# Gaussian Process Model-Based Performance Uncertainty Quantification of a Typical Turboshaft Engine
## 2 Modeling of the Turboshaft Engine Performance
### Turboshaft Engine Performance Simulation Model
In general, given \(N_{com,cr}\) and \(Z_{com}\), the values of \(w_{com,cr}\), \(\pi_{com}\), and \(\eta_{com}\) are obtained by interpolation. Figure 2: (**a**) The pressure ratio–corrected mass flow map; (**b**) the adiabatic efficiency–corrected mass flow map. To adjust the standard map of the compressor, three coupling factors regarding the characteristic parameters are defined as follows:  
\[\left\{ \begin{array}{c}C_{w}=w_{com,r}^{(des)}/w_{com,r}^{(map)}\\ C_{\pi}=\left(\pi_{com}^{(des)}-1\right)/\left(\pi_{com}^{(map)}-1\right)\\ C_{\eta}=\eta_{com}^{(des)}/\eta_{com}^{(map)}\end{array} \right., \tag{2}\]  
where \(C_{w}\), \(C_{\pi}\), and \(C_{\eta}\) denote the coupling factors of corrected mass flow, pressure ratio, and adiabatic efficiency, respectively; \(w_{com,r}^{(des)}\), \(\pi_{com}^{(des)}\), and \(\eta_{com}^{(des)}\) denote the design values of the compressor performance; \(w_{com,r}^{(map)}\), \(\pi_{com}^{(map)}\), and \(\eta_{com}^{(map)}\) denote the corresponding interpolated values obtained from the map at given \(N_{com,r}\) and \(Z_{com}\). The coupling factors reflect the relationships between the design performance of the compressor and the standard map of the compressor. Based on these coupling factors, any points on the standard map can be adapted to the simulation model. The off-design point performance of engine is highly dependent on the component performance. Due to the changes in ambient condition or operating condition of the engine, the operating points of various components may deviate from their corresponding design points, which results in the variation of component performance, especially for the compressor and turbines. For instance, the increase of the ambient temperature often results in the decrease of the corrected spool speed of the compressor, leading to a decrease in efficiency and mass flow. The variation of component performance leads to the rematch of the thermodynamic cycle. In this situation, the component performance and other key thermodynamic parameters are needed to be decided by the equilibrium running principles [26], which are listed as follows:  
1. Flow continuity in the interrelated components;
2. Power balance for the components in a rotor;
3. Rotational speed equality for the components in a rotor;
4. Pressure balance at the flow mixed sections. To simulate the off-design point performance, a vital task is to identify the component performance and the key thermodynamic parameters. In this model, there are four undetermined parameters, as presented in Table 1. \(Z_{com}\), \(Z_{HPT}\), and \(Z_{LPT}\) are used to identify the performance of compressor, HPT, and LPT, respectively. The values of component performance are read from the component maps, which has been explained in the mentioned compressor example. Generally, the relative corrected spool speed \(N_{cr}\) is also an essential parameter to determine the component performance. However, considering the control scheme, the relative corrected spool speed can be directly derived by:  
\[N_{cr}=\frac{N_{act}}{N_{des}}\sqrt{\frac{T_{in}^{des}}{T_{in}^{act}}}, \tag{3}\]  
where \(N_{act}\) is the actual spool speed of the rotational component (including compressor and turbine), \(N_{des}\) is the design spool speed of the rotational component provided by the design module, \(T_{in}^{act}\) is actual total temperature at the inlet of the component, and \(T_{in}^{des}\) is the design total temperature provided by the design module. In this model, the actual spool speed \(N_{act}\) is specified according to the control scheme and the actual total temperature \(T_{in}^{act}\) can be obtained by thermodynamic calculation. The fuel flow \(W_{f}\) is another important thermodynamic parameter to calculate the total temperature at the outlet of combustor and is also affected by the control scheme. Based on equilibrium running principles, four independent constraint equations can be built to recognize the suitable values of the undetermined parameters.