IEEE Power & Energy Magazine - May/June 2022 - 68

a flexible ac transmission system device. The application of
reduced-order model identification normally uses the blackbox
model identification approach because the internal model
structure is usually unknown.
For any control design problem, a plant model describing
the input-output relationship is a necessity. Usually, a
reduced-order plant model is desired. How do we find the
plant model? The measurement-based approach is to perturb
the original system's input and record the output data. From
either the time-domain data or frequency-domain data, the
input-output plant model can be found.
For example, in a power system stabilizer design, the
plant has the reference order of the voltage regulator as an
input and the generator speed as the output. The input can be
perturbed with an impulse signal, and the output response
will be recorded. Subspace methods, e.g., the eigensystem
realization algorithm, may be used to process the output data
and lead to a reduced-order plant model. Based on the plant
model, a controller modulating the voltage regulator's reference
with the generator speed as the input can be designed
and tested for the closed-loop system performance.
Subspace methods such as the eigensystem realization
algorithm can be traced back to the seminal state-space
model realization theory established by Ho and Kalman
in the 1960s. The core message of the theory is that timedomain
dynamic response data can be stacked properly to
form a large-dimension Hankel data matrix, and this Hankel
data matrix is associated with the state-space model's system
matrices. Factorizing the Hankel matrix via singular value
decomposition leads to the system matrices. From there, a
state-space model that matches the input-output relationship
can be recovered. Furthermore, with the system matrices
known, the eigenvalues of the system can also be found.
Subspace methods have been used in other applications, e.g.,
PMU-data-based oscillation mode identification. They may
be viewed as inference methods that relate data to a model.
These techniques also have the capability of differentiating
noise versus meaningful information, a feature of unsupervised
clustering learning.
Application 4: Admittance Model
Identification for SSR Screening
Stability analysis via frequency-domain models has a history
dating back to the 1970s in both the power electronics
and power systems communities. In the power electronics
community, the initial use of impedance models for dc circuit
stability analysis started in 1976. In the power systems
community, dq admittance-based SSR stability analysis
also started in the 1970s, after the Mohave power plant SSR
events in Nevada. For these events, a synchronous generator
radially connected to a series-compensated line experienced
oscillations in its torque shaft, causing shaft damage. The
torsional oscillations were triggered by the electric network
resonance due to the interaction of the series capacitor and
the line inductance. When the frequency of the electric net68
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work resonance is complementary to the frequency of a torsional
model, i.e., the sum of the two frequencies is 60 Hz,
torsional oscillations may become severe.
Compared to dc circuit analysis dealing with a singleinput
and single-output system, stability analysis in power
systems usually deals with three-phase systems. Modeling a
three-phase system in a rotating dq-frame can greatly simplify
the resulting model. Indeed, one of the most influential
modeling technologies of power systems is Park's transformation,
which converts variables in an ABC-frame to those
in a rotor dq-frame. As a result, a synchronous generator
model is expressed from the perspective of the rotating rotor
frame. Besides generators, other components of power systems
may also be expressed in a dq-frame for simplicity.
Thus, dq-frame models are preferred in stability analysis.
For SSR analysis, a circuit of a generator with a seriescompensated
interconnection can be interpreted as a twoinput
and two-output feedback system. The forward unit is the
line's admittance; the feedback unit is the generator's impedance.
A stability analysis can then be carried out by examining
the feedback system via well-established multi-input and
multioutput frequency-domain system analysis theories.
To obtain the generator and network impedance from
computer simulations, frequency scans have since been
popularly used in SSR studies. Recently, frequency scans
have been used in wind farm SSR screening in electromagnetic
transient simulation software environments by the grid
operating industry. In the power electronics field, obtaining
dq impedance/admittance frequency-domain measurement
through hardware setup, perturbation signal injection, and
measurement processing has been a research topic.
Frequency scans lead to frequency-domain measurement.
A benefit of frequency-domain measurement is its use
for stability analysis. Either open-loop system Bode plots or
Nyquist plots can be plotted, and stability predictions can
be made. However, those diagrams have disadvantages compared
to closed-loop system eigenvalues. They may not lead
to accurate stability prediction, or they may not be straightforward
for interpretation.
An eigenvalue, in the form of a complex number, gives a
direct and accurate prediction of the stability of a dynamic
system. The real part of an eigenvalue must be less than zero
for a system to be stable, and the imaginary part of an eigenvalue
implicates the oscillation frequency. Thus, eigenvalues
directly tell if the system is stable or not and what the system's
oscillation modes are. For SSR stability analysis, a generator or
transmission system's frequency-domain admittance/impedance
measurements must be fitted into a model in the form of
a transfer function matrix. From there, eigenvalue calculation
is possible. Though it seems trivial to arrive at eigenvalues
after obtaining the admittance measurements of a subsystem,
note that the frequency-domain data-fitting technology was
not available in the 1970s. This technology became available
only after 2000. Without frequency-domain data fitting, it is
difficult to identify models and compute eigenvalues.
may/june 2022
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