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

System Identification
to satisfy certain constraints or has to exhibit certain characteristic features.
For example, constraints on the low-frequency limit <strong>of</strong> flame transfer functions
can be derived from global conservations laws, as discussed by Polifke
<strong>and</strong> Lawn [100]. According to their analysis, the flame transfer function F u approaches
unity for ω→0. The same behavior holds for transfer functions F φ in
combustion systems with non-stiff <strong>fuel</strong> injection. The unit impulse response,
on the other h<strong>and</strong>, has to reflect the physics <strong>of</strong> the system in showing a realistic
mean time delay <strong>and</strong> time delay distribution <strong>of</strong> the response to the input
signal.
Beside such qualitative evaluation two methods are used in the present work,
which are described in system identification theory (e.g. [73]). In the first
method, the measured 3 input signals <strong>and</strong> the estimated parameters h (i ) are
k
used to determine the predicted response ŷ. The deviation between predicted
<strong>and</strong> measured response y n , the prediction error ǫ (ǫ n = y n − ŷ n ), is then compared
to the variations <strong>of</strong> the measured response by the following relation:
⎛ √ ⎞
∑N
⎜
n=1 (y n− ŷ n ) 2
⎟
Q = 100∗⎝1−
√ ∑N
⎠ [%]. (5.17)
n=1 (y n− ȳ) 2
The numerator <strong>of</strong> the fraction term is the root mean squared prediction error.
The denominator represents the st<strong>and</strong>ard deviation <strong>of</strong> the response measured,
where ȳ denotes the mean value <strong>of</strong> y. Q represents the proportion <strong>of</strong>
the total deviation <strong>of</strong> the measured response that is explained by the model.
A value <strong>of</strong> 100% indicates that the response measured can be completely described
by the input signals <strong>and</strong> the identified model parameters. 100%− Q
can therefore be related to the unexplained deviation <strong>and</strong> can be interpreted
as the proportion <strong>of</strong> the error or white noise in the signal.
A better assessment <strong>of</strong> the quality <strong>of</strong> the model can be achieved in analyzing
the correlation between the prediction error ǫ n <strong>and</strong> the input signals x (i ) , i =
1,2,3
1
C ǫx (i )(k)=
N − O+ 1
N∑
n=O
ǫ n x (i ) , k = 0,...,O; i = 1,2,3. (5.18)
n−k
3 The terms ”measured signal / response” represent in the present context the signals, which are obtained by
the Matlab/Simulink model or the transient CFD simulation.
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