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Research Papers: Gas Turbines: Controls, Diagnostics, and Instrumentation

Quantitative Stability Analysis Using Real-Valued Frequency Response Data

[+] Author and Article Information
Martin Schmid

e-mail: schmid@td.mw.tum.de

Thomas Sattelmayer

Lehrstuhl für Thermodynamik Technische
Universität München Garching,
Garching 85748, Germany

1Corresponding author.

Contributed by the Controls, Diagnostics and Instrumentation Committee of ASME for publication in the JOURNAL OF ENGINEERING FOR GAS TURBINES AND POWER. Manuscript received July 8, 2013; final manuscript received August 13, 2013; published online September 20, 2013. Editor: David Wisler.

J. Eng. Gas Turbines Power 135(12), 121601 (Sep 20, 2013) (8 pages) Paper No: GTP-13-1240; doi: 10.1115/1.4025299 History: Received July 08, 2013; Revised August 13, 2013

Models for the analysis of thermoacoustic instabilities are conveniently formulated in the frequency domain. In this case one often faces the difficulty that the response behavior of some elements of the system is only known at real-valued frequencies, although the transfer behavior at complex-valued frequencies is required for the quantification of the growth rates of instabilities. The present paper discusses various methods for extrapolation of frequency response data at real-valued frequencies into the complex plane. Some methods have been used previously in thermoacoustic stability analysis; others are newly proposed. First the pertinent mathematical background is reviewed, then the sensitivity of predicted growth rates on the extrapolation scheme is explored. This is done by applying different methods to a simple thermoacoustic system, i.e., a ducted premixed flame, for which an analytical solution is known. A short analysis determining the region of confidence of the extrapolated transfer function is carried out to link the present study to practical applications. The present study can be seen as a practical guideline for using frequency response data collected for a set of real-valued frequencies in quantitative linear stability analysis.

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Figures

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Fig. 1

Example of a closed network model of a thermoacoustic system

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Fig. 2

Open network system

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Fig. 3

Closed-loop system

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Fig. 5

Gain of reconstructed F∧ obtained from extrapolating R into the λ-direction by extrusion (method 1), Taylor series expansion (method 2), and filter (method 3)

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Fig. 6

Difference in cycle increment predicted by single frequency (subscript s) versus the error in frequency match (both with respect to the analytical solution indicated by subscript 0). Circles and squares represent knowledge of the relative (option 1) and absolute (option 2) time lag, respectively. (a) Γ0 = 0.1316, i.e., unstable system. (b) Γ0 = 0.00697, i.e., marginally stable system.

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Fig. 7

Gain and phase of F in λ-direction (quantified by cycle increment Γ) at ω = π. Analytical reference solution (--), method 1 (---), method 2 (…), and method 3 (-·-).

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Fig. 8

Gain and phase of R. The dots indicate the frequencies at which gain and phase are assumed to be known.

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Fig. 9

(a) Cycle increment and (b) difference in cycle increment (only shown up to ±0.025) versus flame position obtained by the three methods of reconstructing the flame transfer behavior in comparison to the analytical reference solution (--). Method 1 (---), method 2 (…), and method 3 (-·-).

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Fig. 10

Amplitude of the transfer function as a function of real and imaginary part of the frequency (gray-shaded area). Analytical (—) and fitted solution according to method 3 (---) at ω = π and at λ = 0. Discrete frequencies used for fit are shown as ♦.

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Fig. 11

Relative error between reference and fitted filter: 5% isocontour (—) and region of confidence (---). Discrete frequencies used for fit are shown as ♦. The diameter of the region of confidence increases as the amount of measured points used for the fit is increased (light gray to black: n=[1;3;5;7]). The order of the polynomial fit increases according to (n-1)/2+1.

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Fig. 12

Five percent regions of confidence with respect to reference solution (---) and with respect to filter fit using two extra points for interpolation (—, light gray to black: n = [1;3;5;7]). Discrete frequencies used for fit are shown as ♦. The order of the polynomial fit increases according to (n-1)/2+1.

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