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Technical Brief

Early Warning Signs of Imminent Thermoacoustic Oscillations Through Critical Slowing Down PUBLIC ACCESS

[+] Author and Article Information
Qiang An

Institute for Aerospace Studies,
University of Toronto,
Toronto, ON M3H 5T6, Canada

Adam M. Steinberg

Institute for Aerospace Studies,
University of Toronto,
Toronto, ON M3H 5T6, Canada
e-mail: adam.steinberg@gatech.edu

Sandeep Jella, Gilles Bourque, Marc Füri

Siemens Canada,
Dorval, QC H9P 1A5, Canada

1Corresponding author.

2Present address: School of Aerospace Engineering, Georgia Institute of Technology, Atlanta, GA 30313.

Manuscript received June 29, 2018; final manuscript received October 23, 2018; published online November 28, 2018. Assoc. Editor: Riccardo Da Soghe.

J. Eng. Gas Turbines Power 141(5), 054501 (Nov 28, 2018) (4 pages) Paper No: GTP-18-1405; doi: 10.1115/1.4041963 History: Received June 29, 2018; Revised October 23, 2018

Critical slowing down (CSD) is a phenomenon that is common to many complicated dynamical systems as they approach critical transitions/bifurcations. We demonstrate that pressure signals measured during the onset of thermoacoustic instabilities in a gas turbine engine test exhibit evidence of CSD well before the oscillation amplitude increases. CSD was detected through both the variance and the lag-1 auto-regressive coefficient in a rolling window of the pressure signal. Increasing trends in both metrics were quantified using Kendall's τ, and the robustness and statistical significance of the observed increases were confirmed. Changes in the CSD metrics could be detected several seconds prior to changes in the oscillation amplitude. Hence, real-time calculation of these metrics holds promise as early warning signals of impending thermoacoustic instabilities.

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Considerable progress has been made regarding theoretical, experimental, and computational analysis of thermoacoustic instabilities in gas turbine engines that can help create effective engineering solutions to mitigate high-amplitude oscillations [15]. Nevertheless, unexpected oscillations may arise in practice due to, e.g., off-design operation, off-spec fuel compositions, changes to hardware, and unanticipated perturbations. Early warning of a system moving toward thermoacoustic instability could potentially enable various preventative or mitigative actions.

Several analytical methods have been proposed that may be able to anticipate thermoacoustic instabilities. Lieuwen [6] assessed the dynamic stability margin of a laboratory-scale combustor using the acoustic damping coefficient, which was correlated with the decay rate of the envelope of the pressure signal autocorrelation. Using singular value decomposition to examine transient energy growth of small perturbations, Nagaraja et al. [7] identified regions of potential combustion instability in a linearly stable but non-normal system. Gotoda et al. [8] captured the transition from regular large-amplitude thermoacoustic oscillations in a gas turbine model combustor to irregular low-amplitude pressure fluctuations using translation error as the equivalence ratio was gradually reduced. Rouwenhorst et al. [9] built a low-order model based on state-space representation to monitor the azimuthal acoustic modes in an annular combustor. Sarkar et al. [10] attempted to detect the onset of thermoacoustics based on the change in the D-Markov entropy rates that measured the state complexity.

From a system dynamics point of view, transition of a combustion system from a state exhibiting low-amplitude aperiodic noise to a state of high-amplitude thermoacoustic oscillations occurs when the system switches from a stable attractor to a periodic limit cycle attractor in the phase space. Such a transition could often be realized through a subcritical Hopf bifurcation [11]. In a more general sense, many physical systems exhibit critical transitions at the bifurcation point. Because the post-transition state often is undesirable (e.g., associated with species extinction and stock-market crashes), several generic early warning signals have been developed to anticipate impending bifurcations.

In particular, many systems exhibit a reduced recovery rate to perturbations as a bifurcation is approached, which is referred to as critical slowing down (CSD). Different metrics of CSD have been developed that show different sensitivity in different systems. These include direct measures of the recovery rate, increased autocorrelation, or increased variance, e.g., Refs. [1214].

Gopalakrishnan et al. [15] demonstrated that early warning signals of subcritical Hopf bifurcations can be detected by CSD in a prototypical thermoacoustic system. Their experiment involved flowing air through an electrically heated wire mesh installed in a square duct, with forcing from loud speakers. This configuration allowed systematic variation of the system control parameter (viz., the heater power) to explore bifurcations in the thermoacoustic response. Specifically, they incrementally varied the heater power every 20 s and evaluated critical slowing down via the lag-1 autocorrelation, variance, and conditional heteroskedasticity. The variance was found to provide the most robust early warning sign of impending thermoacoustic oscillations.

Compared to the previously mentioned prediction tools, CSD is a relatively generic and straightforward calculation that could potentially be implemented for real-time engine monitoring. However, the practical applicability of CSD in real combustors has not yet been established. Indeed, most previous investigations on anticipating thermoacoustics were performed on laboratory-scale combustors. Therefore, the objective of this work is to evaluate whether CSD can be used as an early warning signal of thermoacoustic oscillations in a real gas turbine engine. To do so, we utilize pressure signals acquired during spontaneous growth of thermoacoustic oscillations in the initial full-engine testing of a gas turbine combustor at nominally fixed conditions. The oscillations were spontaneous in the sense that they arose unexpectedly during the nominally stationary operation of the engine. It is shown that both the rolling window lag-1 auto-regressive coefficient and the rolling window variance provide robust and early warning of the impending oscillations.

The data used in this paper are pressure fluctuation (p) signals acquired in a combustor during initial testing of a gas turbine engine. While operating at nominally steady-state conditions exhibiting low-amplitude aperiodic noise, the system could undergo a spontaneous transition at an apparently random time, to a state with thermoacoustic oscillations at a particular frequency f. In the present paper, we will focus on the data collected from two engine test runs, designated as case A and case B, respectively. Each case contains two pressure signals recorded by the noise sensors, which only have slight quantitative differences. Figure 1 shows a p time sequence from case A; the raw pressure data from case B were qualitatively similar and are not shown here. The signals were acquired at a frequency of fa ≈ 9 × 102f, but down-sampled by a factor of 10 for subsequent processing. The x-axis of Fig. 1 shows a normalized time T = (t − te) ⋅ f, where te is the end time in the measurement sequence, representing the number of thermoacoustic periods prior to the end of measurement. The y-axis is normalized by the maximum pressure fluctuation amplitude. Sudden growth of the pressure amplitude to a limit cycle state can be clearly seen in the enlarged view at the bottom of the figure.

Methods of extracting metrics of CSD from time signals are well established [16]. The basic steps involve detrending the data to remove any variations in the mean, followed by calculation of the CSD metric in a rolling window containing times prior to the instant being considered. It is also useful to implement metrics to quantify the temporal trends in the CSD metrics, the sensitivity to the calculation parameters, and the statistical significance of the results.

Here, detrending is performed using the Nadaraya–Watson (N–W) estimator [17,18]. The N–W estimator estimates the conditional expectation of the variable (in this case, pressure) relative to another variable (in this case, time) based on the their joint probability distribution function using a Gaussian kernel with a filter bandwidth of tf. Detrending is performed by subtracting the original signal from the N–W estimator for the expectation. The remaining calculations are performed on the resultant residual. Moving forward, p represents the residual after detrending.

The CSD metrics are calculated in a rolling window spanning from the time being considered backward over a user-defined window length, tw. We evaluated two of the most widely used metrics of CSD, namely the variance of p in the rolling window (σ) and the coefficient (α) from a lag-1 auto-regression model for p. To determine α, the residual p is fitted by Display Formula

(1)pt+1=αtpt+θηt+1

where t is the time index and θηt+1 is a vector of random disturbances with standard deviation θ. The auto-regressive coefficient α should be zero for purely white noise. CSD generally is associated with an increasing trend in α and σ with t.

We use the Kendall rank correlation coefficient, or Kendall's τ, as a rigorous mathematical measure to identify increasing trends in α and σ. Kendall's τ is based on the nonparametric Mann–Kendall test, given by Display Formula

(2)τy=i=1n1j=i+1nsign(yjyi)n(n1)/2

where y is either α or σ. The interval [1, n] was taken to be the same rolling window used for the calculation of α and σ. Values of τy greater (less) than zero indicate increasing (decreasing) trends, whereas τy ≈ 0 implies no serial correlation.

Given a value of τy > 0, it is necessary to assess the impact of the user-selected parameters (tf and tw) in the calculation and test the statistical significance of the finding. The former is achieved by repeating the calculation over a range of tf and tw. The latter is achieved by starting with the null hypothesis that a calculated value of τy=τy*>0 is obtained by random chance. We then use bootstrapping with replacement to generate 1000 artificial signals from the measured data, and calculate τy from each of these signals. Statistical significance of τy* is implied by low probability that the τy values calculated from the artificial signals (τy,k) are greater than τy*, e.g., P(τy,k>τy*)0.05. All of the above-mentioned processes were coded and realized in matlab.

Following the procedure described in Sec. 2, the CSD metrics were calculated on the p signal of case A; these are shown in Fig. 2. Only data prior to the visually apparent growth of the oscillation amplitude are shown, corresponding to the time period T ≈ –320 to –33. It is noted that the calculation of α and σ at any time t* only requires information on the time signal at tt*; the shown monitoring period is selected to emphasize the behavior of the CSD metrics prior to the growth in oscillation amplitude. In this calculation, the detrending filter window, tf, was set to 14T (10% of the monitoring duration) based on visual inspection of the resultant data. As the raw p signal at the quiet state was almost statistically stationary, detrending did not significantly affect the results. The rolling window length, tw, was set as 1.4 × 102T (50% of the monitoring duration), as illustrated in Fig. 2. The impact of these selections is assessed below.

The calculated α and σ series show obvious monotonically increasing trends, demonstrated by relatively high values of τα = 0.88 and τσ = 0.82. Hence, while qualitative inspection of the p time series does not indicate any changes prior to the onset of the thermoacoustic oscillations, the signal does exhibit quantifiable CSD. Figure 3 shows the α and σ calculations for a p signal of case B, demonstrating similar behavior. The slightly higher values of τα and τσ for case B were caused by the differences in the raw pressure signals of the two cases. The calculation parameters (e.g., tf and tw in terms of percentage of the monitoring duration) were set to typical values and kept the same between cases to ensure a fair assessment. However, these could be tuned for either case A or case B to obtain higher τ values.

In order to check the sensitivity of the above calculation to the settings of the detrending filter width tf and the rolling window size tw, the same calculation was repeated with various combinations of both parameters. Since the results obtained were qualitatively similar for both cases, here we will use case A as an example; the corresponding τα and τσ values are shown as two-dimensional contour plots in Fig. 4. Both CSD metrics show relatively low sensitivity to the calculation parameters over a reasonable range. Indeed, essentially no sensitivity was found to tf. τα was greater than about 0.9 over most tested values of tw and never fell below 0.6. τσ was consistently lower than τα, but also reasonably robust to changes in the calculation parameters over a more restricted range.

Figure 5 shows the histograms of τα and τσ calculated from the 1000 surrogates that were artificially generated based on the original p signal for case A. According to the significance test, namely P(τα,k>τα*)=0 and P(τσ,k>τσ*)=0.005, both CSD metrics obtained from our previous calculation are statistically significant. To pass the significant test, the calculated τy* has to be greater than about 0.7 according to Fig. 5, which is guaranteed as long as tw is set to a reasonable value (i.e., 40% of monitoring duration), again suggesting the robustness of the detected CSD.

The above results suggest the promise of CSD as an early warning sign of thermoacoustic instabilities in gas turbine combustors. However, for this to be effective, the warning duration would need to be sufficient to perform preventative action or a controlled engine shut-down. Figure 6 shows the calculated α series for case B, based on three durations ranging from approximately 220 to 660 T before the transition to thermoacoustics. The result for the shortest duration (the same as the top subfigure in Fig. 3) is shown here again as a reference. To facilitate the comparison, a fixed tw was adopted for all three durations. Furthermore, τα was also calculated using a rolling window of 0.5tw. The sharp increasing trend of α started around −100 T, regardless of the selected initial point of the monitoring process. τα increased from around 0 at T = −100 and remained nearly 1 after T = −50. These times correspond to a few seconds of real time, which is several orders of magnitude longer than the characteristic time of the engine control system.

In this work, an algorithm based on the system dynamics theory of critical slowing down was developed to predict imminent thermoacoustic instabilities. To the best knowledge of authors, this is the first time this tool is applied to a real-world gas turbine engine. Results showed that large-amplitude combustion instabilities could be robustly detected using the CSD metrics (viz., rolling window lag-1 auto-regressive coefficient and variance), while the combustor was still quiet and no indicative information could be easily inferred from the pressure signals per se. Combined with other tools and techniques such as machine learning, this method could potentially be further developed into an online detector that is integrated into the engine control system to adjust engine operating conditions, once impending thermoacoustic instabilities are detected.

Future work should test the generality of the observations made here by utilizing data from more engine tests in a variety of configurations and conditions. Furthermore, the robustness of the CSD metrics (i.e., τ) must be tested as a function of the calculation parameters in these different tests. Different metrics or simultaneous consideration of multiple metrics may be required to achieve the desired reliability.

This work was supported by Mitacs through the Mitacs Accelerate program and by Siemens Canada Ltd. The authors would like to thank Mr. Sylvain Belanger and Mr. Paul Cloutier-Mathura for their assistance in completing this work.

McManus, K. , Poinsot, T. , and Candel, S. , 1993, “ A Review of Active Control of Combustion Instabilities,” Prog. Energy Combust. Sci., 19(1), pp. 1–29. [CrossRef]
Huang, Y. , and Yang, V. , 2009, “ Dynamics and Stability of Lean-Premixed Swirl-Stabilized Combustion,” Prog. Energy Combust. Sci., 35(4), pp. 293–364. [CrossRef]
Candel, S. , Durox, D. , Schuller, T. , Bourgouin, J.-F. , and Moeck, J. P. , 2014, “ Dynamics of Swirling Flames,” Annu. Rev. Fluid Mech., 46(1), pp. 147–173. [CrossRef]
O'Connor, J. , Acharya, V. , and Lieuwen, T. , 2015, “ Transverse Combustion Instabilities: Acoustic, Fluid Mechanic, and Flame Processes,” Prog. Energy Combust. Sci., 49, pp. 1–39. [CrossRef]
Poinsot, T. , 2017, “ Prediction and Control of Combustion Instabilities in Real Engines,” Proc. Combust. Inst., 36(1), pp. 1–28. [CrossRef]
Lieuwen, T. , 2005, “ Online Combustor Stability Margin Assessment Using Dynamic Pressure Data,” ASME J. Eng. Gas Turbines Power, 127(3), p. 478. [CrossRef]
Nagaraja, S. , Kedia, K. , and Sujith, R. , 2009, “ Characterizing Energy Growth During Combustion Instabilities: Singularvalues or Eigenvalues?,” Proc. Combust. Inst., 32(2), pp. 2933–2940. [CrossRef]
Gotoda, H. , Shinoda, Y. , Kobayashi, M. , Okuno, Y. , and Tachibana, S. , 2014, “ Detection and Control of Combustion Instability Based on the Concept of Dynamical System Theory,” Phys. Rev. E, 89(2), p. 022910. [CrossRef]
Rouwenhorst, D. , Hermann, J. , and Polifke, W. , 2016, “ Online Monitoring of Thermoacoustic Eigenmodes in Annular Combustion Systems Based on a State-Space Model,” ASME J. Eng. Gas Turbines Power, 139(2), p. 021502. [CrossRef]
Sarkar, S. , Chakravarthy, S. R. , Ramanan, V. , and Ray, A. , 2016, “ Dynamic Data-Driven Prediction of Instability in a Swirl-Stabilized Combustor,” Int. J. Spray Combust., 8(4), pp. 235–253. [CrossRef]
Kabiraj, L. , and Sujith, R. I. , 2012, “ Nonlinear Self-Excited Thermoacoustic Oscillations: Intermittency and Flame Blowout,” J. Fluid Mech., 713, pp. 376–397. [CrossRef]
Scheffer, M. , Bascompte, J. , Brock, W. A. , Brovkin, V. , Carpenter, S. R. , Dakos, V. , Held, H. , van Nes, E. H. , Rietkerk, M. , and Sugihara, G. , 2009, “ Early-Warning Signals for Critical Transitions,” Nature, 461(7260), pp. 53–59. [CrossRef] [PubMed]
Dai, L. , Korolev, K. S. , and Gore, J. , 2013, “ Slower Recovery in Space before Collapse of Connected Populations,” Nature, 496(7445), pp. 355–358. [CrossRef] [PubMed]
Guttal, V. , Raghavendra, S. , Goel, N. , and Hoarau, Q. , 2016, “ Lack of Critical Slowing Down Suggests That Financial Meltdowns Are Not Critical Transitions, Yet Rising Variability Could Signal Systemic Risk,” PLoS One, 11(1), p. e0144198. [CrossRef] [PubMed]
Gopalakrishnan, E. A. , Sharma, Y. , John, T. , Dutta, P. S. , and Sujith, R. I. , 2016, “ Early Warning Signals for Critical Transitions in a Thermoacoustic System,” Sci. Rep., 6(1), p. 35310. [CrossRef] [PubMed]
Dakos, V. , Carpenter, S. R. , Brock, W. A. , Ellison, A. M. , Guttal, V. , Ives, A. R. , Kefi, S. , Livina, V. , Seekell, D. A. , van Nes, E. H. , and Scheffer, M. , 2012, “ Methods for Detecting Early Warnings of Critical Transitions in Time Series Illustrated Using Simulated Ecological Data,” PLoS One, 7(7), p. e41010. [CrossRef] [PubMed]
Scott, D. W. , Nadaraya, E. A. , and Kotz, S. , 1990, “ Nonparametric Estimation of Probability Densities and Regression Curves,” J. Am. Stat. Assoc., 85(410), p. 598. [CrossRef]
Bierens, H. J. , 1994, Topics in Advanced Econometrics, Cambridge University Press, Cambridge, UK.
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References

McManus, K. , Poinsot, T. , and Candel, S. , 1993, “ A Review of Active Control of Combustion Instabilities,” Prog. Energy Combust. Sci., 19(1), pp. 1–29. [CrossRef]
Huang, Y. , and Yang, V. , 2009, “ Dynamics and Stability of Lean-Premixed Swirl-Stabilized Combustion,” Prog. Energy Combust. Sci., 35(4), pp. 293–364. [CrossRef]
Candel, S. , Durox, D. , Schuller, T. , Bourgouin, J.-F. , and Moeck, J. P. , 2014, “ Dynamics of Swirling Flames,” Annu. Rev. Fluid Mech., 46(1), pp. 147–173. [CrossRef]
O'Connor, J. , Acharya, V. , and Lieuwen, T. , 2015, “ Transverse Combustion Instabilities: Acoustic, Fluid Mechanic, and Flame Processes,” Prog. Energy Combust. Sci., 49, pp. 1–39. [CrossRef]
Poinsot, T. , 2017, “ Prediction and Control of Combustion Instabilities in Real Engines,” Proc. Combust. Inst., 36(1), pp. 1–28. [CrossRef]
Lieuwen, T. , 2005, “ Online Combustor Stability Margin Assessment Using Dynamic Pressure Data,” ASME J. Eng. Gas Turbines Power, 127(3), p. 478. [CrossRef]
Nagaraja, S. , Kedia, K. , and Sujith, R. , 2009, “ Characterizing Energy Growth During Combustion Instabilities: Singularvalues or Eigenvalues?,” Proc. Combust. Inst., 32(2), pp. 2933–2940. [CrossRef]
Gotoda, H. , Shinoda, Y. , Kobayashi, M. , Okuno, Y. , and Tachibana, S. , 2014, “ Detection and Control of Combustion Instability Based on the Concept of Dynamical System Theory,” Phys. Rev. E, 89(2), p. 022910. [CrossRef]
Rouwenhorst, D. , Hermann, J. , and Polifke, W. , 2016, “ Online Monitoring of Thermoacoustic Eigenmodes in Annular Combustion Systems Based on a State-Space Model,” ASME J. Eng. Gas Turbines Power, 139(2), p. 021502. [CrossRef]
Sarkar, S. , Chakravarthy, S. R. , Ramanan, V. , and Ray, A. , 2016, “ Dynamic Data-Driven Prediction of Instability in a Swirl-Stabilized Combustor,” Int. J. Spray Combust., 8(4), pp. 235–253. [CrossRef]
Kabiraj, L. , and Sujith, R. I. , 2012, “ Nonlinear Self-Excited Thermoacoustic Oscillations: Intermittency and Flame Blowout,” J. Fluid Mech., 713, pp. 376–397. [CrossRef]
Scheffer, M. , Bascompte, J. , Brock, W. A. , Brovkin, V. , Carpenter, S. R. , Dakos, V. , Held, H. , van Nes, E. H. , Rietkerk, M. , and Sugihara, G. , 2009, “ Early-Warning Signals for Critical Transitions,” Nature, 461(7260), pp. 53–59. [CrossRef] [PubMed]
Dai, L. , Korolev, K. S. , and Gore, J. , 2013, “ Slower Recovery in Space before Collapse of Connected Populations,” Nature, 496(7445), pp. 355–358. [CrossRef] [PubMed]
Guttal, V. , Raghavendra, S. , Goel, N. , and Hoarau, Q. , 2016, “ Lack of Critical Slowing Down Suggests That Financial Meltdowns Are Not Critical Transitions, Yet Rising Variability Could Signal Systemic Risk,” PLoS One, 11(1), p. e0144198. [CrossRef] [PubMed]
Gopalakrishnan, E. A. , Sharma, Y. , John, T. , Dutta, P. S. , and Sujith, R. I. , 2016, “ Early Warning Signals for Critical Transitions in a Thermoacoustic System,” Sci. Rep., 6(1), p. 35310. [CrossRef] [PubMed]
Dakos, V. , Carpenter, S. R. , Brock, W. A. , Ellison, A. M. , Guttal, V. , Ives, A. R. , Kefi, S. , Livina, V. , Seekell, D. A. , van Nes, E. H. , and Scheffer, M. , 2012, “ Methods for Detecting Early Warnings of Critical Transitions in Time Series Illustrated Using Simulated Ecological Data,” PLoS One, 7(7), p. e41010. [CrossRef] [PubMed]
Scott, D. W. , Nadaraya, E. A. , and Kotz, S. , 1990, “ Nonparametric Estimation of Probability Densities and Regression Curves,” J. Am. Stat. Assoc., 85(410), p. 598. [CrossRef]
Bierens, H. J. , 1994, Topics in Advanced Econometrics, Cambridge University Press, Cambridge, UK.

Figures

Grahic Jump Location
Fig. 1

A typical pressure signal (recorded by one noise sensor in case A) acquired in the engine test. The bottom subfigure shows an enlarged view of time period around the growth of the oscillations.

Grahic Jump Location
Fig. 2

Results from CSD calculation performed on a p′ signal of case A (the same one shown in Fig. 1). Calculation was only implemented on the time period T ≈ −320 to −33, prior to the growth of oscillation amplitude. Top: pressure signal after detrending. Middle: rolling window α series with the evaluated τα. Bottom: rolling window σ series with the evaluated τσ.

Grahic Jump Location
Fig. 3

Results from CSD calculation performed on a p′ signal of case B

Grahic Jump Location
Fig. 4

Sensitivity of τα and τσ to the settings of Gaussian filter width (tf) and rolling window size (tw) for the calculation shown in Fig. 2. The star symbols represent τα* and τσ* shown in Fig. 2 that were obtained using that particular combination of tf and tw.

Grahic Jump Location
Fig. 5

Histograms of τ for bootstrapping surrogates generated based on the original p′ signal used in Fig. 2. The dashed lines show the originally evaluated τα* and τσ*, respectively.

Grahic Jump Location
Fig. 6

α results calculated for case B using different durations of the pressure signal before the transition to thermoacoustic oscillations, along with the evaluated rolling window τα

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