Page 109 - Modern Control of DC-Based Power Systems

Small-Signal Analysis of Cascaded Systems 73

maximum identifiable frequency is defined by the Nyquist frequency, i.e.,
half the switching frequency of the inverter [47,48]. The lower frequency
boundary, instead, is equal to the inverse of the measured time window
t window during which the PRBS is injected. In practice, the upper bound-
ary depends on the amplitude of the injected PRBS signal. In fact, the
classic output filters of power electronic converters attenuate the high fre-
quency excitation, eventually leading to a signal level that is comparable
with the noise floor ð ,2 60dB) and therefore too small to be identified.
As a consequence, the attenuation at high frequencies introduced by the
output filters dictates the minimum amplitude of the injected PRBS sig-
nal. Therefore, to obtain a good identification as close as possible up to
the Nyquist frequency, despite the attenuation of the output filters, and
maintaining at the same time the small-signal condition, a good choice is
to perturb voltage and current with magnitude between 5% and 10% of
their steady-state value. As an alternative solution, a different perturbation
signal may be used; an alternative to white noise is blue noise which
increases the high-frequency content of the PRBS without affecting the
low-frequency content. The implementation of a blue noise filter for
identification purposes in described in [48].
The choice of the time window is another practical challenge worthy
of discussion. As the time window is directly linked to the number N of
complex data points of the nonparametric impedance according to (2.99),
for a fixed sampling rate, a long enough time window should be selected
in order to have enough data points to capture eventual sharp features of
the identified impedance, such as lightly damped resonances. On the
other hand, the time window should be chosen short enough to avoid
increasing impedance calculation times. Moreover, how short the time
window chosen should be is in relation to the characteristics of the system
under test. For the cascade of power electronic converters, it is important
to set the time window short enough in order to catch with fast changing
impedances, e.g., due to load steps or system reconfiguration.
The finite duration of the time window poses also some limits on the
FFT algorithm. The FFT assumes infinite periodicity of the signal to be
transformed. However, such an assumption is not practical because all the
signal acquisitions are limited in time. During the acquisition, the FFT
returns additional spurious frequency components around the existing
harmonic content because of the discontinuities at the edges of the time
window. This well-known problem is called spectral leakage [50].In