# Developed Design of Battle Royale Optimizer for the Optimum Identification of Solid Oxide Fuel Cell

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## Abstract

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## 1. Introduction

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- They do not produce ${H}_{2}$ fuel, greenhouse gas, or pollution of air.
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- They significantly cause environment enhancement [14].
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- They are more efficient than combustion engines.
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- Unlike co-generation uses, these cells generate heat and electrical power with efficiency of about 80%.
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- FCs generate water and heat without particles, GHGs, or toxins, i.e., these cells generate unpolluted air.
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- They can be used in various sizes from mWs to MWs, such as in buildings, mobile phones, cars, etc.
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- FC supplements are applicable in various energy techniques, such as wind turbines, batteries, solar panels, and super capacitors [8].

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- New optimal parameters estimation of the solid oxide fuel cell system based on metaheuristics.
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- The idea is to minimize the error between the model output and the empirical datapoints.
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- A developed version of Battle Royale algorithm is utilized to minimize the error value.
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- The method is performed on a 96-cell SOFC stack under different temperature and pressure values.

## 2. Modeling of a SOFC

^{2}), ${I}_{L}$ specifies the constraint of current density (mA cm

^{2}), and ${I}_{0}^{c}$ and ${I}_{0}^{a}$ refer to the exchanging flow’s current density of the cathode and anode, respectively. By considering the clarified equations, seven unknown parameters are defined for optimization. The parameters include ${R}_{\mathsf{\Omega}}$, $A$, $B$,${E}_{0}$, ${I}_{L}$, ${I}_{0}^{a}$, and ${I}_{0}^{c}$.

## 3. Objective Function

## 4. Battle Royale Optimization Algorithm (DBRA)

#### 4.1. Strategy of Battle Royale Game

#### 4.2. Battle Royale Optimization Algorithm

#### 4.3. Developed Battle Royal Optimization Algorithm

## 5. Simulation Results

^{−5}in lower temperature, i.e., 550 °C, provided the best confirmation with minimum error value than the other comparative methods. It is clear that by increasing the temperature value, the error value for all of the methods was increased. Because the above values were achieved after 25 runs as mean value of each algorithm, their standard deviation value should be also considered to show their consistency during different independent runs. The standard deviation results of the studied case under different temperature value and 3 atm constant pressure value are given in Figure 3.

^{−5}error in 550 °C provided the minimum value toward the others which shows its higher accuracy than the others. It can also be inferred from Figure 3 that there is an observable difference between the proposed DBRA and the other algorithms in their reliability, which shows the propose method’s higher consistency during 25 independent runs. After parameter estimation of the solid oxide fuel cell system, the value of the unknown variables can be achieved and are reported in Table 7.

^{−5}SSE value in 550 °C provides the highest confirmation with the real value and its results get weaker by increasing the temperature value, where in the highest experimented temperature (750 °C), the maximum SSE value (3.97 × 10

^{−3}) was achieved. The temperature variations provide a strong upshot on the estimator, i.e., ith incrementing of the value of temperature, the density amount of the exchange current for positive and negative is increased, though the voltage is decreased. Moreover, Table 7 shows that ${R}^{2}$ values for both training and testing data were extremely close to 1.00. As a result, we can infer that the proposed approach was flawlessly conducted and could precisely anticipate SOFC voltage with the exception of a few severe border situations. We can also prove the better efficiency of the proposed method from the accuracy (99.03%) results.

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 3.**Standard deviation results of the studied case under different temperature value and 3 atm constant pressure value.

**Figure 8.**Standard deviation results of the studied case under different pressure values and 750 °C constant pressure value.

**Figure 9.**Voltage–current profile of the proposed DBRA and its confirmation with the experimental data by various pressure conditions.

**Figure 10.**Power–current profile of the proposed DBRA and its confirmation with the experimental data by various pressure conditions.

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

$\mathit{T}$ | $\mathbf{1.253}\mathbf{\u2013}\mathbf{2.4516}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{4}}$ |

$\mathit{F}$ | $\mathbf{96,486}Cmo{l}^{\mathbf{-}\mathbf{1}}$ |

$\mathit{R}$ | $\mathbf{8.314}kJ{\mathbf{\left(}kmolK\mathbf{\right)}}^{\mathbf{-}\mathbf{1}}$ |

Algorithm | Parameter | Value |
---|---|---|

Particle Swarm Optimization (PSO) [40] | ${c}_{1}$$\text{}\mathrm{and}\text{}{c}_{2}$ | 1 |

$w$ | 0.7 | |

Whale Optimization Algorithm (WO) [41] | $\overrightarrow{a}$ | 2 |

$\overrightarrow{r}$ | 1 | |

Archimedes Optimization Algorithm (AO) [42] | Protection probability | 10% |

Elimination probability | 25% | |

${c}_{1}$ | 1.5 | |

${c}_{2}$ | 1.5 |

Formulation | Range | ${\mathit{F}}^{*}$ |
---|---|---|

$F1=x\times \mathrm{sin}\left(4x\right)+1.1y\times \mathrm{sin}\left(2y\right)$ | $0<x$$,\text{}y0$ | −18.55 |

$F2=0.5+\frac{{\mathrm{sin}}^{2}\left(\sqrt{{x}^{2}+{y}^{2}}-0.5\right)}{1+0.1\left({x}^{2}+{y}^{2}\right)}$ | $0<x$, $y<2$ | 0.5 |

$F3=\left|x\right|+\left|y\right|+{\left({x}^{2}+{y}^{2}\right)}^{0.25}\times \mathrm{sin}(30{({(x+0.5)}^{2}+{y}^{2})}^{0.1})$ | $[-\infty ,\infty ]$ | −0.25 |

$F4=10n+{\displaystyle \sum _{i=1}^{n}}\left({x}_{i}^{2}-10\mathrm{cos}\left(2\pi {x}_{i}\right)\right),n=9$ | [−5.12, 5.12] | 0 |

Function | Indicator | Algorithm | |||
---|---|---|---|---|---|

PSO [40] | WO [41] | AO [42] | DBRA | ||

${F}_{1}$ | Max | −11.232 | −14.012 | −16.263 | −19.64 |

Min | −15.287 | −15.646 | −14.41 | −14.33 | |

Median | −13.253 | −14.558 | −15.16 | −16.54 | |

Std | 5.355 | 4.839 | 3.64 | 2.34 | |

${F}_{2}$ | Max | 0.647 | 0.637 | 0.325 | 0.315 |

Min | 0.453 | 0.427 | 0.413 | 0.4 | |

Median | 0.55 | 0.537 | 0.48 | 0.476 | |

Std | 0.038 | 0.012 | 0.003 | 0.001 | |

${F}_{3}$ | Max | −0.089 | −0.187 | −0.213 | −0.221 |

Min | −0.212 | −0.234 | −0.246 | −0.297 | |

Median | −0.150 | −0.164 | −0.210 | −0.263 | |

Std | 0.134 | 0.108 | 0.018 | 0.013 | |

${F}_{4}$ | Max | 15.363 | 12.437 | 9.254 | 2.374 |

Min | 1.816 | 1.009 | 0.008 | 0.002 | |

Median | 8.589 | 6.721 | 4.634 | 1.189 | |

Std | 5.234 | 5.054 | 2.372 | 1.062 |

Parameter | Lower Bound | Higher Bound | Unit |
---|---|---|---|

${E}_{OC}$ | 0 | 1.2 | $\mathrm{V}$ |

$A$ | 0 | 1 | $\mathrm{V}$ |

$B$ | 0 | 1 | $\mathrm{V}$ |

${I}_{L}$ | 0 | 10,000 | $\mathrm{mA}\xb7{\mathrm{cm}}^{-2}$ |

${I}_{0,a}$ | 0 | 100 | $\mathrm{mA}\xb7{\mathrm{cm}}^{-2}$ |

${I}_{0,c}$ | 0 | 1 | $\mathrm{mA}\xb7{\mathrm{cm}}^{-2}$ |

**Table 6.**Simulation results of the suggested technique under different temperatures in comparison with some of the latest algorithms.

Algorithms | 550 °C | 600 °C | 650 °C | 700 °C | 750 °C |
---|---|---|---|---|---|

CGWO [47] | 6.28 × 10^{−2} | 7.23 × 10^{−2} | 9.92 × 10^{−2} | 2.19 × 10^{−1} | 6.63 × 10^{−1} |

SBO [48] | 4.15 × 10^{−2} | 5.09 × 10^{−2} | 7.16 × 10^{−2} | 8.80 × 10^{−2} | 2.98 × 10^{−1} |

SCSO [49] | 8.16 × 10^{−3} | 9.11 × 10^{−3} | 2.09 × 10^{−2} | 5.25 × 10^{−2} | 7.46 × 10^{−2} |

TLBO [50] | 5.50 × 10^{−3} | 7.09 × 10^{−3} | 9.05 × 10^{−3} | 1.39 × 10^{−2} | 4.28 × 10^{−2} |

DBRA | 9.41 × 10^{−5} | 6.39 × 10^{−4} | 8.60 × 10^{−4} | 1.63 × 10^{−3} | 3.97 × 10^{−3} |

Parameters | 550 °C | 600 °C | 650 °C | 700 °C | 750 °C |
---|---|---|---|---|---|

${I}_{o,a}\left(\mathrm{mA}\xb7{\mathrm{cm}}^{-2}\right)$ | 13.96 | 15.50 | 19.81 | 22.68 | 24.41 |

${I}_{o,c}\left(\mathrm{mA}\xb7{\mathrm{cm}}^{-2}\right)$ | 7.11 | 7.28 | 7.42 | 7.49 | 7.53 |

${I}_{L}\left(\mathrm{mA}\xb7{\mathrm{cm}}^{-2}\right)$ | 149.73 | 153.35 | 159.66 | 165.85 | 167.26 |

${E}_{oc}\left(\mathrm{V}\right)$ | 1.35 | 1.30 | 1.49 | 1.39 | 1.28 |

$A\left(\mathrm{V}\right)$ | 0.0448 | 0.045 | 0.047 | 0.049 | 0.051 |

$B\left(\mathrm{V}\right)$ | 0.046 | 0.049 | 0.054 | 0.062 | 0.077 |

${R}_{ohm}\left(\mathrm{K}\mathsf{\Omega}\xb7{\mathrm{cm}}^{-2}\right)$ | 0.17 | 0.06 | 0.01 | 0.007 | 0.005 |

SSE | 9.41 × 10^{−5} | 6.39 × 10^{−4} | 8.60 × 10^{−4} | 1.63 × 10^{−3} | 3.97 × 10^{−3} |

${R}^{2}$ value | 0.99991 | 0.99985 | 0.99977 | 0.99635 | 0.99664 |

Accuracy | 99.03 | 97.47 | 97.07 | 95.96 | 93.70 |

Algorithms | 1 atm | 2 atm | 3 atm | 4 atm | 5 atm |
---|---|---|---|---|---|

CGWO [47] | 2.58 | 3.05 | 3.59 | 3.70 | 3.86 |

SBO [48] | 1.43 | 2.10 | 2.54 | 3.09 | 3.85 |

SCSO [49] | 4.18 × 10^{−1} | 5.50 × 10^{−1} | 7.25 × 10^{−1} | 9.12 × 10^{−1} | 9.88 × 10^{−1} |

TLBO [50] | 7.46 × 10^{−2} | 9.80 × 10^{−2} | 1.16 × 10^{−1} | 5.17 × 10^{−1} | 6.94 × 10^{−1} |

DBRA | 9.43 × 10^{−3} | 4.39 × 10^{−2} | 6.82 × 10^{−2} | 8.17 × 10^{−2} | 9.90 × 10^{−2} |

Parameters | 1 atm | 2 atm | 3 atm | 4 atm | 5 atm |
---|---|---|---|---|---|

${I}_{o,a}\left(\mathrm{mA}\xb7{\mathrm{cm}}^{-2}\right)$ | 28.28 | 28.36 | 28.38 | 28.40 | 28.44 |

${I}_{o,c}\left(\mathrm{mA}\xb7{\mathrm{cm}}^{-2}\right)$ | 7.14 | 7.17 | 7.20 | 7.22 | 7.23 |

${I}_{L}\left(\mathrm{mA}\xb7{\mathrm{cm}}^{-2}\right)$ | 161.42 | 161.45 | 161.52 | 161.53 | 161.56 |

${E}_{oc}\left(\mathrm{V}\right)$ | 1.18 | 1.22 | 1.27 | 1.35 | 1.50 |

$A\left(\mathrm{V}\right)$ | 0.043 | 0.043 | 0.043 | 0.043 | 0.043 |

$B\left(\mathrm{V}\right)$ | 0.086 | 0.086 | 0.086 | 0.086 | 0.086 |

${R}_{ohm}\left(\mathrm{K}\mathsf{\Omega}\xb7{\mathrm{cm}}^{-2}\right)$ | 0.016 | 0.016 | 0.016 | 0.016 | 0.016 |

MSE | 1.16 × 10^{−3} | 2.43 × 10^{−3} | 6.95 × 10^{−3} | 8.14 × 10^{−3} | 9.19 × 10^{−3} |

${R}^{2}$ value | 0.9985 | 0.9949 | 0.9918 | 0.9911 | 0.9904 |

Accuracy | 97.18 | 95.14 | 91.66 | 90.98 | 90.35 |

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**MDPI and ACS Style**

Karamnejadi Azar, K.; Kakouee, A.; Mollajafari, M.; Majdi, A.; Ghadimi, N.; Ghadamyari, M.
Developed Design of Battle Royale Optimizer for the Optimum Identification of Solid Oxide Fuel Cell. *Sustainability* **2022**, *14*, 9882.
https://doi.org/10.3390/su14169882

**AMA Style**

Karamnejadi Azar K, Kakouee A, Mollajafari M, Majdi A, Ghadimi N, Ghadamyari M.
Developed Design of Battle Royale Optimizer for the Optimum Identification of Solid Oxide Fuel Cell. *Sustainability*. 2022; 14(16):9882.
https://doi.org/10.3390/su14169882

**Chicago/Turabian Style**

Karamnejadi Azar, Keyvan, Armin Kakouee, Morteza Mollajafari, Ali Majdi, Noradin Ghadimi, and Mojtaba Ghadamyari.
2022. "Developed Design of Battle Royale Optimizer for the Optimum Identification of Solid Oxide Fuel Cell" *Sustainability* 14, no. 16: 9882.
https://doi.org/10.3390/su14169882