*4.1. Artificial Neural Network (ANN) Modelling*

The artificial neural network is the replica of the biological neural network which could be used for the optimization, simulation, modeling, forecasting and performance prediction of various physical systems [25]. The nonlinear relationship between the input and output variables with larger number of data points could be mapped efficiently using the ANN technique [25]. The ANN consists of three layers with the input layer, output layer and one or more than one hidden layer with a suitable number of neurons in each layer [25,26]. The number of neurons in the input layer is equal to the number of input parameters and number of neurons in the output layer is equal to number of output parameters. The number of hidden layers and hidden neurons are decided based on the training error [27]. The neurons of one layer are connected to the other layer using weights and the single weight value is assigned between two neurons [28]. The ANN structure with three layers and various numbers of neurons in each layer is trained using suitable training algorithm [29]. The training algorithm consists of the back-propagation algorithm, training variants and transfer functions [26]. For the training, the maximum training error and the maximum number of epochs are decided. During the training, the weight values get adjusted to predict the desired values of output parameters. If the error between the predicted output and the actual output is lower than the decided training error, the training is stopped; otherwise, further training is done to achieve the desired output [30]. The neural network structure with the desired prediction accuracy is selected as the optimum neural network structure.

In this study, six ANN models are developed to predict the performances of the thermoelectric generator system for waste heat recovery. Figure 3 shows the formulated ANN structure to predict the performance of thermoelectric generator system for waste heat recovery. The hot gas inlet temperature and voltage load conditions are considered as the inputs to the ANN models for predicting the current, power and thermal efficiency of the thermoelectric generator system for waste heat recovery. The back-propagation training algorithm is used to train the six ANN models. The six models are the combinations of three training variants of Levenberg–Marquardt (LM), Scaled Conjugate Gradient (SCG) and Pola–Ribiere Conjugate Gradient (CGP), and two transfer functions of Tan-Sigmoidal and Log-Sigmoidal and number of hidden neurons (N) of 10, 15, 20, and 25. The maximum number of epochs is set to 1000 and the maximum training error is set to 10−<sup>6</sup> for training to confirm the prediction accuracy of the tested model for the thermoelectric generator system for waste heat recovery. The experiments are conducted on the thermoelectric generator system for waste heat recovery at the hot gas inlet temperatures and voltage load conditions to collect the data for training. A total of 931 data points of the input and output parameters are used to train the six models. For each ANN model, the training is done until the error becomes steady and the outputs predicted by that trained model are recorded. The predicted output values of the current, power and thermal efficiency are compared with the corresponding experimental values based on three statistical parameters of the coefficient of determination (R<sup>2</sup> ), root mean square error (RMSE), and coefficient of variance (COV). The ANN model with the highest value of R<sup>2</sup> and the lowest values of RMSE and COV, respectively, is selected as the optimum ANN model to predict the current, power and thermal efficiency of the thermoelectric generator system for waste heat recovery for the hot gas inlet temperatures range of 315.12 to 621.61 ◦C and voltage load conditions range of 0 to 10 V.

*Symmetry* **2020**, *12*, x FOR PEER REVIEW 7 of 30

**Figure 3.** Formulated ANN structure to predict the performances of the thermoelectric generator **Figure 3.** Formulated ANN structure to predict the performances of the thermoelectric generator system for waste heat recovery.
