 × 10^{−3} | |

GA | 0.759 | 1.29 × 10^{−7} | 46.427 | 0.0399 | 1.3938 | 2.248309383635 × 10^{−3} | |

TLBO | 0.761 | 3.40 × 10^{−7} | 55.608 | 0.0362 | 1.4865 | 9.917684200620 × 10^{−4} | |

SEDE | 0.761 | 3.36 × 10^{−7} | 54.054 | 0.0362 | 1.4852 | 9.902825250634 × 10^{−4} | |

JAYA | 0.762 | 9.73 × 10^{−7} | 88.523 | 0.0312 | 1.6013 | 2.589835639165 × 10^{−3} | |

PGJAYA | 0.761 | 3.23 × 10^{−7} | 53.722 | 0.0364 | 1.4812 | 9.860220454267 × 10^{−4} | |

WSO | 0.761 | 3.23 × 10^{−7} | 53.719 | 0.0364 | 1.4812 | 9.860218778915 × 10^{−4} | |

GWO | 0.769 | 4.43 × 10^{−6} | 24.455 | 0.0200 | 1.8059 | 9.281563258264 × 10^{−3} | |

SSA | 1.000 | 8.72 × 10^{−7} | 1.098 | 0.0007 | 1.6512 | 1.525312427660 × 10^{−1} |

n | Algorithm | ${\mathit{I}}_{\mathit{p}}\left(\mathbf{A}\right)$ | ${\mathit{I}}_{1}\left(\mathbf{A}\right)$ | ${\mathit{I}}_{2}\left(\mathbf{A}\right)$ | ${\mathit{R}}_{\mathit{s}}(\Omega )$ | ${\mathit{R}}_{\mathit{s}\mathit{h}}(\Omega )$ | ${\mathit{u}}_{1}$ | ${\mathit{u}}_{2}$ | RMSE |
---|---|---|---|---|---|---|---|---|---|

40 | ICO | 0.760781 | 7.46 × 10^{−7} | 2.26 × 10^{−7} | 0.036740 | 55.456 | 2.000 | 1.4511 | 9.82486099138 × 10^{−4} |

CO | 0.760781 | 7.50 × 10^{−7} | 2.26 × 10^{−7} | 0.036741 | 55.486 | 2.000 | 1.4510 | 9.82484882272 × 10^{−4} | |

DE | 0.764966 | 2.55 × 10^{−6} | 2.40 × 10^{−6} | 0.022457 | 100.000 | 1.752 | 1.9806 | 6.28139269321 × 10^{−3} | |

PSO | 0.760733 | 1.71 × 10^{−7} | 1.46 × 10^{−6} | 0.036757 | 61.054 | 1.429 | 2.0000 | 1.00247341473 × 10^{−3} | |

GA | 0.763271 | 0.0000000 | 4.23 × 10^{−6} | 0.022775 | 97.844 | 1.670 | 1.7963 | 5.99279194424 × 10^{−3} | |

TLBO | 0.760090 | 9.73 × 10^{−8} | 2.76 × 10^{−6} | 0.036975 | 100.000 | 1.387 | 1.9994 | 1.30300020067 × 10^{−3} | |

SEDE | 0.760769 | 2.14 × 10^{−7} | 8.07 × 10^{−7} | 0.036790 | 55.795 | 1.447 | 1.9869 | 9.82753663536 × 10^{−4} | |

JAYA | 0.759873 | 5.29 × 10^{−7} | 4.00 × 10^{−11} | 0.034438 | 70.729 | 1.532 | 1.8894 | 1.93867560984 × 10^{−3} | |

PGJAYA | 0.760782 | 2.45 × 10^{−7} | 2.90 × 10^{−7} | 0.036477 | 54.289 | 1.999 | 1.4720 | 9.84193519571 × 10^{−4} | |

WSO | 0.759500 | 4.52 × 10^{−7} | 0.0000000 | 0.035285 | 100.000 | 1.516 | 2.0000 | 1.43847589737 × 10^{−3} | |

GWO | 1.000000 | 0.0000000 | 1.16 × 10^{−5} | 0.000000 | 2.179 | 1.000 | 2.0000 | 1.54903625180 × 10^{−1} | |

SSA | 0.834308 | 0.0000000 | 0.0000000 | 0.000000 | 1.152 | 1.000 | 1.0000 | 2.22868413284 × 10^{−1} | |

80 | ICO | 0.760780 | 6.63 × 10^{−7} | 2.36 × 10^{−7} | 0.036695 | 55.257 | 2.000 | 1.4547 | 9.82538943274 × 10^{−4} |

CO | 0.760781 | 2.22 × 10^{−7} | 7.72 × 10^{−7} | 0.036757 | 55.539 | 1.450 | 1.9969 | 9.82528425982 × 10^{−4} | |

DE | 0.763865 | 5.19 × 10^{−8} | 9.13 × 10^{−6} | 0.023974 | 99.963 | 1.407 | 1.9937 | 6.73013079580 × 10^{−3} | |

PSO | 0.760797 | 6.03 × 10^{−7} | 2.03 × 10^{−7} | 0.036852 | 54.797 | 1.900 | 1.4430 | 9.84648707354 × 10^{−4} | |

GA | 0.760727 | 0.0000000 | 9.74 × 10^{−7} | 0.031507 | 100.000 | 1.681 | 1.6015 | 2.39573932360 × 10^{−3} | |

TLBO | 0.760754 | 3.22 × 10^{−7} | 5.04 × 10^{−17} | 0.036453 | 55.423 | 1.481 | 1.0230 | 9.95677091382 × 10^{−4} | |

SEDE | 0.760178 | 8.63 × 10^{−7} | 2.07 × 10^{−7} | 0.034961 | 82.980 | 1.806 | 1.4577 | 1.47037750973 × 10^{−3} | |

JAYA | 0.761997 | 1.39 × 10^{−6} | 0.0000000 | 0.028824 | 100.000 | 1.644 | 2.0000 | 3.57709882707 × 10^{−3} | |

PGJAYA | 0.760851 | 5.18 × 10^{−7} | 2.30 × 10^{−7} | 0.036667 | 54.633 | 1.917 | 1.4533 | 9.84200147988 × 10^{−4} | |

WSO | 0.760776 | 0.0000000 | 3.23 × 10^{−7} | 0.036377 | 53.719 | 2.000 | 1.4812 | 9.86021877892 × 10^{−4} | |

GWO | 0.999003 | 0.0000000 | 5.29 × 10^{−6} | 0.000514 | 1.373 | 2.000 | 1.8772 | 1.38743574369 × 10^{−1} | |

SSA | 0.836762 | 1.17 × 10^{−9} | 0.0000000 | 0.000071 | 1.149 | 1.121 | 1.4507 | 1.57126305055 × 10^{−1} |

n | Algorithm | ${\mathit{I}}_{\mathit{p}}\left(\mathbf{A}\right)$ | $\mathit{I}\left(\mathbf{A}\right)$ | ${\mathit{R}}_{\mathit{s}\mathit{h}}(\Omega )$ | ${\mathit{R}}_{\mathit{s}}(\Omega )$ | $\mathit{u}$ | RMSE |
---|---|---|---|---|---|---|---|

40 | ICO | 1.03051 | 3.48 × 10^{−6} | 27.277 | 0.0334 | 1.3512 | 2.425074868095030 × 10^{−3} |

CO | 1.03051 | 3.48 × 10^{−6} | 27.277 | 0.0334 | 1.3512 | 2.425074868094980 × 10^{−3} | |

DE | 1.02991 | 1.49 × 10^{−5} | 1065.617 | 0.0284 | 1.5261 | 5.266650305240960 × 10^{−3} | |

PSO | 1.02677 | 5.98 × 10^{−6} | 88.261 | 0.0318 | 1.4111 | 2.864391667859010 × 10^{−3} | |

GA | 1.02370 | 1.52 × 10^{−5} | 1944.805 | 0.0278 | 1.5294 | 6.099240455880790 × 10^{−3} | |

TLBO | 1.02611 | 4.78 × 10^{−6} | 75.270 | 0.0325 | 1.3855 | 2.700403640152360 × 10^{−3} | |

SEDE | 1.03051 | 3.48 × 10^{−6} | 27.277 | 0.0334 | 1.3512 | 2.425074868095090 × 10^{−3} | |

JAYA | 1.02742 | 8.89 × 10^{−6} | 911.208 | 0.0305 | 1.4586 | 3.697656950234140 × 10^{−3} | |

PGJAYA | 1.03052 | 3.48 × 10^{−6} | 27.250 | 0.0334 | 1.3511 | 2.425077305006140 × 10^{−3} | |

WSO | 1.03051 | 3.48 × 10^{−6} | 27.277 | 0.0334 | 1.3512 | 2.425074868095050 × 10^{−3} | |

GWO | 1.04843 | 5.00 × 10^{−5} | 3.016 | 0.0000 | 1.7509 | 5.383466416567090 × 10^{−2} | |

SSA | 1.15116 | 5.00 × 10^{−5} | 2.191 | 0.0129 | 1.7224 | 5.130174319081860 × 10^{−2} | |

80 | ICO | 1.03051 | 3.48 × 10^{−6} | 27.277 | 0.0334 | 1.3512 | 2.425074868095010 × 10^{−3} |

CO | 1.03051 | 3.48 × 10^{−6} | 27.277 | 0.0334 | 1.3512 | 2.425074868094990 × 10^{−3} | |

DE | 1.02868 | 2.27 × 10^{−5} | 1968.622 | 0.0264 | 1.5866 | 6.921126739647050 × 10^{−3} | |

PSO | 1.02664 | 6.66 × 10^{−6} | 115.721 | 0.0314 | 1.4238 | 3.029451135597340 × 10^{−3} | |

GA | 1.03138 | 2.87 × 10^{−5} | 2000.000 | 0.0252 | 1.6215 | 7.712963596737400 × 10^{−3} | |

TLBO | 1.02522 | 5.63 × 10^{−6} | 881.405 | 0.0321 | 1.4034 | 3.244452302450550 × 10^{−3} | |

SEDE | 1.03013 | 3.56 × 10^{−6} | 28.820 | 0.0333 | 1.3536 | 2.427164258722220 × 10^{−3} | |

JAYA | 1.02758 | 1.51 × 10^{−5} | 1713.306 | 0.0277 | 1.5278 | 5.590302366807740 × 10^{−3} | |

PGJAYA | 1.02922 | 4.29 × 10^{−6} | 33.745 | 0.0327 | 1.3739 | 2.518017787843970 × 10^{−3} | |

WSO | 1.03051 | 3.48 × 10^{−6} | 27.277 | 0.0334 | 1.3512 | 2.425074868095060 × 10^{−3} | |

GWO | 1.07157 | 5.00 × 10^{−5} | 5.528 | 0.0187 | 1.7213 | 2.019064583084870 × 10^{−2} | |

SSA | 1.06056 | 4.31 × 10^{−5} | 12.652 | 0.0238 | 1.6888 | 1.554532543514840 × 10^{−2} |

n | Algorithm | Min | Mean | Max | SD | Mean Rank | Sum Rank | Significance |
---|---|---|---|---|---|---|---|---|

40 | ICO | 9.86021877891 × 10^{−4} | 9.86021877892 × 10^{−4} | 9.86021877892 × 10^{−4} | 3.091 × 10^{−17} | 1.633 | 49 | |

CO | 9.86021877891 × 10^{−4} | 9.86021877892 × 10^{−4} | 9.86021877893 × 10^{−4} | 2.299 × 10^{−16} | 1.600 | 48 | $\approx $ | |

DE | 5.27402841551 × 10^{−3} | 7.00472269280 × 10^{−3} | 8.61400240318 × 10^{−3} | 1.008 × 10^{−3} | 8.200 | 246 | $+$ | |

PSO | 1.04990884301 × 10^{−3} | 2.59824261345 × 10^{−3} | 5.43861383375 × 10^{−3} | 1.215 × 10^{−3} | 5.700 | 171 | $+$ | |

GA | 5.02871519763 × 10^{−3} | 1.85106717646 × 10^{−1} | 3.05981702986 × 10^{−1} | 1.178 × 10^{−1} | 10.933 | 328 | $+$ | |

TLBO | 1.06139448736 × 10^{−3} | 3.05558085709 × 10^{−3} | 7.98832352445 × 10^{−3} | 1.489 × 10^{−3} | 5.867 | 176 | $+$ | |

SEDE | 9.86021877891 × 10^{−4} | 9.86021877892 × 10^{−4} | 9.86021877892 × 10^{−4} | 4.368 × 10^{−17} | 2.867 | 86 | $\approx $ | |

JAYA | 1.59630328617 × 10^{−3} | 4.42292722271 × 10^{−3} | 6.90548939964 × 10^{−3} | 9.777 × 10^{−4} | 6.933 | 208 | $+$ | |

PGJAYA | 9.86021933233 × 10^{−4} | 9.86276195755 × 10^{−4} | 9.89060476576 × 10^{−4} | 6.385 × 10^{−7} | 4.133 | 124 | $+$ | |

WSO | 9.86021877892 × 10^{−4} | 1.58438200609 × 10^{−1} | 6.30741696212 × 10^{−1} | 1.452 × 10^{−1} | 8.733 | 262 | $+$ | |

GWO | 2.22869916120 × 10^{−1} | 2.23053219785 × 10^{−1} | 2.23414777753 × 10^{−1} | 1.541 × 10^{−4} | 10.600 | 318 | $+$ | |

SSA | 2.22876227179 × 10^{−1} | 2.23093108473 × 10^{−1} | 2.23798438512 × 10^{−1} | 1.976 × 10^{−4} | 10.800 | 324 | $+$ | |

80 | ICO | 9.86021877891 × 10^{−4} | 9.86021877892 × 10^{−4} | 9.86021877892 × 10^{−4} | 5.21 × 10^{−17} | 1.933 | 58 | |

CO | 9.86021877891 × 10^{−4} | 9.86021877891 × 10^{−4} | 9.86021877892 × 10^{−4} | 1.02 × 10^{−16} | 1.267 | 38 | $\approx $ | |

DE | 3.54168798753 × 10^{−3} | 7.44468352091 × 10^{−3} | 8.66642059402 × 10^{−3} | 8.63 × 10^{−4} | 8.233 | 247 | $+$ | |

PSO | 1.00153064773 × 10^{−3} | 2.60127712828 × 10^{−3} | 4.82228315158 × 10^{−3} | 1.30 × 10^{−3} | 5.633 | 169 | $+$ | |

GA | 2.24830938364 × 10^{−3} | 1.73964495048 × 10^{−1} | 2.97093810571 × 10^{−1} | 9.96 × 10^{−2} | 10.767 | 323 | $+$ | |

TLBO | 9.91768420062 × 10^{−4} | 5.34155103461 × 10^{−3} | 1.97944235719 × 10^{−2} | 4.83 × 10^{−3} | 6.600 | 198 | $+$ | |

SEDE | 9.90282525063 × 10^{−4} | 1.01400153629 × 10^{−3} | 1.09363977975 × 10^{−3} | 2.20 × 10^{−5} | 4.033 | 121 | $+$ | |

JAYA | 2.58983563916 × 10^{−3} | 5.68192621557 × 10^{−3} | 9.00499477318 × 10^{−3} | 1.14 × 10^{−3} | 7.067 | 212 | $+$ | |

PGJAYA | 9.86022045427 × 10^{−4} | 1.01574698584 × 10^{−3} | 1.18949290801 × 10^{−3} | 4.93 × 10^{−5} | 3.767 | 113 | $+$ | |

WSO | 9.86021877892 × 10^{−4} | 3.86485383212 × 10^{−2} | 2.99953326338 × 10^{−1} | 8.23 × 10^{−2} | 7.233 | 217 | $+$ | |

GWO | 9.28156325826 × 10^{−3} | 2.08662190383 × 10^{−1} | 2.22887009586 × 10^{−1} | 5.41 × 10^{−2} | 10.650 | 319.5 | $+$ | |

SSA | 1.52531242766 × 10^{−1} | 1.76192424924 × 10^{−1} | 2.22861399093 × 10^{−1} | 2.22 × 10^{−2} | 10.817 | 324.5 | $+$ |

n | Algorithm | Min | Mean | Max | SD | Mean Rank | Sum Rank | Significance |
---|---|---|---|---|---|---|---|---|

40 | ICO | 9.8248609913822 × 10^{−4} | 9.8726627184106 × 10^{−4} | 1.0056534591025 × 10^{−3} | 5.0 × 10^{−6} | 2.400 | 72 | |

CO | 9.8248488227226 × 10^{−4} | 9.9001427702291 × 10^{−4} | 1.0209237817020 × 10^{−3} | 9.2 × 10^{−6} | 2.667 | 80 | $\approx $ | |

DE | 6.2813926932069 × 10^{−3} | 8.0881324167144 × 10^{−3} | 8.9374240928482 × 10^{−3} | 5.8 × 10^{−4} | 7.867 | 236 | $+$ | |

PSO | 1.0024734147331 × 10^{−3} | 2.424626758664 × 10−3 | 4.7097299966204 × 10^{−3} | 1.2 × 10^{−3} | 5.233 | 157 | $+$ | |

GA | 5.9927919442382 × 10^{−3} | 9.4034625176630 × 10^{−2} | 3.1453144926695 × 10^{−1} | 9.2 × 10^{−2} | 9.533 | 286 | $+$ | |

TLBO | 1.3030002006649 × 10^{−3} | 5.3139563821659 × 10^{−3} | 1.9545185280400 × 10^{−2} | 3.5 × 10^{−3} | 6.533 | 196 | $+$ | |

SEDE | 9.8275366353636 × 10^{−4} | 9.9691619739783 × 10^{−4} | 1.1517616060394 × 10^{−3} | 3.4 × 10^{−5} | 2.133 | 64 | $\approx $ | |

JAYA | 1.9386756098406 × 10^{−3} | 5.3049157008462 × 10^{−3} | 1.9545234801188 × 10^{−2} | 3.1 × 10^{−3} | 6.567 | 197 | $+$ | |

PGJAYA | 9.8419351957116 × 10^{−4} | 1.0033692701556 × 10^{−3} | 1.2637028049522 × 10^{−3} | 5.6 × 10^{−5} | 2.867 | 86 | $\approx $ | |

WSO | 1.4384758973674 × 10^{−3} | 1.8658457526494 × 10^{−1} | 6.3074169621190 × 10^{−1} | 1.5 × 10^{−1} | 9.967 | 299 | $+$ | |

GWO | 1.5490362517966 × 10^{−1} | 2.2110084129396 × 10^{−1} | 2.2456299512895 × 10^{−1} | 1.3 × 10^{−2} | 11.067 | 332 | $+$ | |

SSA | 2.2286841328421 × 10^{−1} | 2.2344376300210 × 10^{−1} | 2.2475077282769 × 10^{−1} | 5.3 × 10^{−4} | 11.167 | 335 | $+$ | |

80 | ICO | 9.8253894327 × 10^{−4} | 9.8641995737 × 10^{−4} | 9.9981923040 × 10^{−4} | 3.134 × 10^{−6} | 1.633 | 49 | |

CO | 9.8252842598 × 10^{−4} | 9.9825780367 × 10^{−4} | 1.0827441957 × 10^{−3} | 2.432 × 10^{−5} | 1.900 | 57 | $\approx $ | |

DE | 6.7301307958 × 10^{−3} | 8.0618449324 × 10^{−3} | 8.8108931833 × 10^{−3} | 6.169 × 10^{−4} | 7.933 | 238 | $+$ | |

PSO | 9.8464870735 × 10^{−4} | 2.3813391278 × 10^{−3} | 5.5055357165 × 10^{−3} | 1.184 × 10^{−3} | 4.533 | 136 | $+$ | |

GA | 2.3957393236 × 10^{−3} | 2.2114853037 × 10^{−2} | 1.1924927342 × 10^{−1} | 3.269 × 10^{−2} | 7.800 | 234 | $+$ | |

TLBO | 9.9567709138 × 10^{−4} | 1.8966303725 × 10^{−2} | 6.2679374194 × 10^{−2} | 1.887 × 10^{−2} | 7.700 | 231 | $+$ | |

SEDE | 1.4703775097 × 10^{−3} | 2.6141249460 × 10^{−3} | 4.0397037616 × 10^{−3} | 8.191 × 10^{−4} | 4.833 | 145 | $+$ | |

JAYA | 3.5770988271 × 10^{−3} | 6.7480847825 × 10^{−3} | 9.7931015684 × 10^{−3} | 1.726 × 10^{−3} | 7.067 | 212 | $+$ | |

PGJAYA | 9.8420014799 × 10^{−4} | 1.0315913230 × 10^{−3} | 1.3731176270 × 10^{−3} | 7.747 × 10^{−5} | 2.700 | 81 | $\approx $ | |

WSO | 9.8602187789 × 10^{−4} | 1.1327669684 × 10^{−1} | 2.9995332634 × 10^{−1} | 1.185 × 10^{−1} | 9.700 | 291 | $+$ | |

GWO | 1.3874357437 × 10^{−1} | 1.6797247572 × 10^{−1} | 2.2219565068 × 10^{−1} | 1.317 × 10^{−2} | 11.167 | 335 | $+$ | |

SSA | 1.5712630505 × 10^{−1} | 1.6563118085 × 10^{−1} | 1.7878158052 × 10^{−1} | 5.962 × 10^{−3} | 11.033 | 331 | $+$ |

n | Algorithm | Min | Mean | Max | SD | Mean Rank | Sum Rank | Significance |
---|---|---|---|---|---|---|---|---|

40 | ICO | 2.425074868095 × 10^{−3} | 2.425074868095 × 10^{−3} | 2.425074868095 × 10^{−3} | 4.998 × 10^{−17} | 1.917 | 57.5 | |

CO | 2.425074868095 × 10^{−3} | 2.425074868095 × 10^{−3} | 2.425074868095× 10^{−3} | 1.105 × 10^{−16} | 1.983 | 59.5 | $\approx $ | |

DE | 5.266650305241 × 10^{−3} | 7.256622699267 × 10^{−3} | 9.637970829668 × 10^{−3} | 8.375 × 10^{−4} | 8.433 | 253 | $+$ | |

PSO | 2.864391667859 × 10^{−3} | 5.046272325408 × 10^{−3} | 6.503471754949 × 10^{−3} | 1.123 × 10^{−3} | 6.800 | 204 | $+$ | |

GA | 6.099240455881 × 10^{−3} | 1.746223736535 × 10^{−1} | 2.954738924287 × 10^{−1} | 1.059 × 10^{−1} | 11.000 | 330 | $+$ | |

TLBO | 2.700403640152 × 10^{−3} | 3.836378114785 × 10^{−3} | 9.005315197585 × 10^{−3} | 1.277 × 10^{−3} | 5.933 | 178 | $+$ | |

SEDE | 2.425074868095 × 10^{−3} | 2.425074868095 × 10^{−3} | 2.425074868095 × 10^{−3} | 2.319 × 10^{−17} | 2.833 | 85 | $\approx $ | |

JAYA | 3.697656950234 × 10^{−3} | 1.607949812692 × 10^{−2} | 3.296667881870 × 10^{−1} | 5.923 × 10^{−2} | 7.333 | 220 | $+$ | |

PGJAYA | 2.425077305006 × 10^{−3} | 2.441975159647 × 10^{−3} | 2.489534368567 × 10^{−3} | 1.780 × 10^{−5} | 4.467 | 134 | $+$ | |

WSO | 2.425074868095 × 10^{−3} | 7.524779448546 × 10^{−2} | 4.435604586495 × 10^{−1} | 1.357 × 10^{−1} | 6.100 | 183 | $+$ | |

GWO | 5.383466416567 × 10^{−2} | 9.374561478453 × 10^{−2} | 2.759007717317 × 10^{−1} | 6.334 × 10^{−2} | 10.633 | 319 | $+$ | |

SSA | 5.130174319082 × 10^{−2} | 1.152556758930 × 10^{−1} | 2.769662944201 × 10^{−1} | 8.570 × 10^{−2} | 10.567 | 317 | $+$ | |

80 | ICO | 2.425074868095 × 10^{−3} | 2.425074868095 × 10^{−3} | 2.425074868095 × 10^{−3} | 3.193 × 10^{−17} | 2.083 | 62.5 | |

CO | 2.425074868095 × 10^{−3} | 2.425074868095 × 10^{−3} | 2.425074868095 × 10^{−3} | 4.371 × 10^{−17} | 1.300 | 39 | $\approx $ | |

DE | 6.921126739647 × 10^{−3} | 8.220549984670 × 10^{−3} | 9.164483090113 × 10^{−3} | 5.053 × 10^{−4} | 7.733 | 232 | $+$ | |

PSO | 3.029451135597 × 10^{−3} | 5.684960029725 × 10^{−3} | 7.416901298303 × 10^{−3} | 1.083 × 10^{−3} | 6.267 | 188 | $+$ | |

GA | 7.712963596737 × 10^{−3} | 1.843031971911 × 10^{−1} | 4.075884497235 × 10^{−1} | 1.287 × 10^{−1} | 10.533 | 316 | $+$ | |

TLBO | 3.244452302450 × 10^{−3} | 5.331121609895 × 10^{−3} | 9.273482738970 × 10^{−3} | 1.343 × 10^{−3} | 6.033 | 181 | $+$ | |

SEDE | 2.427164258722 × 10^{−3} | 2.498740919096 × 10^{−3} | 2.654316961320 × 10^{−3} | 5.638 × 10^{−5} | 3.533 | 106 | $+$ | |

JAYA | 5.590302366807 × 10^{−3} | 6.191413986379 × 10^{−2} | 1.279747793100 × 10^{−1} | 3.926 × 10^{−2} | 9.800 | 294 | $+$ | |

PGJAYA | 2.518017787844 × 10^{−3} | 2.811575663308 × 10^{−3} | 3.090779454847 × 10^{−3} | 1.572 × 10^{−4} | 4.533 | 136 | $+$ | |

WSO | 2.425074868095 × 10^{−3} | 9.170708861046 × 10^{−2} | 4.440040747715 × 10^{−1} | 1.376 × 10^{−1} | 5.983 | 179.5 | $+$ | |

GWO | 2.019064583085 × 10^{−2} | 8.420780466394 × 10^{−2} | 2.742293345820 × 10^{−1} | 9.724 × 10^{−2} | 9.767 | 293 | $+$ | |

SSA | 1.554532543515 × 10^{−2} | 1.440832570166 × 10^{−1} | 2.742483329712 × 10^{−1} | 1.240 × 10^{−1} | 10.433 | 313 | $+$ |

Model | Algorithm | Min | Mean | Max | SD |
---|---|---|---|---|---|

SDM | ICO | 6.324406 × 10^{−4} | 6.324406 × 10^{−4} | 6.324406 × 10^{−4} | 4.359430 × 10^{−17} |

CO | 6.324406 × 10^{−4} | 6.324406 × 10^{−4} | 6.324406 × 10^{−4} | 4.514867 × 10^{−17} | |

DE | 2.548243 × 10^{−3} | 3.429082 × 10^{−3} | 4.156655 × 10^{−3} | 4.593907 × 10^{−4} | |

PSO | 1.593396 × 10^{−3} | 2.868491 × 10^{−3} | 3.775047 × 10^{−3} | 5.243550 × 10^{−4} | |

GA | 4.691761 × 10^{−3} | 8.714732 × 10^{−3} | 2.787222 × 10^{−2} | 5.653593 × 10^{−3} | |

TLBO | 1.664059 × 10^{−3} | 1.999367 × 10^{−3} | 2.559745 × 10^{−3} | 2.180581 × 10^{−4} | |

SEDE | 6.324406 × 10^{−4} | 6.324406 × 10^{−4} | 6.324406 × 10^{−4} | 3.991527 × 10^{−13} | |

JAYA | 2.418342 × 10^{−3} | 3.531127 × 10^{−3} | 5.851238 × 10^{−3} | 6.100849 × 10^{−4} | |

PGJAYA | 6.333531 × 10^{−4} | 7.058293 × 10^{−4} | 1.014990 × 10^{−3} | 8.519482 × 10^{−5} | |

WSO | 6.324406 × 10^{−4} | 7.772328 × 10^{−3} | 1.389641 × 10^{−1} | 2.497353 × 10^{−2} | |

GWO | 1.960570 × 10^{−3} | 5.136671 × 10^{−2} | 1.390015 × 10^{−1} | 6.306585 × 10^{−2} | |

SSA | 3.066949 × 10^{−2} | 8.840933 × 10^{−2} | 1.161650 × 10^{−1} | 2.092938 × 10^{−2} | |

AVO | 1.732400 × 10^{−3} | 4.348500 × 10^{−3} | 6.433800 × 10^{−3} | 1.084200 × 10^{−3} | |

TSO | 1.921900 × 10^{−3} | 4.622900 × 10^{−3} | 6.116800 × 10^{−3} | 9.666900 × 10^{−4} | |

AHT | 1.729800 × 10^{−3} | 1.729800 × 10^{−3} | 1.730000 × 10^{−3} | 5.392300 × 10^{−8} | |

DDM | ICO | 4.770147 × 10^{−4} | 5.670894 × 10^{−4} | 6.310191 × 10^{−4} | 4.266204 × 10^{−5} |

CO | 4.782418 × 10^{−4} | 5.509457 × 10^{−4} | 6.223188 × 10^{−4} | 4.826332 × 10^{−5} | |

DE | 3.053124 × 10^{−3} | 4.242541 × 10^{−3} | 4.891955 × 10^{−3} | 4.969994 × 10^{−4} | |

PSO | 5.569682 × 10^{−4} | 2.279931 × 10^{−3} | 3.391297 × 10^{−3} | 7.329564 × 10^{−4} | |

GA | 4.466775 × 10^{−3} | 9.483070 × 10^{−3} | 2.733149 × 10^{−2} | 6.105827 × 10^{−3} | |

TLBO | 1.428260 × 10^{−3} | 1.960379 × 10^{−3} | 2.680988 × 10^{−3} | 2.592217 × 10^{−4} | |

SEDE | 6.275536 × 10^{−4} | 7.240450 × 10^{−4} | 1.215475 × 10^{−3} | 1.323627 × 10^{−4} | |

JAYA | 3.080909 × 10^{−3} | 4.479366 × 10^{−3} | 7.592475 × 10^{−3} | 1.207246 × 10^{−3} | |

PGJAYA | 6.319029 × 10^{−4} | 8.466414 × 10^{−4} | 1.804906 × 10^{−3} | 3.018222 × 10^{−4} | |

WSO | 5.007168 × 10^{−4} | 1.992023 × 10^{−2} | 1.578042 × 10^{−1} | 4.683434 × 10^{−2} | |

GWO | 2.207844 × 10^{−3} | 1.101719 × 10^{−2} | 1.390014 × 10^{−1} | 2.460597 × 10^{−2} | |

SSA | 1.306709 × 10^{−2} | 9.820823 × 10^{−2} | 1.328993 × 10^{−1} | 2.030729 × 10^{−2} | |

AVO | 1.702800 × 10^{−3} | 4.316300 × 10^{−3} | 5.413700 × 10^{−3} | 8.339100 × 10^{−4} | |

TLSBO | 2.684300 × 10^{−3} | 4.688800 × 10^{−3} | 6.167000 × 10^{−3} | 8.990200 × 10^{−4} | |

AHT | 1.704900 × 10^{−3} | 1.728700 × 10^{−3} | 1.762900 × 10^{−3} | 9.851200 × 10^{−6} |

Model | SDM | ||||||||

Algorithm | ICO | CO | hARS-PS | APLO | CMM-DE/BBO | DE/BBO | BLPSO | CLPSO | MSSA |

SD | 3.09 × 10^{−17} | 2.30 × 10^{−16} | 3.01 × 10^{−7} | 1.60 × 10^{−16} | 8.17 × 10^{−5} | 2.51 × 10^{−4} | 2.12 × 10^{−4} | 7.49 × 10^{−5} | 3.01 × 10^{−7} |

Mean | 9.86 × 10^{−4} | 9.86 × 10^{−4} | 9.85 × 10^{−4} | 9.86 × 10^{−4} | 1.05 × 10^{−3} | 1.29 × 10^{−3} | 1.31 × 10^{−3} | 1.06 × 10^{−3} | 9.86 × 10^{−4} |

Max | 9.86 × 10^{−4} | 9.86 × 10^{−4} | 9.87 × 10^{−4} | 9.86 × 10^{−4} | 1.35 × 10^{−3} | 2.23 × 10^{−3} | 1.79 × 10^{−3} | 1.32 × 10^{−3} | 9.87 × 10^{−4} |

Min | 9.86 × 10^{−4} | 9.86 × 10^{−4} | 9.84 × 10^{−4} | 9.86 × 10^{−4} | 9.86 × 10^{−4} | 9.99 × 10^{−4} | 1.03 × 10^{−3} | 9.96 × 10^{−4} | 9.86 × 10^{−4} |

Algorithm | TLABC | GOTLBO | STLBO | IJAYA | SATLBO | MVO | QMVO | Rcr-JADE | IWOA |

SD | 1.19 × 10^{−5} | 5.02 × 10^{−5} | 1.8602 × 10^{−5} | 1.40 × 10^{−5} | 2.30 × 10^{−5} | 4.20 × 10^{−4} | 1.94 × 10^{−4} | 5.12 × 10^{−16} | 1.13 × 10^{−5} |

Mean | 9.94 × 10^{−4} | 1.05 × 10^{−3} | 9.8607 × 10^{−4} | 9.92 × 10^{−4} | 9.95 × 10^{−4} | 1.44 × 10^{−3} | 1.14 × 10^{−3} | 9.86 × 10^{−4} | 1.03 × 10^{−3} |

Max | 1.03 × 10^{−3} | 1.21 × 10^{−3} | 9.8655 × 10^{−4} | 1.06 × 10^{−3} | 9.88 × 10^{−4} | 1.18 × 10^{−3} | 1.03 × 10^{−3} | 9.86 × 10^{−4} | 9.95 × 10^{−4} |

Min | 9.86 × 10^{−4} | 9.89 × 10^{−4} | 9.8602 × 10^{−4} | 9.86 × 10^{−4} | 9.86 × 10^{−4} | 1.14 × 10^{−3} | 9.88 × 10^{−4} | 9.86 × 10^{−4} | 9.86 × 10^{−4} |

Model | DDM | ||||||||

Algorithm | ICO | CO | hARS-PS | APLO | CMM-DE/BBO | DE/BBO | BLPSO | CLPSO | MSSA |

SD | 3.13 × 10^{−6} | 2.43 × 10^{−5} | 1.45 × 10^{−7} | 7.80 × 10^{−5} | 2.94 × 10^{−4} | 3.63 × 10^{−4} | 1.78 × 10^{−4} | 1.44 × 10^{−4} | 1.49 × 10^{−6} |

Mean | 9.86 × 10^{−4} | 9.98 × 10^{−4} | 9.84 × 10−4 | 1.02 × 10^{−3} | 1.55 × 10^{−3} | 1.56 × 10^{−3} | 1.48 × 10^{−3} | 1.15 × 10^{−3} | 9.94 × 10^{−4} |

Max | 1.00 × 10^{−3} | 1.08 × 10^{−3} | 9.87 × 10−4 | 1.34 × 10^{−3} | 2.06 × 10^{−3} | 2.40 × 10^{−3} | 1.74 × 10^{−3} | 1.55 × 10^{−3} | 9.99 × 10^{−4} |

Min | 9.83 × 10^{−4} | 9.83 × 10^{−4} | 9.82 × 10−4 | 9.83 × 10^{−4} | 1.01 × 10^{−3} | 1.03 × 10^{−3} | 1.06 × 10^{−3} | 9.99 × 10^{−4} | 9.83 × 10^{−4} |

Algorithm | TLABC | GOTLBO | STLBO | IJAYA | SATLBO | MVO | QMVO | Rcr-JADE | IWOA |

SD | 2.11 × 10^{−4} | 1.13 × 10^{−4} | 2.90 × 10^{−4} | 9.83 × 10^{−5} | 1.95 × 10^{−5} | 7.55 × 10^{−4} | 4.02 × 10^{−4} | 9.86 × 10^{−5} | 1.93 × 10^{−5} |

Mean | 1.21 × 10^{−3} | 1.15 × 10^{−3} | 1.06 × 10^{−3} | 1.03 × 10^{−3} | 1.05 × 10^{−4} | 1.58 × 10^{−3} | 1.37 × 10^{−3} | 9.86 × 10^{−4} | 1.09 × 10^{−3} |

Max | 1.98 × 10^{−3} | 1.39 × 10^{−3} | 2.45 × 10^{−3} | 1.41 × 10^{−3} | 9.98 × 10^{−4} | 1.21 × 10^{−3} | 1.04 × 10^{−3} | 9.83 × 10^{−4} | 9.97 × 10^{−4} |

Min | 1.00 × 10^{−3} | 9.87 × 10^{−4} | 9.83 × 10^{−4} | 9.83 × 10^{−4} | 9.83 × 10^{−4} | 1.02 × 10^{−3} | 9.83 × 10^{−4} | 9.82 × 10^{−4} | 9.83 × 10^{−4} |

Model | PVMM | ||||||||

Algorithm | ICO | CO | hARS-PS | APLO | CMM-DE/BBO | DE/BBO | BLPSO | CLPSO | MSSA |

SD | 3.19 × 10^{−17} | 4.37 × 10^{−17} | 1.38 × 10^{−5} | 5.96 × 10^{−17} | 3.55 × 10^{−7} | 2.93 × 10^{−5} | 1.37 × 10^{−5} | 2.58 × 10^{−5} | 1.75 × 10^{−5} |

Mean | 2.43 × 10^{−3} | 2.43 × 10^{−3} | 2.43 × 10^{−3} | 2.43 × 10^{−3} | 2.43 × 10^{−3} | 2.46 × 10^{−3} | 2.44 × 10^{−3} | 2.45 × 10^{−3} | 2.54 × 10^{−3} |

Max | 2.43 × 10^{−3} | 2.43 × 10^{−3} | 2.50 × 10^{−3} | 2.43 × 10^{−3} | 2.43 × 10^{−3} | 2.53 × 10^{−3} | 2.49 × 10^{−3} | 2.54 × 10^{−3} | 2.78 × 10^{−3} |

Min | 2.43 × 10^{−3} | 2.43 × 10^{−3} | 2.42 × 10^{−3} | 2.43 × 10^{−3} | 2.43 × 10^{−3} | 2.43 × 10^{−3} | 2.43 × 10^{−3} | 2.43 × 10^{−3} | 2.42 × 10^{−3} |

Algorithm | TLABC | GOTLBO | STLBO | IJAYA | SATLBO | MVO | QMVO | Rcr-JADE | IWOA |

SD | 8.75 × 10^{−7} | 2.94 × 10^{−5} | 6.90 × 10^{−2} | 3.78 × 10^{−6} | 7.41 × 10^{−7} | 3.31 × 10^{−4} | 1.58 × 10^{−4} | 2.90 × 10^{−17} | 2.24 × 10^{−6} |

Mean | 2.43 × 10^{−3} | 2.48 × 10^{−3} | 2.06 × 10^{−2} | 2.43 × 10^{−3} | 2.43 × 10^{−3} | 2.57 × 10^{−3} | 2.54 × 10^{−3} | 2.43 × 10^{−3} | 2.43 × 10^{−3} |

Max | 2.43 × 10^{−3} | 2.56 × 10^{−3} | 2.74 × 10^{−1} | 2.44 × 10^{−3} | 2.43 × 10^{−3} | 2.48 × 10^{−3} | 2.46 × 10^{−3} | 2.43 × 10^{−3} | 2.43 × 10^{−3} |

Min | 2.43 × 10^{−3} | 2.43 × 10^{−3} | 2.43 × 10^{−3} | 2.43 × 10^{−3} | 2.43 × 10^{−3} | 2.46 × 10^{−3} | 2.43 × 10^{−3} | 2.43 × 10^{−3} | 2.43 × 10^{−3} |

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## Share and Cite

**MDPI and ACS Style**

Memon, Z.A.; Akbari, M.A.; Zare, M.
An Improved Cheetah Optimizer for Accurate and Reliable Estimation of Unknown Parameters in Photovoltaic Cell and Module Models. *Appl. Sci.* **2023**, *13*, 9997.
https://doi.org/10.3390/app13189997

**AMA Style**

Memon ZA, Akbari MA, Zare M.
An Improved Cheetah Optimizer for Accurate and Reliable Estimation of Unknown Parameters in Photovoltaic Cell and Module Models. *Applied Sciences*. 2023; 13(18):9997.
https://doi.org/10.3390/app13189997

**Chicago/Turabian Style**

Memon, Zulfiqar Ali, Mohammad Amin Akbari, and Mohsen Zare.
2023. "An Improved Cheetah Optimizer for Accurate and Reliable Estimation of Unknown Parameters in Photovoltaic Cell and Module Models" *Applied Sciences* 13, no. 18: 9997.
https://doi.org/10.3390/app13189997