## 3 Principles Demonstration through Application Example
Comparing the values calculated for the step change to the ones calculated for the ramp, the latter exhibits lower magnitudes. This is consistent with the observation made earlier, which is that the deviations in the equations are due to first derivative terms. Such terms are of a larger magnitude during the steeper fuel input. It is also noted that when the gas path variables, for example the ones shown in Figure 2, approach their final steady-state value, the health factors approach zero deviation from the nominal, which correctly reflects the condition of the engine (healthy). The calculated factors asymptotically approach their reference value. This means that they are within a range of this value when the time derivatives are small enough. In this test case, the factor with the largest deviation all along the transient is the one for high-pressure compressor flow SW25. We proceed now to examine how the magnitude of the time derivatives of the measured quantities varies along the transient. The magnitude of such derivatives is represented by the slope of the time variation of the quantity. We select the exhaust gas temperature Tt5 as the variable to observe. The exhaust gas temperature reflects all the changes happening along the engine gas path and is thus suitable for reflecting whatever deviates from nominal conditions along the gas path, be it a change in the physical condition of the components or a change in the working medium process, for example due to transient phenomena. We also note that this temperature is more appropriate than another global variable. For example, in a constant-rotational-speed industrial gas turbine, the load demand is closely followed. A \"step\" change in load will take place, so that the load demand is satisfied, while the exhaust gas temperature will then settle at a slower pace, as the result of heat transfer effects. This is why quantities, such as load, are deemed less appropriate, despite the fact that they have been used by other investigators (e.g., [16]). An example slope of the exhaust gas temperature corresponding to the variations observed in Figure 2 is shown in Figure 4. The slopes have been calculated by the local best fit of a straight line on a window of successive data points along the time variation of the quantity. Figure 3: Variation in all health factors derived by feeding the measurements to a steady-state diagnostic method during the speed-up transient: (**a**) Fuel “step” change; (**b**) Fuel ramp. The derivatives calculated this way are a good approximation of the actual local slope only when the slope does not change within the width of the window chosen. This approach (instead of some other finite difference approach, for example) is chosen, however, so that it mimics the application to actual measured data. In such data, noise is always present and thus a local best-fit line produces values less sensitive to this noise. This is a compromise against the loss of fidelity when variability in the actual time derivative occurs within the window. More comments on this matter are offered later in the paper, when the method's application to data representing realistic noisy situations is studied. The initial phase of the transient corresponds to the highest slope values, an observation that is in line with the form of time traces in Figure 2. Comparing these pictures to the variation in health indices in Figure 3, we observe that large values of the slope correspond to large values of the obtained indices. The interrelation of the slope with all health factors is shown in \[\text{Flow factor}:\ SW_{i}=\left(W_{i}\cdot\sqrt{Tt_{i}}/Pt_{i}\right)/\left( W_{i}\cdot\sqrt{Tt_{i}}/Pt_{i}\right)_{ref}\]0 This interrelation can be used to establish a criterion for a slope magnitude that should be a limit; this is for the characterization of a point as steady state.