# Application of the Residue Method in Steam Superheater Fault Detection
## 3 A Mathematical Modelling Method of the Proposed System
### Testing Equivalent Models of Steam Superheaters for Actuator Faults
#### 3.1.1 Steam Superheaters Model
It has been noted that various faults can occur in the installation or control structures, which is why it is vital to develop new detection strategies. These strategies should allow for the type of fault and its location to be specified based solely on current measurements collected from the process, with a minimum of additional equipment required for direct management. Another issue that needs to be addressed is how to handle faults when they occur. Should the plant be shut down, or should operations continue with a new control algorithm? These decisions need to be made while keeping in mind external connections but staying within the safety limits of the power unit. It can be quite challenging to assess the effectiveness of the intricate control structure that regulates the steam boiler. This is because it relies on mathematical and/or physical models of electronic superheaters, which are not easily observable. Fortunately, there is a wealth of helpful literature available that presents a range of models for the steam superheaters commonly used in steam power plants. These models can provide valuable insights into the functioning of the superheaters and help ensure that they operate safely and reliably. In the following, a simplified scheme, as shown in Figure 6, will be used to develop the mathematical model of a superheater. It can be seen how the heating pipe groups _SA1_, _SA2_, and _SA3_ are located, and these heating pipes have the respective gas zones _SG1_, _SG2_, and _SG3_. Due to complex mathematical models, it is easier to approximate these modules with concentrated parameter equations. To determine the mathematical model, it is necessary to make some clarifications on the quantities involved in the processes, namely:  
* The temperatures \(T_{a1}\), \(T_{a2}\), and \(T_{a3}\) at the outputs of every overheated area were measured with their adequate transducers. * The adjusting and regulation of the temperature \(T_{a2}\) and \(T_{a3}\) was performed with the help of the injection flow \(W_{inj1}\) and \(W_{inj2}\);
* \(F_{t}\) is steam flow at the input of the turbine and represents the primary measurable disturbance;
* The temperature of the burning gas \(T_{gi}\) and the gas flow \(F_{g}\), at the input of the overheaters area, hides immeasurable disturbances but can give information on the faults in the fuel-burning process. Based on the block diagram, it is clear that there are certain pairs that are directly influenced by fault-measurable output magnitude, such as \(W_{inj}\)-\(T_{a2}\) and \(W_{inj2}\)-\(T_{a3}\). Additionally, one has access to the pair \(T_{g}\)-\(T_{a1}\), although it may also simultaneously affect \(T_{a2}\) and \(T_{a3}\). It is important to note that disturbances in the vaporization system, such as \(T_{ai}\) and  
Figure 6: Block diagram of steam superheaters. \(F_{ai}\), can also have an impact on \(T_{a1}\). The mathematical model of this system is described by Equations (18)-(23), which were calculated using the steady state for a 420 t/h steam boiler working at 13.7 MPa and 823 K, and using coal and oil as fuels. By using the mass and heat transfer balance equations for each heat exchanger and injector, one can derive equations that correspond to the lumped parameter model of the superheater's area. The heat exchange surfaces found on the steam circuit's boiler are utilized for the initial steam superheat stage. The heat exchange process is accomplished through convection, with a steam flow rate of 407.83 t/h. The steam flows through the superheater pipes in opposition to the flue gas. Within these surfaces, the steam is superheated from saturation temperature up to 396.7 \({}^{\circ}\)C.  
The entirety of the superheater is comprised of pipes with an internal volume of 25 [m\({}^{3}\)], a total steam flow of 407.03 [t/h], and a flue gas volume of 52 [m\({}^{3}\)].