Page 108 - Modern Control of DC-Based Power Systems
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72 Modern Control of DC-Based Power Systems
where t window is the observation window during which PRBS is injected
and voltage and current are measured and f sampling is the sampling fre-
quency of v½n and i½n.
2.6.2.1.5 Fitting Routine
The fitting routing consists of a real-time algorithm able to return the
parametric impedance from the nonparametric data set. The fitting con-
sists of two steps: data thinning and least squares fitting. The thinning
enforces equal weighting over the full frequency window of the nonpara-
metric data. The thinning technique is used to obtain a logarithmically
spaced subset of the data points. In order to do so, the lower and upper
frequency boundaries for the data thinning routine have to be specified.
All data points outside this window are disregarded for further processing.
The least squares fitting routine matches the thinned data to a polynomial
function and it is an algorithm based on Levy [49]. The order of numera-
tor n and denominator m must be selected. If they are not known a
priori, the order can be tentatively estimated by increasing n and m until
the fitting result matches the nonparametric impedance. The result of the
fitting routine is the parametric impedance, given by the coefficients of a
polynomial function below.
a 0 1 a 1 s 1 ? 1 a n21 s n21 1 a n s n
Z sðÞ 5 (2.100)
b 0 1 b 1 s 1 ? 1 b m21 s m21 1 b m s m
2.6.2.2 Performance of the WSI Technique and Overcoming Practical
Challenges
The real-time performance of the WSI technique is given by the execution
time of all the routines described in the previous subsection starting from
when the PRBS injection is activated and ending when the parametric
impedance result is available. The execution time of the implemented WSI
technique consists of the chosen time window and the impedance calcula-
tion time. The impedance calculation time t calculation comprises the execution
time of the FFT algorithm, which is of complexity OðNUlogðNÞÞ, and the
fitting routine, which depends on exit conditions of the least squares fitting
algorithm [49]. Therefore, to measure the impedance of the system under
test, the online WSI technique takes t window 1 t calculation .
A major challenge in impedance identification consists of maintaining
the small-signal condition and, at the same time, guaranteeing a measur-
able perturbation within the frequency range of interest. In theory, the
