# Gas Turbine Off-Design Performance Adaption Based on Cluster Sampling
## 2 Methodology
### Performance Adaptation  
The gas turbine model was introduced in the previous section, and the component characteristic maps play an important role in the model computation. Therefore, the performance adaption method is proposed to modify component characteristic maps for enhancing the model precision. The basic method of performance adaption used in this paper contains two procedures: (1) Map scaling for design point, which introduces a set of fixed scaling factors to modify the original components' characteristic maps. This procedure is not described in this paper as it is not its focus. More details can be obtained from [PERSON] et al. (2017). (2) Performance adaption for off-design points, in which a set of scaling factors are introduced to improve the prediction accuracy of the simulation model during off-design conditions. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**No.** & **Measurable Parameters** & **No.** & **Measurable Parameters** \\ \hline
1 & Compressor inlet temperature \(T_{1}\) & 7 & Fuel pressure \(P_{f}\) \\
2 & Compressor inlet pressure \(P_{1}\) & 8 & Turbine outlet temperature * \(T_{4}\) \\
3 & Compressor outlet temperature * \(T_{2}\) & 9 & Turbine outlet pressure \(P_{4}\) \\
4 & Compressor outlet pressure * \(P_{2}\) & 10 & Power output * \(P_{w}\) \\
5 & Fuel mass flow \(m_{f}\) & 11 &
\begin{tabular}{c} Rotor speed \(n\) \\ \end{tabular} \\
6 & Fuel temperature \(T_{f}\) & & \\ \hline \hline \end{tabular}  
* Performance parameters used to test the predicted result.  
\end{table} 
Table 1: Measurable parameters of the E-class gas turbine. Figure 2: Gas-path schematic of an E-class gas turbine. In this paper, the component characteristic maps were adapted according to multiple off-design operation data. As shown in Figure 3, the off-design speed lines in the initial component characteristic map (solid line) are different from the actual map (dotted line). To modify the maps, three characteristic parameters (corrected mass flow rate, pressure ratio, and isentropic efficiency) were calibrated at different speed lines. Taking the compressor as an example, the off-design scaling factors of the corrected mass flow rate, pressure ratio, and isentropic efficiency were introduced and defined as shown in Equations (2)-(4). It can be observed that the scaling factors at each speed line are different, and the variation of the scaling factors with the speed lines is nonlinear. In this paper, a quadratic form (Equation (5)) was applied to describe the nonlinearity of the corrected relative non-dimensional rotational speed (CN) [17; 18]. \[SF_{Comp,WAC}=\frac{WAC_{Comp,OD}^{*}}{WAC_{Comp,OD}} \tag{2}\]  
\[SF_{Comp,PR}=\frac{PR_{Comp,OD}^{*}-1}{PR_{Comp,OD}-1} \tag{3}\]  
\[SF_{Comp,ETA}=\frac{ETA_{Comp,OD}^{*}}{ETA_{Comp,OD}} \tag{4}\]  
\[SF_{x}=1+b_{x}\bigg{(}1-\frac{\overline{n}_{OD}}{\overline{n}_{DP}}\bigg{)}+c _{x}\bigg{(}1-\frac{\overline{n}_{OD}}{\overline{n}_{DP}}\bigg{)}^{2} \tag{5}\]  
In Equations (2)-(4), terms with the superscript \"*\" mean the modified characteristic parameters and terms without \"**\" are the original characteristic parameters. The subscript \"\(Comp\)\" means compressor. _WAC_, _PR_, and _ETA_ represent the corrected mass flowrate, pressure ratio, and isentropic efficiency, respectively. _OD_ denotes the off-design condition. Equation (5) defines those scaled factors as a quadratic form of CN. \(x\) represents one of the characteristic parameters, _WAC_, _PR_, and _ETA_. _DP_ denotes the design point. _b\({}_{x}\)_ and _c\({}_{x}\)_ represent the first-order and second-order coefficients in the correlation function, respectively. \(n\) denotes CN, and \(\left(1-\frac{\overline{n}_{OD}}{\overline{n}_{DP}}\right)\) represents the difference of CN between any OD point and design point. Figure 3 shows the errors between the original component map and the actual map. The error is larger at a lower speed. Thus, the performance adap  
Figure 3: Changes in component characteristic maps in performance adaption. points needs to be conducted on the speed lines.