Determining Steady-State Operation Criteria Using Transient Performance Modelling and Steady-State Diagnostics

Abstract

Data from the steady-state operation of gas turbine engines are used in gas path diagnostic procedures. A method to identify steady-state operation is thus required. This paper initially explains and demonstrates the factors that cause a deviation in engine health when transient data are used for diagnosis and shows that there is a threshold in the slope of time traces, below which the variation in engine health parameters is acceptable. A methodology for deriving a criterion for steady-state operation based on actual flight data is then presented. The slope of the exhaust gas temperature variation with time and the size of its time-series window, from which this slope is determined, are the required parameters that must be specified when applying this criterion. It is found that the values of these parameters must be selected so that a sufficient number of steady-state points are available without compromising the accuracy of the diagnostic procedure.

1. Introduction

Diagnostic methods using aero-thermodynamic data, known as gas path analysis methods, use data obtained from steady-state operation to identify the condition of a gas turbine. Numerous studies obtaining diagnoses using transient operation data have been presented; some very recent ones, for example, are presented in [1,2,3,4,5], with reviews of the existing literature extensively covered in [4,5]. The capacity and robustness of these methods are limited, and their applicability is demonstrated on simulated data. It is noted that discrepancies in the findings of steady-state and transient diagnostics are observed in [5], while the final conclusion of the paper states that “the combined effect of physical faults in a transient state remains an unexplored problem”.

When a robust diagnostic approach is needed, the data that are to be fed to a gas path diagnostic procedure need to have been obtained from steady-state operation. This is a well-recognized fact, as can be concluded from publications originating from the personnel of engine manufacturers [6,7,8]. A question that arises then is: how is it ensured that this requirement is fulfilled at the operating point the data come from?

The importance of identifying the steady-state operation of engineering systems in general has been recognized, and efforts towards developing algorithms that are able to identify such operation have been presented for many years (for example, [9]) using statistical concepts. The features of the problem of determining whether an operating condition can be characterized as steady-state, for the particular case of gas turbines, is also well known, and methods for its solution have been presented in the literature until very recently [10,11,12,13,14,15,16,17,18]. The motivation of exploiting test data for the purpose of engine condition monitoring was the driver of earlier methods [10,12]. Optimizing engine test cell trials was another motive [11], as the ability to characterize the operation reduces the testing time (and thus cost) while leading to better-quality data. Although the operational features of jet engines make them more relevant to the problem at hand, it is also relevant to ground gas turbines [15]. The fact that steady-state operation constitutes the basis for reliable diagnosis using flight data has led the efforts aiming to characterize states as “quasi steady-state”, as reported in [6,7]. The early method introduced in [10] was used as the basis for the very recent (2022) work of [8]. Excellent reviews on the usefulness of gas turbine steady-state detection and methods, developed until recently, are presented in [8,16].

All methods presented in the literature are based on the collection of data, which are processed in a variety of ways to determine whether the operating conditions are steady state or not. The criteria used also rely on the data themselves. In [16], for example, measurement data from operation are statistically processed in order to derive the relevant criteria. In [18], test data are used to identify clusters that are indicative of the mode of operation, with steady state being one of the modes of interest. Although the underlying physical principles are understood regarding the reasons why steady-state diagnostic methods using data from transient operation do not produce accurate information on the condition of the components, no quantification of such effects has been presented; neither has a procedure of a general nature been presented for tackling this problem in gas turbines.

The focus of the method introduced in the present paper is on applications to gas turbine performance monitoring and fault diagnostics. The purpose is to analyze and quantify how steady-state diagnostic methods produce information that is not representative of the actual engine condition when fed with data from non-steady-state operating conditions. Following a brief theoretical reasoning, a quantitative analysis of the effect of the use of transient operation data on a steady-state diagnostic method will be presented. This analysis then provides the basis for proposing a generalized approach to identifying steady-state operating points, which will then be validated through application to a realistic data set from a civil turbofan used in an actual flight mission.

2. Theoretical Background

An operating gas turbine engine is a system that relates quantities that define its operating point (inputs) to measured quantities that describe its operation (outputs). This interrelation depends on the condition of its components and can be expressed as a mathematical function. A diagnostic method using steady-state data for the generation of the health indices of the components of a gas turbine uses, in one way or another, the inversion of the nonlinear relation, $F_{SS}$, between measured quantities $\left(Y\right)$, the engine operating condition $\left(u\right),$ and component health parameters $\left(f\right)$:

$$Y=F_{SS}\left(u,f\right)\to f=F_{SS}^{-1}\left(u,Y\right)$$

(1)

When the engine operates in a transient condition, the equations that formulate the vector function (F) change. The following relationships change:

• The energy balance equation between components on the same shaft, due to the existence of the shaft acceleration or deceleration power;
• The energy equation of the components, due to the presence of heat transfer between the fluid and the component solid walls;
• The continuity and momentum equation of the components, due to the inertia and compressibility of the fluid within the component.

The relative significance in terms of the magnitude of these different factors depends on the particular engine (for example, the component size and type) and the form of transient operation. For example, the inertia of the fluid within blade passages contributes measurably to the response when the transient is “fast”.

In any case, the interrelation between measurements and component health factors is represented by a different vector function, $F_{TR}$:

$$Y^{\prime }=F_{TR}\left(u,f\right)$$

(2)

This implies that if the steady-state function ${(F}_{SS})$ is inverted instead of the transient function ${(F}_{TR})$ for calculating the health coefficients, a set that is different from the actual one will be derived:

$$f^{\prime }=F_{SS}^{-1}\left(u,Y^{\prime }\right)$$

(3)

Deviations in the health indices of the components could thus mistakenly be interpreted as changes in the component condition.

We note here that all the terms that are modifying the steady-state functional interrelation, to make it represent the transient behavior, are time derivatives. In the power balance equation, for example, the derivative of angular velocity is contained in the acceleration or deceleration power term:

$$P_{\mathit{tur}}=P_{\mathit{comp}}+P_{\mathit{load}}+P_{\mathit{acc}}=P_{\mathit{comp}}+P_{\mathit{load}}+\frac{1}{2}I\frac{d\left({\omega }^2\right)}{dt}$$

(4)

Similarly, when considering the heat transfer between metal and gas, the metal temperature is described by the following dynamic equation [19]:

$$\frac{d\left(T_m\right)}{dt}=\frac{h·A}{m·Cp}·({Tt}_g-Tm)$$

(5)

This means that when the values of these derivatives become comparatively small, then the transient behavior will be well approximated by the steady-state relations.

This observation will form the basis of the criteria that will be presented later in the paper, which will then be used for deciding whether an operating condition can be approximated by the steady-state model with sufficient accuracy.

Before continuing, it should be noted that gas turbine and jet engine transient performance modelling capabilities have reached a high level of maturity [19,20]. A particular aspect that is very important when control or monitoring applications are to be supported is the ability to adapt the models to specific engine units. An example of such an approach, taking advantage of modern data-driven technologies, has been presented in [21].

3. Principles Demonstration through Application Example

Application to Typical Transients

We will elucidate this behavior by using the example of a mixed-flow turbofan, with a layout as shown in Figure 1. The analysis is carried out using an engine performance model of this configuration [22]. The model is generated in the PROOSIS v6.4.0 simulation environment [23] using its standard library of engine components, which is based on the industry-accepted performance modelling methodology. The off-design performance of all turbomachinery components is obtained from suitable maps, scaled accordingly during the design calculation. The model is adapted to the known engine performance using the procedure described in [22].

The health factors used to characterize the condition of the individual components, as well as the measurements used for monitoring this engine, are shown in the same figure. It is noted here that the health condition of the engine components is represented by the “health parameters”. Two health parameters per engine component are defined. For a component with entrance at station i along the engine, these parameters are as follows:

$$\mathrm{Flow}\mathrm{factor}:{SW}_i=\left(W_i·\sqrt{{Tt}_i}/{Pt}_i\right)/{\left(W_i·\sqrt{{Tt}_i}/{Pt}_i\right)}_{ref}$$

$$\mathrm{Efficiency}\mathrm{factor}:{SE}_i=n_i/{\left(n_i\right)}_{ref}$$

$W_i$ is the gas mass flow rate at station i, ${Tt}_i$ is the total temperature at station i, ${Pt}_i$ is the total pressure at station i, $n_i$ is the efficiency of the component with entrance at station i, while the subscript ref indicates the reference values, i.e., the values of a healthy engine.

The use of such factors for describing the health condition of engine components has been discussed by Stamatis et al. [24]. The deviation of an engine component health parameter from its nominal value indicates the presence of a fault in the corresponding component. Moreover, the more severe the fault, the greater this deviation is. The deviation of a health parameter f is defined as $\mathsf{\Delta }f\left(\mathrm{\%}\right)=\frac{f-fo}{fo}·100\mathrm{\%}$, where $f$ is the value of the health parameter and $fo$ is its nominal value.

To start with, the transient maneuvers of a fuel flow step change and a fuel ramp-up are simulated. The fuel flow and the corresponding variation in some performance parameters are shown in Figure 2. The well-known behavior of the measured quantities that vary significantly over the steep part of the transient and then asymptotically tend towards the steady-state condition is observed.

A steady-state diagnostic problem is then solved to determine the values of the 8 health parameters of the turbomachinery components of the engine. The measurements used as input are NH, Wf, Pt13, Tt13, Pt31, Tt31, Tt5 and Pt45. A non-linear square diagnostic problem is solved, where these eight measurements are given and the eight health parameters are the unknowns. The obtained values for the health parameters are shown in Figure 3.

The following observations can be made: although the engine components’ condition has not changed, a significant variation in the health parameters over a part of the transient is observed. Moreover, this variation is not physically consistent, as it indicates that the performance of all the components in terms of efficiency and flow capacity increase. It is well known that such behavior cannot occur when the engine condition changes. Engine deterioration or faults always lead to a reduction in the values of the efficiency factors, while the flow capacity of compressors always reduces also. Additionally, the magnitude of deviations reaches values that would mean catastrophic changes, if they were of the right physical significance.

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.

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 Figure 5. 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.

Bearing in mind that, in practice, estimated health factors are always characterized by uncertainty, as a result of the presence of noise in the measured quantities, when the deviation caused by the non-steady operation is well within this uncertainty interval, it could be considered acceptable, as it would have the same effect as measurement uncertainty. This means that when the slope is below a certain value, even though the operating condition is not perfectly steady state, it could be acceptable in terms of uncertainty in the estimated health coefficients. The picture for all health indices is shown in Figure 5. The health index with the largest deviation should be used to establish the interrelation with the slope (i.e., SW25). For the data shown in Figure 6, if we take, for example, a value of 0.25% as the maximum acceptable deviation for steady state, then the largest corresponding slope is 1 Kelvin/min.

A question that arises now is whether the limit defined through this procedure is the same in different types of transients. What we have seen above is that for speed increase, whether it is steep or more gradual, the transient behavior in terms of the equivalent health factors is similar. Speed decrease is another representative type of transient (the entire operation of the engine consists of speed decrease and increase variations).

The equivalent health factors calculated for the ramp down are shown in Figure 7. We observe a pattern similar in form to the one in Figure 3, but the deviations are negative in this case. This is because the time derivatives disturbing the steady-state equation have an opposite sign (we have now deceleration instead of acceleration). We also observe that the asymptotic approach to the steady-state values happens at longer intervals, which we should expect to behave somewhat differently due to the fact that the operating lines move along different paths on the component maps.

If we now observe the interrelation between slopes and the equivalent health factor deviations, the difference observed above is reflected in different lines, as shown in Figure 8. A comparison with Figure 6 shows that, for the same threshold in virtual factor deviation, a smaller slope threshold has to be applied. For example, for virtual factors deviating less than 0.25%, a slope with a maximum of 0.4 K/min has to be enforced.

4. Discussion

The results presented above have been derived from the solution of a square diagnostic problem, because such a problem represents the interrelation between health parameters and measurements with no approximations or assumptions. If a particular aero-thermodynamic technique is employed, it could be used to formulate the procedure presented, using the data of a transient performance model.

Since the deviation from steady state is determined by the magnitude of the physical properties of the engine structure, some trends, such as the influence on the deviations caused by the transient data fed into a steady-state model, can be studied by observing the sensitivity to the magnitude of such properties. For example, the way the magnitude of the thermal inertia of components influences conclusions about the slope requirements is shown in Figure 9, for SW25. It is seen that when the heat capacity is smaller, a threshold in the value of the deviations of the virtual health factors can have slope values larger than those for a larger heat capacity. This implies that the slope criterion for an industrial gas turbine will require a smaller rate of variation in the exhaust gas temperature before the data are exploitable through a steady-state diagnostic model, compared to an aircraft engine.

The numerical examples discussed above show that for the transients considered, it is possible to establish a limit of the rate of variation in the exhaust gas temperature, to use it as a criterion for applying a diagnostic technique. In an aircraft engine operating in the field, a variety of changes in the operating point are encountered, changes that could always be decomposed to speed-up and slow-down parts. Still, what we have studied may not apply to such situations, where the succession of slow-down and speed-up transients occurs before the proceeding one settles. For this reason, being guided by the study presented above, when data from field operation are available, criteria that can be applied in practice can be derived, as discussed in the following section.

5. Steady-State Criterion Derivation from Flight Data

The procedure proposed for establishing the criteria to be used on data collected from an operating engine is based on insights gained from the preceding discussion and the elaboration of data collected from an engine of interest. Data from a typical operating envelope will now be used to assess the different ways of applying a criterion for steady-state characterization.

Data from a flight mission will be used to demonstrate the applicability of the criteria previously established to actual engine operation. The dataset used has been generated from a set of actual measured values on a civil engine, on an aircraft executing a flight among European destinations [25]. The actual data have been scaled due to confidentiality reasons. The engine model supporting the diagnosis has been scaled accordingly.

The flight profile for the mission used is shown in Figure 10. In order to show the existence of non-steady operating conditions even during flight phases that may be considered as steady state, the time variation in the engine low-spool speed is plotted on the same figure.

The deltas of the measured quantities have been evaluated, using the values calculated by a steady-state engine model as a reference at each operating point. The model is run using the low-spool rotational speed as a setting parameter. When the engine operates at steady state, those deltas are close to zero. During unsteady operation, the deltas obtain values that can be quite large. The evolution of ΔTt5 along the mission path is shown in Figure 11, where the evolution of the low-spool speed is also plotted. It is observed that when abrupt changes in the rotational speed occur, denoting transient operation, the deltas obtain also high values.

The deltas time series is now employed to obtain a diagnosis. A diagnosis of whether the engine is in healthy or faulty condition is sought. A diagnostic technique employing probabilistic neural networks (PNNs) [22] is applied on the time series of the measurement deltas. The PNN is a classification method, and the outcome of the diagnosis is whether the engine is healthy or faulty. It is a three-layer feed forward network that allows statistical pattern recognition based on Bayes’ decision rule [26]. The nodes of the first layer represent the available measurement deltas, which are input to the network. The nodes of the hidden layer each represent a set of measurement deltas for a known engine health condition. The set of nodes of this layer defines the set of training patterns of the network. The nodes of the output layer represent the set of possible engine health conditions that may occur. Each considered engine health condition is represented by the deviation of a group of health parameters from their nominal value due to engine fault. The set of training patterns is generated with the aid of the engine performance model. In total, 165 training patterns corresponding to specific engine health conditions that may be met in practice for this type of engine are generated. Given a set of measurement deltas, the PNN assigns a probability to each considered engine health condition; this is the node of the output layer. The engine health condition is defined as the one with the maximum estimated probability. The efficiency of the network and the adequacy of the training set are examined over a number of simulated fault cases—other than the training patterns—that show that the considered architecture can correctly discriminate patterns among the considered engine health conditions. For more information, the reader is referred to [22].

At first, the algorithm is applied to all the data points available. Although the engine is healthy, the algorithm provides a correct diagnosis for only about 75% of data points. We then use a steady-state criterion to filter the data before applying the diagnostic algorithm. The criterion employed is that the absolute value of the slope of variation in the exhaust gas temperature over time should be less than a given threshold. The slope at any given point is considered to be the one derived from a best-fit line applied to a window of points preceding that point. The idea of how this slope is calculated is demonstrated in Figure 12. Two points are used as an example, one for which the operation is transient and one for which steady state has been achieved.

When this criterion is applied, two parameters must be chosen by the user:

• The magnitude of the slope that is used as a threshold, a choice that has been substantiated in our analysis in the preceding sections.
• The size of the window for calculating the slope, since the data collected contain noise.

An example of how the window size affects the magnitude of the slope is shown in Figure 13. It can be claimed that when the slope does not change with time, a larger number of points will produce less uncertainty in the slope value estimated. Additionally, the size of the window may interact with the form of the transient. For example, a window that is very long may indicate that the operation is still transient, even though steady state has been achieved, because the data used extend far back to incorporate part of the transient. In this respect, short windows are desirable for avoiding such overlaps.

Taking these comments into account, the result of the application of the slope criteria for different magnitudes of the parameters involved is examined.

The success rate of the diagnosis applied to points filtered with different slopes as thresholds and with different window sizes for obtaining the slopes is shown in Figure 14. Short time windows produce lower success rates, reflecting the fact that the slope obtained is more sensitive to noise content. The trend reverses for windows larger than a certain size (30 s for the figure), now reflecting the overlap of transient and steady-state regions.

It is interesting to see how many data points are found to obey the steady-state criterion, which thus forms the population in which diagnosis can be affected. The number of points both in absolute numbers and as a percentage is shown in Figure 15. It is observed that when a “tight” slope criterion is applied, the number of points decreases significantly. A trade-off between the requirements for a high success rate and a substantial number of points that can be diagnostically exploited thus has to be decided.

The procedure of selecting steady-state data points can be further improved if the criterion used is modified. A variant that immediately becomes a possibility is not only examining the slope itself, but also the way this slope varies along the time series. If, for example, it is required that this slope is not allowed to change abruptly from point-to-point, the scores presented above using just the slope magnitude are modified, as shown in Figure 16 and Figure 17. The results of these figures have been derived by adding the requirement that the slope magnitude does not vary from point-to-point more than half the maximum allowed slope value. It is observed that scores of 100% are now achievable with certain combinations of the window and slope threshold. At the same time, however, the number of available points significantly decreases. Inspecting these figures shows that, for the example taken here, an optimum combination of a window 30 s wide and a slope threshold of 0.5 to 1 K/min exists.

6. Conclusions

The reasons for health parameter deviations occurring when an aero-thermodynamic diagnostic method, based on steady-state modelling, is fed with data from transient operation have been presented.

The use of a combination of transient performance modelling and a steady-state diagnostic technique in order to derive criteria for steady-state operating points has also been presented. It was shown that the thresholds of the slopes of time traces can be determined, in function of the uncertainty in health indices that can be considered acceptable.

An experimental approach for determining the size of the time window and slope that are appropriate for use as steady-state criteria, on the basis of the actual operation profiles, has been presented. It was shown that simple or composite slope criteria can be used to increase the effectiveness of steady-state point determination.

It was found that the window size and slope threshold values have to be chosen in a way that reaches a compromise between the requirement of highly successful diagnostics and a sufficient number of useful data points when applying the criteria.