1. Introduction
The high pressure control valve is one of the most important pieces of equipment in the steam turbine, which governs the steam flow and regulates the power generation.
Figure 1 illustrates the working mechanism of the control valve. When the valve is opening, the pressure oil enters into the pressure chamber and pushes the valve stem downwards; when the valve is closing, the oil is discharged from the solenoid valve and the return spring drives the stem upwards.
Due to frequent opening and closing of the valve in the process of daily operation, the valve stem and valve body are easy to wear, resulting in the stiction, such as valve dead zone. The dead zone is an insensitive area where the valve position does not change with the command. The control valve dead zone can easily cause system oscillations. These oscillations will lead to increase energy consumption and increased wear and tear of equipment along with poor product quality [
1,
2]. Hence, detecting the valve stiction becomes imperative for the stable and economic operation and power generation for the steam turbine.
Much literature has revealed the fault detection and diagnosis in actuators, such as the state observer based method [
3], the Kalman filter method [
4], and the artificial neural network [
5]. Among a large number of fault detection methods for the control valve, the graph model is a promising one due to its strong reasoning ability, such as bond graphs [
6]. The premise of graph model application is to establish the graph topology accurately. On the one hand, the dead zone detection based on graph model requires accurate graph topology to characterize the industrial equipment, subsystem, and system. On the other hand, the fault of dead zone will propagate to other equipment, subsystem, and system along the related paths. The propagation paths are represented as the edges in the graph topology. Hence, completed and accurate graph edges or relationships between variables of graph nodes are crucial for the fault detection for the control valve dead zone.
However, obtaining the completed and accurate graph topology for the steam turbine control valve is never an easy task, since with the deepening of the research, the graph topology construction faces several difficulties. Above all, as the steam turbine system becomes more complex, it is never an easy task to find all the relationships according to the mechanism. Then, edges in the graph topology not only appear as physical connections, but also as cross-correlation dependencies, which is difficult to analyze by pure mechanism. Last but not least, only limited knowledge related to steam turbines can be obtained, leading to the inaccuracy of the graph topology. Hence, it is necessary to develop a method to estimate the relationship and predict the graph topology for the steam turbine graph model.
In complex networks or graph theory, the problem of relationship prediction for the steam turbine control valve is equivalent to the link prediction problem for the graph. The basic idea for link prediction is to reveal the relationship between graph nodes by analyzing the graph topology and the attributes of nodes and edges. Typical link prediction methods mainly include similarity-based algorithm [
7], maximum likelihood methods [
8], and probabilistic models [
9], and they are well summarized in [
10,
11,
12]. To the best of the authors’ knowledge, little literature implements the link prediction in the graph of the industrial system, and none of them studies the link prediction and relationship prediction for the steam turbine control valve.
In this study, a novel method for the relationship prediction based on graph model for steam turbine control valve is proposed. First of all, the uncompleted graph which may have missing edges is established for the steam turbine control valve and its surrounding equipment. Each node in the graph corresponds to the physical variable of the equipment in the steam turbine, along with its measurement. Next, graph convolution is implemented iteratively to learn the low-level representations for graph nodes. In the meantime, the dead zone detection is finished. Afterwards, a score function for the edge, relying on the low-level representations for the linked graph nodes, is defined to predict the links. Ultimately, the accuracy for the dead zone detection and link prediction are over 98%. Moreover, the results of the link prediction follow the principles of thermodynamics. The proposed method is suitable for the relationship prediction for the steam turbine system. Doubtlessly, the relationship prediction method can also be applied to other inter-connected industrial system. In this paper,
Section 2 includes a more detailed definition of the problem, the mathematical preliminaries for the graph convolutional network, and the description of the link prediction algorithm.
Section 3 shows a numerical examples for the fault detection and link prediction for the control valve of the steam turbine. Finally,
Section 4 gives a conclusion for the whole paper.
3. Numerical Examples
Consider a steam turbine simulation system, which consists of the boiler, the control valve, the high, intermediate, low pressure turbine, the condenser, and the two stage steam extractions, etc. The simulation is conducted under the Matlab/Simscape environment. Matlab/Simscape supports a steam turbine physical system based on Rankine cycle [
18]. For the simulation of the fault, a dead zone block is connected between the PID controller output and the opening of the high pressure control valve, and a 15% dead zone is injected into the valve, with the simulation time of 2400 s. Consequently, the time series of node variables in
Figure 2 is obtained, and
together with dead zone illustration are exhibited in
Figure 4.
Figure 4a shows the inlet pressure of HP in the process of turbine power regulation. The inlet pressure of HP is nearly stable at 4100 kPa. It can be inferred that the control valve dead zone does cause the HP oscillation. To some extent, the HP oscillation will lead to system oscillation, which will affect the safe and stable operation of steam turbine.
To feed the time series data into the proposed link prediction model, the data are pre-processed. Above all, the data is scaled into 0∼1 using standard normalization. Then, data under normal and dead zone condition are labeled with 0 and 1 respectively. Next, they are combined and randomly shuffled, with 70% for training the model and 30% for testing the model. The total layers sum up to 4, and the dimensions of the layers are 4, 8, 8, 4. The coefficient of the L2 normalization is 0.001. The learning rate of the batched gradient descent algorithm is 0.01. Each batch contains 10 samples, and the batched gradient descent algorithm utilizes all the 10 samples in one batch to update the parameters of the model at one training step. The training batches for dead zone detection and link prediction are 31 and 63, respectively. Finally, the training accuracy and loss are shown in
Figure 5a. Moreover, the test accuracy for dead zone detection reaches 98.8%.
After dead zone detection, the representations for graph nodes are obtained. The link prediction can be conducted based on the node’s representations. Regard the edges
and
as the positive samples, and randomly selected another two unconnected edges as the negative samples. The training results are illustrated in
Figure 5b. The test accuracy for the link prediction reaches 99.2%.
Since the link prediction model is tested with high accuracy, it can be adopted to predict the unknown edges. In
Figure 6, the score histograms for all of the predicted existent edges and parts of the nonexistent are exhibited. Each histogram shows the score distribution of the corresponding links, and the average score is attached above each picture, indicated by
. For the positive samples, i.e., the existent links, it can be inferred that the predicted average scores for the nine types of links are bigger than 0.5. For the negative samples, that is the nonexistent links, the predicted average scores for the three types of links are smaller than 0.5. Obviously, the link prediction based on the score function for the steam turbine system is accurate.
What is more, the completed graph
is shown in
Figure 7, which conforms to the results in Equation (
1). The red lines are the predicted existent edges, labeled with the score of the link prediction model.
The link prediction results mainly reveal two kinds of relations: the relation between the steam pressure and the steam mass flow rate, and the relation between the steam pressure and the steam enthalpy. On the one hand, according to the thermodynamics of fluid, when the cross-sectional area of the flow is fixed, the larger the flow rate is, the greater the pressure is. On the other hand, the enthalpy
H has the following relations with the intensity of pressure
P:
where
U and
V represent the system internal energy and the volume, respectively. Obviously, the enthalpy is directly related to the pressure. Therefore, the link prediction results are convincing. The proposed method is suitable for the relationship prediction for the steam turbine and other inter-connected industrial system.
4. Conclusions Remarks
To solve the problem of inaccurate and uncompleted graph topology while detecting the fault of dead zone for the steam turbine control valve based on the graph model, a link prediction technology is proposed to estimate the relationships in this study. First of all, the uncompleted graph topology for the steam turbine control valve, which may lack some edges, is established according to the limited mechanism knowledge. Then, graph nodes representations are obtained using the graph convolution network. Finally, scores for edges are calculated utilizing pairs of connected graph nodes. The edges with scores larger than 0.5 indicate that there exist relationships between the corresponding graph nodes. Results exhibit the test accuracy of 99.2%, and follow the principles of thermodynamics. Moreover, in addition to the steam turbine control valve, other industrial system and even other disciplines, such as social networks and recommendation systems, must also have the same issue of link prediction and relationship prediction. The proposed method can also take these areas into account, with a good application prospect.