# Gaussian Process Model-Based Performance Uncertainty Quantification of a Typical Turboshaft Engine
## 4 Results and Discussion
### Uncertainty Quantification of the Turboshaft Engine Performance
#### 4.3.3 Uncertainty Analysis of the Turboshaft Engine Performance  
In this section, the effects of six uncertain parameters on three PMs of the turboshaft engine is analyzed considering the independence assumption of the parameters. As presented in Figure 5, an apparent phenomenon is that the scatter of each PM becomes larger as the standard deviations of the uncertain parameters increase. In this paper, the performance scatter is defined as six times the standard deviation of the PM, which corresponds to the range \([\mu-3\sigma,\mu+3\sigma]\) of the PM distribution. The \(\pm 3\sigma\)-quantile can be approximated as the maximum (positive and negative) deviations from the nominal design value of the PM since the mean value of the PM distribution is very close to the design value. From Table 5, it can be found that the uncertain parameters have a larger influence on \(P_{e}\) than on \(T_{45}\) as well as SFC. Specifically, when the standard deviations of the uncertain  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Performance Parameters** & **Distribution Parameters** & \begin{tabular}{c} **Relative Deviation** \\ **(\(\sigma\) = 0.002)** \\ \end{tabular} & \
\begin{tabular}{c} **Relative Deviation** \\ **(\(\sigma\) = 0.003)** \\ \end{tabular} &
\begin{tabular}{c} **Relative Deviation** \\ **(\(\sigma\) = 0.004)** \\ \end{tabular} \\ \hline \(T_{45}\) & Mean & \(-0.000175\%\) & \(-0.000876\%\) & \(-0.001247\%\) \\  & Std * & \(-2.337830\%\) & \(-4.218916\%\) & \(-3.548779\%\) \\  & Mean & \(0.001091\%\) & \(0.002451\%\) & \(0.003923\%\) \\  & Std * & \(-6.624252\%\) & \(-8.693671\%\) & \(-8.350732\%\) \\  & Mean & \(-0.000679\%\) & \(-0.001874\%\) & \(-0.003072\%\) \\  & Std * & \(3.716896\%\) & \(2.997669\%\) & \(3.738209\%\) \\ \hline \hline \end{tabular}  
* Std denotes the standard deviation of the parameter.  
\end{table}
Table 5: The relative deviations between the two scenarios for three PMs under different standard deviations of the uncertain parameters.  
parameters is set to the low level (0.002), the performance scatter of \(P_{e}\) is 0.04348, whereas those of \(T_{45}\) and SFC are 0.02653 and 0.01678, respectively. Furthermore, the performance scatters almost double when the standard deviations of the uncertain parameters increase to the high level (0.004). For turboshaft engine designers, the minimum \(P_{e}\), maximum \(T_{45}\), and maximum SFC are three important performance indices that need to be paid attention to in the overall performance design. The results of this study indicate that the uncertainties (standard deviation of 0.002) in the component performance may lead to a 1.3% increase in \(T_{45}\), a 0.84% increase in SFC, and a 2.2% decrease in \(P_{e}\). These deviations nearly double as the uncertainties become greater (standard deviation of 0.004). Therefore, the engine designers should reserve sufficient performance margins for engine performance (\(T_{45}\), \(P_{e}\), and SFC) to compensate for the effect of the component performance uncertainty.