# A Data-Knowledge Hybrid Driven Method for Gas Turbine Gas Path Diagnosis
## 3 Methodology
### Physical Model-Based GPA Method  
The schematic of the physical model-based GPA method is shown in Figure 2. The inputs of the GPA model are boundary conditions \(\mathbf{x}\) and measurement data \(\mathbf{y}\), and its output is the fault feature \(\mathbf{D}\). As shown in Figure 2, the whole solving process of the GPA model is an iterative process. In the \(i\)-th iterative process, the iterative process \(\mathbf{D}^{i}\) and boundary condition \(\mathbf{x}\) are inputted into the gas turbine physical model \(f(\,)\) to obtain the simulation data \(\overset{\mathbf{\kappa}}{\mathbf{y}}\) of the measurement parameters. The iterative process ends when the simulation data \(\overset{\mathbf{\lambda}}{\mathbf{y}}\) and actual measurement data \(\mathbf{y}\) satisfy the tolerance condition.  
The GPA solving process is the reverse of the simulation process of the physical model. Therefore, the gas turbine physical model \(f(\,)\) can simulate the measurement parameters based on the boundary condition \(x\) and the fault feature \(D\). It is a steady-state model which is developed by the energy and mass balance equations. The simulation model can be expressed as follows.  
\[\overset{\alpha}{\boldsymbol{y}}=f(\boldsymbol{x},\boldsymbol{D}) \tag{1}\]  
The gas turbine used in this study is a GE LM 2500 PLUS gas turbine engine [27], shown in Figure 2. It is a two-shaft gas turbine, which consists of a compressor, a combustor, a high pressure turbine, and a power turbine.  
The gas path fault affects the aerodynamic performance in the compressor and turbines. In other word, the characteristic maps derivate from the healthy maps when the gas path fault occurs. The gas path fault feature of each component is defined as follows:  
\[DG=\left(1-G_{f}/G_{h}\right)\cdot 100\% \tag{2}\] \[DE=\left(1-\eta_{f}/\eta_{h}\right)\cdot 100\%\]  
where \(DG\) is the component flow rate degradation, \(G\) is the reduced flow rate in the component characteristic map, \(DE\) is the component efficiency degradation, and \(\eta\) is the efficiency in the component characteristic map. Subscript \(f\) denotes the fault state; \(h\) denotes the healthy state.  
Figure 2: Schematic of the physical model-based GPA method for an LM2500 PLUS gas turbine [27].  
Three components (including compressor, high pressure turbine, and power turbine) are analyzed in the GPA method. Consequently, six gas path fault features of the above three components are applied to evaluate the degradation degree of the three components. The gas path fault feature \(\mathbf{D}\) = {_DGC_, _DEC_, _DGT_, _DET_, _DGP_, _DEP_} is summarized in Table 1. The boundary condition \(\mathbf{x}\) = {\(p_{1}\), \(T_{1}\), \(P\), \(n_{2}\)} includes ambient pressure \(p_{1}\) and temperature \(T_{1}\), output power \(P\), and power turbine rotational speed \(n_{2}\). The measurement parameter is \(\mathbf{y}\) = {\(p_{2}\), \(T_{2}\), \(p_{34}\), \(T_{34}\), \(p_{4}\), \(T_{4}\), \(Q_{f}\), \(n_{1}\)}. The \(T\) and \(p\) indicate the temperature and pressure. Subscript 2 indicates compressor outlet, 34 indicates high pressure turbine outlet, and 4 indicates power turbine outlet. \(Q_{f}\) is the fuel flow rate; \(n_{1}\) is the high pressure rotational speed.