IEEE Electrification Magazine - December 2017 - 66

Stringent
assessment
techniques are
required to ensure
the stability of the
electrical network
for the MEA.

loads. Thus, it can be safely argued
that, in practice, nominal system mod-
els are bound to contain uncertainties.
From another perspective, even
though a nominal model is deemed to
be accurate, it may not truly represent
the actual system, which is generally
subject to various operating condition
uncertainties. For instance, in aero-
space applications, PE-based systems
may be exposed to temperatures
ranging from -40 °C to 125 °C (Solleci-
to and Swann 1960). These large varia-
tions in temperature may have
considerable effect on the properties of system compo-
nents. Aging is another factor that brings uncertainty to
the system elements over time. Although an EPS is
assessed as stable based on fixed parameters and condi-
tions, it is questionable whether it continues to be stable in
the face of all of the aforementioned possible types of
uncertainties.
Even though exact values of system components, sys-
tem loads, or operating conditions may not be known accu-
rately, their range of variation can generally be estimated to
good accuracy. For instance, the tolerance of most compo-
nents can be obtained from data sheets. The variation of
resistances can be computed from the range of change in
operating temperatures. Uncertainty sets of power supply
and filter impedances may be obtained based on possible
make and type. Given that uncertainties seem to be inher-
ent in EPS, it may be more natural to work around uncer-
tain system models. In contrast with nominal models,
uncertain models define both the nominal values and the
possible range of variation of their parameters. The uncer-
tain model is thus closer to the physical system. While clas-
sical methods are applied for stability analysis of nominal
system models, a robust approach is needed for the stabili-
ty assessment of uncertain system models. The structural
singular value (SSV)-based n approach is a robust stability
method that incorporates all sources of uncertainties with-
in the system (Doyle 1982, Doyle et al. 1992, Green and
Limebeer 2012, Skogestad and Postlethwaite 2005).

3.0
µ Bounds

µ Bounds

3.0
2.0
1.0
0.0
715

µ
µ
725
Frequency (Hz)
(a)

735

µ
µ

2.5
720.00
720.15
Frequency (Hz)
(b)

Figure 7. A single uncertain parameter system: (a) a n chart to
predict critical Pin ; (b) the zoomed-in area near the peak of the n chart.

66

I E E E E l e c t r i f i cati o n M a gaz ine / DECEMBER 2017

It can be argued that uncertainties
can be incorporated when using clas-
sical methods. However, applying
single-input,  single-output methods
to multiple-input, multiple-output
systems may not produce reliable
results, as reported in a number of
studies (Kuhn et al. 2007, Young
et al. 1991).
The n approach is a deterministic
method that can provide a direct mea-
sure of stability robustness to a system
with respect to its uncertain elements.
The robust stability measure n should
be less than 1 for a system to be robustly stable (Skogestad
and Postlethwaite 2005, Zhou et al. 1996). The n method is
founded on the concept of the uncertain system model.
Hence, by working directly on an uncertain model, n analy-
sis eliminates the burden of performing exhaustive parame-
ter iterations and system linearization (Sumsurooah et al.
2016, Sudhoff et al. 2000). The n approach has proven to pro-
duce reliable results in the robust stability analysis of power
systems subject to multiple simultaneous uncertainties
(Doyle 1982, Skogestad and Postlethwaite 2005, Young et al.
1991, Zhou et al. 1996, Ferreres 1999).
It is therefore evident that there is a need to ensure
that an EPS is not only stable but robustly stable, i.e., it
must remain stable in the face of all system uncertainties.
This is especially important for safety-critical applications.

CPL
Robust stability domains can be viewed as subsets of the
much wider stability domains in the multidimensional
parametric space. To illustrate the concept of stability
robustness, the n tool is employed to identify the robust
stability domains of the representative EPS connected to
an ideal CPL, as shown in Figure 6, when it is subject to sin-
gle and multiple parametric uncertainties. In this section,
the n results, which are generated in the frequency
domain, have been translated to the more perceivable
uncertain parameters domain to better illustrate the con-
cept of robust stability domains.
The first study in this section evaluates the robust sta-
bility of the analyzed system when it is exposed to a sin-
gle parametric uncertainty. The input power Pin can vary
within 10.4 W ! 33% of its nominal value. The values of
the line resistance R in, input filter capacitance C in, and
input filter inductance L in are kept fixed at their nominal
values of 160 mX, 95 nF and 511.8 nH, respectively.
n analysis of the uncertain system yields the n chart in
Figure 7, from which it can be seen that the peak values
of the lower bound n- and the upper bound nr are equal
to 3.02 (S. Sumsurooah 2017). The critical destabilizing
frequency is 720 Hz, which corresponds to the resonant
frequency of the inductor-capacitor (LC) filter. From the n
value, the critical destabilizing value of the input power



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