research highlights
Measurement-based Adaptive Wide-Area Damping Controller (continued)
Figure 2: SISO model for damping control
Figure 3: Damping control performance comparison
Frequency-amplitude curve – a new analytical tool for characterization of nonlinear oscillations
in power systems
by Dr. Kai Sun ([email protected])
Inter-area oscillations have been threatening system operations and stability since the beginning of
interconnecting power systems over weak tie lines in the 1960s. An inter-connected power system is
essentially a nonlinear oscillator network, so its electromechanical oscillations (EOs) have inherent
nonlinearities and are different from oscillations with a linear system. Power system oscillations have been
analyzed in the past several decades. However, most methods do not take the nonlinearities with a power
system into account and only use a linearized system model to study oscillations from a small-signal
stability point of view. The measurement-based methods, such as the Prony method and Hilbert-Huang
Transformation, either assume an EO to be harmonic with its constant frequency, damping and phasing
during a specific time period or purely rely on signal processing without considering the nonlinear nature
with a power system. As a result, if the modal frequency fluctuates during the measurement window, an
existing measurement-based method detects two or more separate modes.
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Based on our previous studies, a power system EO mode can change its frequency all over the swings,
which can be captured by a phase-locked loop based method applied to the measurements. This effort has
focused on analytical studies addressing the nonlinear mechanism of an EO mode. Both model-based and
measurement-based approaches have been developed to study nonlinearities of EOs mainly due to the
networking of generator swing equations. First, the oscillation frequency of an undamped Single-MachineInfinite-Bus (SMIB) system is analytically formulated to discover the frequency-energy dependency with
a single EO mode under disturbances around a given stable system equilibrium. Accordingly, a new tool
named frequency-amplitude (F-A) curve, as a comparative concept of the power-voltage (P-V) curve for
voltage stability analysis, is proposed to characterize EOs regarding a specific mode. We also discovered
the existence of such an F-A curve for each of the dominant EO modes with a multi-machine power system.
A measurement-based method is proposed to estimate the F-A curve for any EO mode of interest and
accordingly calculate an associated angular stability margin index. Such a measurement-based method is
valuable for online angular stability monitoring regarding any EO mode of interest and for taking preventive
control actions if the F-A curve shows a transition of the mode to an instability mode. From the tests on the
CURENT WECC large-scale testbed (LTB) system, it is found that the modeling of more details of generators
(e.g. increasing the orders of models and adding excitation systems) does not impact the shapes of
formulated F-A curves. Figure 1 below gives the F-A curve about the only EO model of an SMIB system. The
curve shows the nonlinearity of the mode due to its frequency-energy dependency. The function of the F-A
curve can be analytically solved by means of elliptic integrals about the swing equation. Figure 2 shows the
same post-disturbance trajectory of the IEEE 9-bus system drawn respectively in the phase plane and the
F-A plane: in the phase plane, the trajectory is tangled while in the F-A plane, we may clearly see the stability
margins of four segments TW1-TW3 on the trajectory. The F-A curves estimated for selected EO modes of
the WECC system are shown in Figure 3. Each star on an F-A curve is the real-time location of the actual
system state seen from that EO mode, whose distance to the nose-point tells the real-time angular stability
margin.