## 2 Materials and Methods
### Neural Models of Temperature Distribution  
The neural models for predicting the temperature recorded during the storage phase of the accumulator were developed in the Matlab environment, using the Neural Network Toolbox add-on. The network's structures consist of an Multi-Layer Perceptron (MLP) [17; 27; 28; 29; 30] with 4 inputs, a diverse number of neurons in the hidden layer (5 or 10 neurons) and 1 neuron in the output layer.  
The one-way multilayer networks of the multilayer perceptron type are among the most studied and most commonly used network topologies. ANN-type MLP represents the so-called parametric neural model class. For example, they are characterized by the fact that the number of neurons forming their structure is markedly lower than the learning file size. The defined input variables and output variable are shown in Table 1.  
The collection of data necessary to develop the neural model was, by default, divided, in the ratio 2:1:1, into the following files: training, validation and testing. These files included, respectively: 1008, 504 and 504 cases. The testing file was not used in the network training process, as this was essential for the final evaluation of the neural model. A fragment of the training file (cases from 331 to 337) is shown in Table 2.  
The developed ANNs were trained using an error back propagation algorithm (BP) with Bayesian regularisation. Mean Squared Error (MSE) was used as the error function. The neural networks generated were designated as MLP 4-5-1, MLP 4-7-1 and MLP 4-10-1. The sigmoid function was used as the activation function for the hidden layer neurons, and a linear function was used for the output layer neurons. The structure of the neural model MLP 4-7-1 is shown in Figure 3.  
In the network training process in the Matlab environment, the following algorithm stoppage criteria were adopted:  
* Maximum training epochs (epochs parameter): 1500;
* Maximum subsequent epochs, in which the error function value may deteriorate for the validation file (max_fail parameter): 100;
* Minimum error function gradient (min_grad parameter): \(1\times 10^{-7}\).  
\begin{table}
\begin{tabular}{c c c c} \hline \hline  & **Name** & **Designation** & **Unit** \\ \hline \multicolumn{4}{c}{Input signals (4 variables)} \\
1 & x coordinates of the location & X & Non dimensional \\
2 & y coordinates of the location & Y & Non dimensional \\
3 & z coordinates of the location & Z & Non dimensional \\
4 & Time & t & [h] \\ \hline \multicolumn{4}{c}{Output signal (1 variable)} \\
5 & Temperature & T & [\({}^{\circ}\)C] \\ \hline \hline \end{tabular}
\end{table}
Table 1: Input and output variables.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**No.** & **X** & **Y** & **Z** & **t [h]** & **T [\({}^{\circ}\)C]** \\ \hline
331 & 0.6511 & 0.5 & 0.5 & 7.133 & 42.55 \\
332 & 0.6511 & 0.5 & 0.5 & 7.300 & 42.53 \\
333 & 0.6511 & 0.5 & 0.5 & 7.466 & 42.50 \\
334 & 0.6511 & 0.5 & 0.5 & 7.633 & 42.48 \\
335 & 0.6511 & 0.5 & 0.5 & 7.800 & 42.46 \\
336 & 0.6511 & 0.5 & 0.5 & 7.966 & 42.46 \\
337 & 0.6511 & 0.5 & 0.5 & 8.150 & 42.43 \\ \hline \hline \end{tabular}
\end{table}
Table 2: A fragment of the training file for the ANN simulator (T—output variable).