Abstract: This paper introduces the basic principles of artificial neural networks and the theory of backpropagation (BP) algorithm, and studies the application of BP algorithm in boiler system fault diagnosis. An example training process and test results for a boiler sample are presented. Finally, error analysis confirms that this method can achieve good results in boiler fault diagnosis. Keywords: Artificial neural network; BP algorithm; Fault diagnosis; Boiler system 1 Introduction Large thermal power units have numerous boiler system parameters, complex systems, and most equipment operates in high-temperature and high-pressure environments, resulting in a high failure rate and significant hazards. Therefore, real-time monitoring of the thermal parameters of the thermal power unit during operation and fault diagnosis and prediction based on abnormal changes in these parameters are crucial issues in the operation of thermal power units. Fault Diagnosis (FD) technology addresses this problem by studying the occurrence and development mechanisms of boiler faults after understanding the boiler operation process, enabling early detection of faults and their causes, and prediction of fault development trends. During the diagnosis process, it is essential to utilize various useful information exhibited by the object being diagnosed, and after appropriate processing and analysis, to draw correct diagnostic conclusions. 2. Basic Principles of Boiler Fault Diagnosis Based on Neural Networks Artificial Neural Networks (ANNs) are artificially constructed neural networks capable of performing certain functions based on people's understanding of the neural network of the human brain. They are actually complex networks composed of a large number of simple interconnected components, exhibiting high nonlinearity and capable of performing complex logical operations and nonlinear relationships. Figure 1 shows a general model of a neuron that makes up a neural network. Figure 1 Artificial Neuron Model An artificial neuron is the basic computational unit of a neural network, a nonlinear unit with multiple inputs and one output, possessing a threshold, , which is the transfer function of neuron activity, and , which is the weight value for each input. The mathematical model of a neuron is: [img=159,77]http://www.chuandong.com/uploadpic/THESIS/2007/12/2007120510001060968V.jpg[/img] where is the signal transmitted from the previous layer of neurons. For convenience and uniformity, the net input of the neuron in the above formula is also simplified to: Where, in the field of fault diagnosis, The transfer function f(x) is usually taken as the Sigmoid function: [img=184,43]http://www.chuandong.com/uploadpic/THESIS/2007/12/2007120510050359470M.jpg[/img] or the hyperbolic tangent function [img=176,64]http://www.chuandong.com/uploadpic/THESIS/2007/12/2007120510053291710J.jpg[/img]. The Sigmoid function has a shape resembling the letter S and is often called an S-shaped function. In the field of fault diagnosis, it is suitable for the confidence region of [0,1]. The hyperbolic tangent function is suitable for the confidence region of [-1, 1]. Among the many artificial neural network models, the most commonly used is the BP (Back Propagation) model, which is a multilayer feedforward neural network model solved using the error backpropagation algorithm. Backpropagation (BP) networks have been widely used in pattern recognition, image recognition, and management systems. This paper will utilize the BP model in neural networks for fault diagnosis of boiler systems. A typical neural network structure is shown in Figure 2: Figure 2 Basic Structure of a BP Neural Network. First, knowledge acquisition is required. Experts provide examples of various boiler system fault phenomena (symptom set) and corresponding fault causes (fault set) as learning samples. The data is divided into two parts: one for training the network and the other for testing. The training network data is encoded in a certain order and assigned to the network input and output nodes. The samples are learned through a neural network learning algorithm, and the weights are continuously adjusted by an internal adaptive algorithm until the required learning accuracy is achieved. At this point, expert knowledge and experience are distributed across the weights connecting a large number of neurons. After training, the test network data is used to reason backward from the initial state. The displayed fault results are compared with the actual test data results. If the error is small, the network weights are correctly established; if the error is large, the network weights are incorrect, and the network needs to be retrained. The neural network is described as follows: Output layer: [img=200,75]http://www.chuandong.com/uploadpic/THESIS/2007/12/2007120510065758587Q.jpg[/img] Hidden layer: [img=200,68]http://www.chuandong.com/uploadpic/THESIS/2007/12/2007120510071990228L.jpg[/img] Here, the transfer function is taken as: [img=178,41]http://www.chuandong.com/uploadpic/THESIS/2007/12/2007120510073890799R.jpg[/img] where is called the steepness factor, which is used to adjust the shape of the Sigmoid function. 3. BP Network Learning Algorithm 3.1 Derivation of BP Network Learning Algorithm The essence of the BP algorithm is to solve the problem of minimizing the error function. It adopts the gradient descent method in nonlinear programming and corrects the weights according to the negative gradient direction of the error function. The main idea is to calculate the error E of the metric function of the training network: [img=174,43]http://www.chuandong.com/uploadpic/THESIS/2007/12/2007120510082542646T.jpg[/img] Figure 3 Flowchart of the BP learning algorithm. Then, the weight correction amount is calculated using E, as follows: [img=200,86]http://www.chuandong.com/uploadpic/THESIS/2007/12/200712051009488506419.jpg[/img] Where: [img=200,56]http://www.chuandong.com/uploadpic/THESIS/2007/12/2007120510101348311Q.jpg[/img] n is the learning rate. Finally, the correction relationship of the network weights is obtained as follows: [img=200,54]http://www.chuandong.com/uploadpic/THESIS/2007/12/2007120510104536566Q.jpg[/img] 3.2 Selection of Network Input/Output Vectors and Parameters The fault type X = (x1, x2, x3, x4) is used as input, and the fault cause O = (o1, o2, ..., o12) is used as output. Fault modes caused by different fault causes are used as training samples for learning, thereby establishing a mapping relationship between fault modes and fault causes. x1, x2, x3, x4 represent four fault phenomena: boiler water shortage, boiler full water, steam-water surging, and boiler water hammer, respectively; o1, o2, ..., o12 represent 12 fault causes, such as water level gauge malfunction. 3.3 Network Structure and Selection of Training Samples The neural network model for boiler system fault diagnosis consists of three layers: an input layer, a hidden layer, and an output layer. Based on the sample pattern pairs, the number of input layer nodes is determined to be 4, corresponding to 4 fault phenomena, and the number of output layer nodes is 12, corresponding to 12 fault causes. The magnitude of the output node value reflects the probability of the fault occurring. See Table 1 for details. Table 1: Analysis of Fault Types and Causes in Boiler Systems. The number of hidden layer nodes can be selected using an empirical formula: where is the number of output nodes, is the number of input nodes, and is a constant from 1 to 10. Here, we tentatively select 10 hidden layer nodes. Thus, the network has a total of 26 nodes. 3.4 Network Training and Testing Table 2: Sample Training Table for BP Network. We use a given fault phenomenon as the network input, requiring the network to adjust all connection weight coefficients and the thresholds of each neuron to obtain ideal results on the output layer neurons; then, another phenomenon is given, requiring the network to continue learning this pair of patterns. Training is complete when the system's average error E meets the requirements. Based on the flowchart and the BP algorithm, a program is written in Matlab language for sample training. The test results of the training samples are shown in Table 3 below. [b]4 Error Analysis[/b] [align=center]Figure 4 Relationship between Training Times and Error[/align] [align=center]Figure 5 Relationship between Number of Hidden Layer Nodes and Error[/align] Table 4 shows the relationship between the number of hidden nodes and training error. It can be seen that the training error of the BP network initially decreases rapidly, then gradually stabilizes; when the number of hidden nodes is less than 3, the error is extremely large, then rapidly decreases to a very small value, and then changes slowly. After experiencing a "suitable" number of hidden nodes, it begins to fluctuate and rise again. This indicates that if the number of hidden layer nodes is too small, the network cannot summarize the patterns reflected in the training sample set; if it is too large, over-adjustment occurs, reducing the generalization ability. Therefore, constructing a good network requires not only utilizing experience but also undergoing hundreds or thousands of trials. 5 Conclusion In summary, it can be seen that, unlike traditional diagnostic methods, artificial neural networks are a very effective intelligent information processing method. They are self-learning and trainable, possessing many excellent characteristics. Introducing fault diagnosis technology based on the neural network backpropagation (BP) algorithm into boiler systems can play a significant role in the monitoring and diagnosis of boiler systems. With appropriate improvements to the BP algorithm, or by combining it with other methods to leverage the strengths of each approach, even more ideal results can be obtained.