# A Single-Stack Output Power Prediction Method for High-Power, Multi-Stack SOFC System Requirements

^{1}

^{2}

^{3}

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## Abstract

**:**

## 1. Introduction

- The prediction for SOFC output power utilizes the combustion chamber temperature, stack temperature, and electrical current as input parameters, with the output power being the output parameter. This approach is more comprehensive than traditional approaches to predictive modeling that rely solely on the inputs and outputs of the stack.
- The suggested GA-BP algorithm notably enhances the precision of output power prediction in a SOFC system, thereby providing valuable references for future investigations on forecasting system operating conditions.

## 2. SOFC System Architecture

^{2}, and platinum is used as the catalyst. A typical IV curve of the stack is shown in Figure 2.

## 3. Problem Description and Methodology

#### 3.1. Problem Description

#### 3.2. Methodology

## 4. Model Building

#### 4.1. BP Neural Network

#### 4.2. Genetic Algorithm

#### 4.3. GA-BP Algorithm Model

- Choice of encoding

- 2.
- Population initialization

- 3.
- Initialize the BP neural network model

- 4.
- Designing the Fitness Function

_{i}and o

_{i}represent the actual value of the neurons and the predicted output of the neurons, respectively. In the output power prediction model of the SOFC system, y

_{i}is the true value of SOFC system output power, and o

_{i}is the predicted value of the output power. Additionally, the coefficient k is defined as 1/n.

- 5.
- Selection operation

_{i}) is the relative fitness of individual x

_{i}, which indicates the probability of that individual being selected. Meanwhile, f(x

_{i}) is the original fitness of an individual, and ${\sum}_{j=1}^{N}f\left({x}_{i}\right)$ is the cumulative fitness of the population.

_{i}), the roulette wheel is divided into N sectors. Equation (5) represents the central angle of the i-th sector:

- 6.
- Cross operation

_{i}and the j chromosome a

_{j}at position r, is illustrated by Equation (6):

- 7.
- Mutation operation

_{max}and a

_{min}represent the maximum and minimum values of the gene a

_{ij}, respectively. r

_{2}represents a random number, with a value ranging between [0, 1]. g denotes the current iteration number, while G

_{max}represents the maximum number of evolutions. Additionally, r is another random number, with a value ranging between [0, 1].

- 8.
- Calculate fitness

- 9.
- Termination criterion:
- a.
- Reach the maximum number of iterations.
- b.
- The current best solution has either remained unchanged or changed very little for several consecutive steps.
- c.
- The algorithm has found an acceptable best solution, achieving the performance goal.

- 10.
- Train

## 5. Model Settings

#### 5.1. Data Sample

_{max}and x

_{min}are the maximum and minimum values of the sample data, respectively. x and x* are the values before and after normalization, respectively. Reverse normalization is performed after obtaining the experimental results to obtain the true value.

#### 5.2. Parameter Settings of BP and GA-BP

## 6. Results and Discussion

^{2}of the GA-BP model is 0.9832, indicating a strong correlation between the predicted value of GA-BP and the true value. Furthermore, it can be observed that out of the forty test sample points, only one point is outside the prediction interval, indicating a low likelihood of the predicted value deviating from the true value, and in most cases, the predicted values are close to the true values.

^{2}of the GA-BP model is greater than these two models. Furthermore, as shown in Table 3 and Figure 10, the relative error of the predicted values of the GA-BP model is also smaller in most cases than these two models.

^{2}, maximum relative error, minimum relative error, average error, and training duration of the three models are shown in Table 5.

## 7. Conclusions

^{2}of the enhanced GA-BP prediction model increases from 0.9573 to 0.9832. In addition, the MSE decreases from 92.67 to 35.94, the maximum relative error decreases from 15.068% to 9.724%, the minimum relative error decreases from 0.043% to 0.021%, and the average relative error value decreases from 1.7% to 1%. Meanwhile, compared to the various indicators of the LSTM prediction results, it also demonstrates superior performance. The reduction of these relative errors helps to improve the accuracy of the average prediction results. The high-precision prediction and modeling of stack output power in the SOFC system provides an important basis for the development of high-power multi-stack SOFC systems in the future.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 9.**Fit curves and prediction intervals at the 95% confidence level. (

**a**) GA-BP prediction model. (

**b**) BP prediction model. (

**c**) LSTM prediction model.

Hidden Layer Neuron | MSE | Iterations |
---|---|---|

3 | 0.00061 | 20 |

5 | 0.00052 | 17 |

7 | 0.00074 | 16 |

9 | 0.00085 | 15 |

11 | 0.00085 | 14 |

Parameter of BP | Setting |

Training times | 1000 |

Neurons in the input layers | 3 |

Neurons in the hidden layers | 5 |

Neurons in the output layers | 1 |

Activation function | tansig |

Training function | LM |

Learning rate | 0.1 |

Training goal | 0.0001 |

Parameter of GA | Setting |

Size of the population | 50 |

Maximum number of evolutions | 50 |

Crossover probability | 0.8 |

Mutation probability | 0.05 |

Generation gap | 0.9 |

Sample | GA-BP/% | BP/% | LSTM/% | Sample | GA-BP/% | BP/% | LSTM/% |
---|---|---|---|---|---|---|---|

1 | 0.194 | 0.546 | 1.278 | 21 | 0.305 | 0.277 | 0.929 |

2 | 0.678 | 0.095 | 1.072 | 22 | 0.389 | 0.678 | 0.595 |

3 | 0.141 | 0.043 | 0.196 | 23 | 0.142 | 0.201 | 0.24 |

4 | 2.826 | 3.444 | 3.128 | 24 | 0.041 | 0.331 | 0.129 |

5 | 1.721 | 0.565 | 1.658 | 25 | 9.724 | 15.068 | 11.053 |

6 | 0.461 | 1.652 | 1.431 | 26 | 0.224 | 6.14 | 2.859 |

7 | 1.189 | 0.239 | 0.631 | 27 | 1.292 | 0.452 | 0.709 |

8 | 0.269 | 0.603 | 1.821 | 28 | 1.998 | 2.248 | 2.049 |

9 | 0.622 | 1.553 | 1.246 | 29 | 1.63 | 2.335 | 2.271 |

10 | 1.085 | 0.794 | 0.137 | 30 | 0.359 | 0.499 | 0.192 |

11 | 0.867 | 2.703 | 0.876 | 31 | 0.353 | 1.247 | 0.323 |

12 | 0.884 | 1.669 | 0.983 | 32 | 0.286 | 1.282 | 0.344 |

13 | 1.598 | 2.189 | 0.641 | 33 | 0.121 | 0.505 | 0.108 |

14 | 0.75 | 0.095 | 1.57 | 34 | 0.737 | 1.164 | 1.839 |

15 | 0.439 | 0.184 | 1.186 | 35 | 0.517 | 1.443 | 1.438 |

16 | 0.578 | 0.488 | 0.14 | 36 | 0.021 | 1.385 | 1.09 |

17 | 1.052 | 1.523 | 0.837 | 37 | 0.345 | 0.276 | 0.522 |

18 | 2.822 | 4.207 | 2.801 | 38 | 0.035 | 1.575 | 2.027 |

19 | 1.476 | 1.556 | 1.903 | 39 | 0.282 | 2.45 | 2.286 |

20 | 1.345 | 2.09 | 2.445 | 40 | 0.213 | 2.175 | 1.517 |

Parameter | Setting |
---|---|

Batch size | 30 |

Neurons in the input layers | 3 |

Number of hidden layers | 1 |

Neurons in the hidden layers | 16 |

Neurons in the output layers | 1 |

Learning rate | 0.01 |

Training times | 1000 |

Optimizer | Adam |

MSE | R^{2} | Maximum Error | Minimum Error | Average Error | Training Duration | |
---|---|---|---|---|---|---|

BP | 92.67 | 0.9573 | 15.068% | 0.043% | 1.7% | 4 s |

GA-BP | 35.94 | 0.9832 | 9.724% | 0.021% | 1% | 83 s |

LSTM | 56.50 | 0.9733 | 11.053% | 0.108% | 1.5% | 61 s |

MSE | R^{2} | Maximum Error | Minimum Error | Average Error | |
---|---|---|---|---|---|

GA-BP | 43.82 | 0.9805 | 4.325% | 0.032% | 1.3% |

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## Share and Cite

**MDPI and ACS Style**

Zhang, D.; Hu, J.; Zhao, W.; Lai, M.; Gao, Z.; Wu, X.
A Single-Stack Output Power Prediction Method for High-Power, Multi-Stack SOFC System Requirements. *Inorganics* **2023**, *11*, 474.
https://doi.org/10.3390/inorganics11120474

**AMA Style**

Zhang D, Hu J, Zhao W, Lai M, Gao Z, Wu X.
A Single-Stack Output Power Prediction Method for High-Power, Multi-Stack SOFC System Requirements. *Inorganics*. 2023; 11(12):474.
https://doi.org/10.3390/inorganics11120474

**Chicago/Turabian Style**

Zhang, Daihui, Jiangong Hu, Wei Zhao, Meilin Lai, Zilin Gao, and Xiaolong Wu.
2023. "A Single-Stack Output Power Prediction Method for High-Power, Multi-Stack SOFC System Requirements" *Inorganics* 11, no. 12: 474.
https://doi.org/10.3390/inorganics11120474