# Application of the Residue Method in Steam Superheater Fault Detection
## 3 A Mathematical Modelling Method of the Proposed System
### The Equivalent Electronic Model of Superheat  
To conduct certain experiments on power plant subsystems, it is sometimes necessary to search for equivalent systems and subsystems based on mathematical models. The first solution that has been proposed is to create an equivalent electrical/electronic system of a steam superheater that can be used to simulate various faults. This would help researchers better understand the performance of different subsystems under different conditions, and could ultimately lead to better engineering solutions for power plants. Developing an equivalent electronic circuit begins with a thorough analysis of the superheater structure depicted in Figure 7. From there, the mathematical model outlined by Equations (18)-(26) is extracted to ensure that the circuit functions efficiently and accurately. As a result of this approach, an exact representation of the circuit that meets all requirements was created. It is necessary to establish a correspondence between the actual (measured) temperature inside the superheater and the voltage at the equivalent point of the electronic circuit simulating the superheater. The first step in such an experiment is to establish a correspondence between the measurement scales of the quantities involved in the description of each circuit. In this case, it is necessary to create a relationship between temperatures and electric currents, i.e., voltages that intervene when analyzing an electronic circuit. To achieve this, one can see in Table 2 the relationship between all the quantities involved in the experiment. Due to the orders of magnitude of the magnitudes one is working with, the temperature scale 0-1000 \({}^{\circ}\)C and the 0-10 V equivalent voltage range of the electronic simulation modules with a conversion factor of 1/100 [V/\({}^{\circ}\)C] will be considered. Figure 7: Electrical diagram of the electronic simulator. Based on the given information, it can be observed that the temperature of the steam and gas varies at different sections. For instance, the temperature of the steam at the output of the drum is 345 \({}^{\circ}\)C, which increases to 396 \({}^{\circ}\)C at the output of the first superheater. Similarly, the temperature at the output of the second superheater is 453 \({}^{\circ}\)C, and the third superheater's output temperature is approximately 570 \({}^{\circ}\)C, which is the input temperature of the turbine. The temperature of the gas also varies along the three sections, as shown in Table 2.  
In order to simulate the first-order differential equations in the mathematical model, an electronic circuit consisting of an RC group and an operational amplifier in a summing assembly will be utilized. Depending on the sign of the terms of the equations, it will be utilized as either direct or inverse action. To eliminate inter-influences between the RC cells, it is necessary to introduce operational amplifiers. Choosing the appropriate feedback resistors will ensure the necessary amplification factors from one module to another. Additionally, by adequately biasing the non-inverting input or inverters, it will be able to establish the steady-state voltages at specific points of the circuit that correspond to the equivalent values of the temperatures in the superheaters of the steam boiler. This enabled the study of the actual superheating process qualitatively and quantitatively on the electronic simulator. Furthermore, each module of the electronic simulator corresponds to Equations (18)-(23) for the steam and gas zones of the superheaters and (24)-(26) for the injection blocks. The delays that occur during the overheating of the steam and gas corresponding to the (18)-(26) are entered using an RC circuit assembly, as can be seen in Figure 8. All the modules corresponding to the block diagram in Figure 6 have been realized in this way. For the injectors, an electronic structure has been chosen around an operational amplifier simulating the block diagram.