Neural network-based modeling of solid oxide fuel cells for marine applications

As the solid oxide fuel cell (SOFC) experimental test is still quite cost-effective and time-consuming, there is a growing need for developing effective simulation tools to reduce the time and cost of the marine SOFC performance test and optimization. The present paper is aimed to study the modeling and simulation of marine solid oxide fuel cells by artificial intelligence method. A neural network based on a particle swarm optimization algorithm is used to establish a marine solid oxide fuel cell model for voltage/current characteristic analysis. The model is also compared with BP neural network and Hopfield neural network methods. The simulation results compared with experimental data show that the effectivity of the particle swarm optimization neural network algorithm is best, which can accurately predict the voltage/current characteristic curves of a SOFC under different fuel flow-air volume ratios. The model study can provide support for SOFC performance characteristics analyses and has significant potential in SOFC optimization applications.


Introduction
Solid oxide fuel cell (SOFC) is a low-pollution energy conversion device.It has become a research hotspot and development direction of fuel cell technology due to its advantages of not using precious metal catalysts and having lower requirements on fuel type and quality [1].Demonstration performance proves that SOFC is one of the ideal choices for future fossil fuel power generation technology [2].
However, due to the high cost and time-consuming nature of SOFC experimental testing, the development of effective simulation tools to reduce the time and cost of marine SOFC performance testing and optimization has become increasingly important.In 2011, Song and Zhong applied the HDP algorithm, a heuristic dynamic planning algorithm based on BP neural network, to the control of the SOFC system, which effectively improved the utilization of fuel gas [3].In 2014, Razbani and Assadi utilized a BP neural network to develop a SOFC model that incorporated variables like fuel flow, battery voltage, and temperature.The research outcomes indicated an inverse correlation between the projected and observed results, with an average deviation of 0.2% [4].
The particle swarm optimization algorithm (PSO) is a random optimization algorithm that combines global search and local optimization strategies.Compared with traditional neural networks (BP, Hopfield), PSO effectively circumvents local optima, exhibiting superior convergence speed and precision.Therefore, this paper adopts a neural network based on the PSO algorithm to establish a more accurate and efficient SOFC stack model, which can accurately predict the electrical behavior of SOFC when subjected to varying fuel ratios.In addition, this paper provides insights into the application of artificial intelligence methods in the field of renewable energy at sea, which is an important research direction for sustainable development.Through this research, we hope to contribute to the development of ship energy conversion technology [5].

Theoretical foundations of the PSO algorithm
Particle swarm optimization (PSO) is one of the most famous swarm-based optimization techniques inspired by nature [6].In the PSO algorithm, it is necessary to define the objective function, which is the problem to be optimized by the PSO algorithm.For the SOFC model, the objective function can be defined as a mean squared error (MSE).In the PSO algorithm, each particle's position represents a potential solution to the problem being optimized, and each particle also has a velocity vector that guides its movement.The updated formulas for the particle's velocity and position are as follows: , , , where represents the velocity of dimension j for particle i, represents the position of dimension j for particle i, j i p represents the best position achieved by particle i in its history, j g represents the best position achieved by the whole swarm in its history, w represents the inertia weight, 1 c and 2 c represent the learning factors regarding the best positions of particles and the swarm, individually, and 1 r and 2 r are random numbers between 0 and 1.

SOFC discriminating model structure based on PSO
SOFCs are complex nonlinear systems, and the identification model for SOFCs adopts a concise model structure known as the Nonlinear AutoRegressive with exogenous inputs (NARX) model [7][8], which can be described using nonlinear difference equations: where represent the orders of the input and output, respectively.[.] f represents the nonlinear function of the system to be identified, [.] represents the regression vector composed of the output and input before time k, and represents the output voltage of the fuel cell stack at the time 1 + k [9].The structure of the SOFC stack model identified based on PSO is shown in Figure 1, where ) ( 2 k q H and ) (k l are the inputs of the SOFC stack, is the actual output of the SOFC stack, is the predicted output of the RBF model, is the mean squared error between , and X(k) represents the input of the PSO identification model through the time delay line (TDL) [10] and

Simulation experiment and analysis
According to the dynamic model of the SOFC stack, the voltage/current density curve is affected by multiple factors such as stack temperature, different fuel ratios, and gas pressures.In this study, data collection only considered the effect of different fuel ratios on the stack performance.According to the least squares support vector regression machine [11], the dynamic model of the SOFC was established, and data was collected.To improve predictive model generalization, the voltage and current data of the SOFC under different fuel ratios were collected under the condition of T=973 K, and the data were divided into training and testing sets.
In this study, the voltage/current data of the SOFC under fuel ratios of 2-5, 4-10, 8-20, and 32-80 were selected as the training set, and the voltage/current data under a fuel ratio of 16-40 was selected as the testing set, as shown in Figure 2.  In this experiment, we trained the SOFC model using the particle swarm optimization (PSO) algorithm and validated the trained PSO model on the training set, as shown in Figure 3.As shown in Figure 4, we can see that the PSO algorithm's predicted data is very close to the actual data on the training set, and although there are slight deviations in the testing set, the overall fitting effect is still good.To conduct a more comprehensive evaluation of the particle swarm optimization (PSO) algorithm's effectiveness, we compared it with the BP algorithm and the Hopfield algorithm on the test set data.As shown in Figures 5 and 6, it can be seen that compared with the BP algorithm and the Hopfield algorithm, the PSO algorithm has higher prediction accuracy on the test set data and is closer to the actual data.
To better illustrate the superiority of the PSO algorithm, we plotted the mean square error (MSE) graph as shown in Figure 7. Analyzing the results of the experiment from a quantitative point of view, it can be seen that on the test set data, the PSO algorithm has a smaller MSE, indicating better fitting and higher prediction accuracy.In summary, the ship-based SOFC model based on the particle swarm optimization neural network algorithm proposed in this study has good predictive performance and can provide a reference value for research in related fields.

Conclusion
The findings of this study demonstrate the successful application of a neural network model, optimized using the particle swarm optimization algorithm, in accurately predicting the performance of marine solid oxide fuel cell (SOFC) models.Through comprehensive training, the model exhibited precise voltage output predictions, and subsequent evaluations confirmed its high accuracy and stability in forecasting capabilities.Therefore, this study provides an effective method for predicting the modeling of ship-based SOFC models and has high application prospects and research value.
Looking ahead, there is room for further exploration into alternative optimization algorithms and more advanced neural network structures, aiming to enhance the model's overall performance.Additionally, the inclusion of a larger volume of experimental and real-world test data would provide valuable insights, validating the model's reliability and applicability.
In conclusion, this study introduces an effective methodology for the predictive modeling of marine SOFC systems, offering promising prospects for practical implementation and carrying significant research value.

Figure 1 .
Figure 1.The model structure of the SOFC stack based on PSO identification.

Figure 4 .
Figure 4. Predicted results from the PSO model.Figure 5. Predicted results from the BP model.

Figure 5 .
Figure 4. Predicted results from the PSO model.Figure 5. Predicted results from the BP model.

Figure 6 .
Figure 6.Predicted results from the Hopfield model.Figure 7. Mean square error plots of PSO, BP, and Hopfield modeling.

Figure 7 .
Figure 6.Predicted results from the Hopfield model.Figure 7. Mean square error plots of PSO, BP, and Hopfield modeling.