Impact of fuel supply impedance and fuel staging on gas turbine ...

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System Identification The second term on the right h<strong>and</strong> side can be estimated as the crosscorrelation vector <strong>of</strong> the input <strong>and</strong> output signals: 1 C x y = N− M+ 1 N∑ n=M x (i ) n y n−k; k = 0,..., M; i = 1,2,3. (5.11) Compared to the basic CFD/SI method, which identifies SISO <strong>and</strong> MIMO models (see [34, 98, 102]), the auto-correlation matrix contains now as well non-diagonal entries as the input signals are correlated. This fact requires a discussion about the choice <strong>of</strong> excitation signals <strong>and</strong> the validation methods to judge the quality <strong>of</strong> the results obtained. Equation (5.9) can be rewritten as follows ˆθ= Γ −1 xx C x y, (5.12) which is the optimal linear least square estimator for the unit impulse response. It is also known as the Wiener-Hopf equation or rather the Wiener- Hopf inversion [40, 52, 73]. After the auto-correlations <strong>and</strong> cross-correlations are determined, the unknown parameters are solved using a method which is based on a least-square root algorithm for sparse linear equations [86]. 5.3 Excitation signals The identification method presented above processes data from simulations with broadb<strong>and</strong> excitation. Several types <strong>of</strong> broadb<strong>and</strong> excitation signals are known, <strong>and</strong> it is worthwhile to investigate how the signal type influences the results <strong>of</strong> the identification. Insufficient excitation strength will yield poor identification results, especially in the presence <strong>of</strong> noise. On the other h<strong>and</strong>, because the identification procedure is limited to linear systems, it has to be assured that maximum signal amplitudes do not generate non-linear behavior. A maximum utilization <strong>of</strong> the amplitude limit over a wide range <strong>of</strong> frequencies is therefore beneficial. This can be judged in analyzing the so-called crest factor or peak-to-average ratio, proposed by Ljung [73]. The crest factor is a property <strong>of</strong> the waveform <strong>and</strong> is equal to the peak amplitude <strong>of</strong> the waveform divided by the mean square <strong>of</strong> 86
5.3 Excitation signals the zero mean signal: C 2 r = max n(x 2 n ) 1 N ∑ N n=1 x2 n (5.13) According to [73] a good signal waveform is one with a small crest factor. A similar definition can be found in Isermann [51]. In the next section, three different excitation signals are shortly explained as they are used in the following. Overlaid multi-frequency signals A broadb<strong>and</strong> excitation signal x ex can be generated by a superposition <strong>of</strong> sine waves (SINE): x ex,n = ∑N F ν=0 u 0 sin(2πf ν (n∆t+ φ ν )), (5.14) where NF denotes the number <strong>of</strong> single frequencies, u 0 the basic amplitude <strong>and</strong> ∆t is the sampling interval <strong>of</strong> the signal. Broadb<strong>and</strong> white noise (BBWN) A broadb<strong>and</strong> white noise signal can be generated from pseudo-r<strong>and</strong>om numbers ”r<strong>and</strong>(n)” distributed uniformly between 0 <strong>and</strong> 1: { 2u0 (r<strong>and</strong>(m)−0.5), n= m T c /∆t , x ex,n = x ex,(n−1) , n≠ m T c /∆t , (5.15) for m = 0,1,..., N ∆t /T c . T c represents the cycle time or rather the cut-<strong>of</strong>f frequency F c = 1/T c , which determines the frequency content <strong>of</strong> the signal. Discrete r<strong>and</strong>om binary signal (DRBS) A discrete r<strong>and</strong>om binary signal can take on only two values±u 0 . The change between the two values takes place at a r<strong>and</strong>om multiple <strong>of</strong> the time step 87
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Technische Universität München In
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und ihre Geduld besonders für die
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Contents 1 Introduction 1 1.1 Techn
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CONTENTS 7.2.2 Impact</stro
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Nomenclature G stretch factor [-] h
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Nomenclature λ air-fuel</s
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1 Introduction 1.1 Technology backg
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1.2 Thermo-acoustic instabilities 1
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1.2 Thermo-acoustic instabilities d
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1.3 Control of the
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1.3 Control of the
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1.4 Intention and
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1.5 Terminology and</strong
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1.5 Terminology and</strong
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2 Numerical analysis of</st
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2.1 Overview of me
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2.1 Overview of me
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2.1 Overview of me
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2.2 Flame dynamics of</stro
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2.3 Identification of</stro
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Page 57 and 58: 3 Acoustics Before analyzing the ac
Page 59 and 60: 3.1 Basic acoustic equations as a f
Page 61 and 62: 3.2 Linear acoustic equations Here,
Page 63 and 64: 3.2 Linear acoustic equations p ′
Page 65 and 66: 3.3 Acoustic network model p ′ (x
Page 67 and 68: 3.3 Acoustic network model matrix e
Page 69 and 70: 3.3 Acoustic network model The <str
Page 71 and 72: 3.3 Acoustic network model where A
Page 73 and 74: 3.3 Acoustic network model coming c
Page 75 and 76: 3.3 Acoustic network model Z i Air
Page 77 and 78: 3.3 Acoustic network model pressure
Page 79 and 80: 3.3 Acoustic network model The eige
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Page 83 and 84: 4 Modeling of turb
Page 85 and 86: 4.1 Theory and num
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Page 89 and 90: 4.2 Theoretical and</strong
Page 95 and 96: 5 System Identification Flame trans
Page 97 and 98: 5.1 Model structures and</s
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Page 103 and 104: 5.4 A posteriori validation <strong
Page 105 and 106: 5.4 A posteriori validation <strong
Page 107 and 108: 5.5 Proof
Page 119 and 120: 6 Numerical simulation MISO model i
Page 121 and 122: 6.1 Practical premixed combustor co
Page 129 and 130: 6.2 Steady-state simulations <stron
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Page 133 and 134: 6.3 Identifiction of</stron
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6.4 Identification of</stro
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7 Acoustic analysis Once the acoust
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7.1 Setup of the a
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7.2 Acoustic network model results
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8 Summary and Conc
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The flame element of</stron
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Bibliography [1] Verband</s
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BIBLIOGRAPHY [18] L. Crocco. Theore
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BIBLIOGRAPHY [36] A. Giauque, L. Se
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BIBLIOGRAPHY [54] J.J. Keller, W. E
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BIBLIOGRAPHY [72] T. Lieuwen <stron
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BIBLIOGRAPHY Number 98-GT-269 in In
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BIBLIOGRAPHY [112] T. Sattelmayer.
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BIBLIOGRAPHY [130] A. Widenhorn. pr
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A.2 Estimation of
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A.3 Main flow parameters an
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