# Research on Model-Based Fault Diagnosis for a Gas Turbine Based on Transient Performance
## 2 Methodology
### Modeling of Gas Turbine  
The dynamic system of a gas turbine plant consists of working fluid and rotary parts. The system behavior can be derived by conservation laws and the equations of motion [19]. To analyze the dynamic characteristics of a gas turbine, unsteady three-dimensional calculations can be used. However, applying the unsteady three-dimensional simulation requires vast computational resources. It has been proved that one-dimensional simulation gives sufficiently accurate results [20]. In this study, the modular modeling method is applied to build the model. A two-shaft industrial gas turbine, which consists of one compressor, one combustor, one high pressure turbine, and one power turbine, is cited as an example for gas turbine modelling and fault diagnosis, as shown in Figure 1.  
Based on the principles of the modular modeling method, the entire plant is divided into four parts: compressor, combustor, turbine (high pressure turbine and power turbine) and rotating shaft. The input and output parameters of each module need to be defined. The modeling flowchart and modular inputs and outputs are shown in Figure 2.  
**Compressor and Turbine. ** The compressor and two turbines (high pressure turbine and power turbine) are modelled using the characteristic map that describes the correlation between pressure ratio, mass flow rate, rational speed, and efficiency. For example, compressor speed and isentropic efficiency can be determined using the characteristic map. Since the design pressure ratio of the compressor  
Figure 1: Gas turbine configuration of this case study. Figure 2: Modeling flowchart based on the modular modelling method for gas turbine. is known, and the mass flow rate of the air is determined from the plant operation requirement, mainly referring to power output, the compressor speed and isentropic efficiency can be read off the characteristic map. The following lists their performances:  
\[\left(\frac{Q\sqrt{T_{in}}}{p_{in}},\eta\right)=f\Big{(}\pi,n/\sqrt{T_{in}} \Big{)} \tag{1}\]  
where \(\pi\), \(n\) and \(\eta\) stand for pressure ratio, rotational speed and component efficiency, and \(T_{in}\) is the inlet temperature. From steady-state characteristics map generated by the field operating data [21], compressor mass flow \(G_{c}\) and component efficiency \(\eta\) can be calculated by Equation (1), which can be used to calculate the power required and the temperature of the discharge air. \[T_{2}=\left(1+\frac{\pi^{\frac{R}{G_{p}}}-1}{\eta_{c}}\right)T_{1} \tag{2}\]  
\[P_{c}=G_{c}C_{p}(T_{2}-T_{1}) \tag{3}\]  
where \(C_{p}\) is the heat capacity and \(R\) is the gas constant. The degradation of gas path components can be mathematically determined by scaled component maps. It is assumed that the degraded maps of the compressor, combustor, and turbine are down-scaled versions of the corresponding original maps, due to the fact that the geometries do not change significantly. More details on the definitions of the degradation parameters of gas turbine components can be found in reference [22; 23]. **Combustor. ** The combustor model involves the prediction of the dynamic response of pressure and temperature inside the combustor. The pressure is determined by the combustor model, while the flow rate is determined by the compressor and turbine model using the characteristics maps. Therefore, the state equation of pressure can be written as:  
\[\frac{dp}{dt}=\frac{R_{g}T_{g}}{V}\Big{(}G_{f}+G_{c}-G_{t}\Big{)} \tag{4}\]  
where the subscripts \(f\), \(c\) and \(t\) stand for the fuel, compressor and turbine, respectively. \(V\) is the volume of the combustor, \(t\) is the time. The turbine inlet temperature is the most important parameter for the gas turbine, which is also calculated in the combustor model.