Conference PaperPDF Available

Designing Model-Free Time Derivatives in the Frequency Domain for Ambient PMU Data Applications

Authors:
  • Dominion Energy

Abstract

Model-free derivatives are essential to several synchrophasor applications. The standard approach to estimate them is to combine a smoothing operation with an ideal derivative computation. However, because the derivative operation increases the signal's content in higher spectral frequencies which can undo the effect of smoothing. Ambient data also brings unique challenges, as mere visual inspection of the estimate in the time domain does not provide insight into the quality of the final estimate. In this regard, the underlying signal's frequency spectrum can provide valuable information for designing a good derivative estimate. This paper introduces a framework for designing a model-free derivative estimate in the frequency domain that accounts for the system's underlying dynamics. The approach is demonstrated on two classic synchrophasor analytics problems on measurements from the Dominion Energy system.
Designing Model-Free Time Derivatives in the
Frequency Domain for Ambient PMU Data
Applications
Chetan Mishra, Kevin D. Jones, R. Matthew Gardner
Dominion Energy, Richmond, VA
chetan.mishra@dominionenergy.com
Luigi Vanfretti
Electrical, Computer and Systems Engineering
Rensselaer Polytechnic Institute, Troy, NY
luigi.vanfretti@gmail.com
Abstract Model-free derivatives are essential to several
synchrophasor applications. The standard approach to estimate
them is to combine a smoothing operation with an ideal derivative
computation. However, because the derivative operation
increases the signal’s content in higher spectral frequencies which
can undo the effect of smoothing. Ambient data also brings
unique challenges, as mere visual inspection of the estimate in the
time domain does not provide insight into the quality of the final
estimate. In this regard, the underlying signal’s frequency
spectrum can provide valuable information for designing a good
derivative estimate. This paper introduces a framework for
designing a model-free derivative estimate in the frequency
domain that accounts for the system’s underlying dynamics. The
approach is demonstrated on two classic synchrophasor analytics
problems on measurements from the Dominion Energy system.
Index Terms Robust Derivative, Inertial Response,
Synchrophasors
I. INTRODUCTION
Reliably estimating derivatives from measurements is a
common task in many measurement data-driven applications in
power systems. Estimating frequency is fundamentally a
problem of estimating the phase angle derivative, and in
practice, it involves a smoothing operation to remove fast
transients [1]. While Phasor Measurement Units (PMUs)
provide a frequency estimate derived from the measured angle,
these estimates can vary drastically between manufacturers,
even in ideal testing conditions and can be affected by time
errors [2]. To this end, alternative means to compute these
estimates can be useful for PMU applications that require
reliable n-th order derivatives. For instance, [3] proposes using
the rate of change of frequency (ROCOF) to detect PMU time
errors. In addition, data-driven dynamic stability assessment
applications involve operating on signal values at successive
time steps. With the increasing penetration of renewable
generation, coupled with the retirement of conventional
generation, there is a need to track inertia, which captures the
relationship between power imbalance and ROCOF [4].
Detecting voltage collapse using Lyapunov exponents [5]
reduces to finding average voltage magnitude derivative at any
given time compared to its value in the past. For detecting
generator angular instability, the single and double derivatives
of slower electromechanical angular dynamics are often used
[6]. Similar ideas apply to frequency stability [7].
Steady-state applications also focus on incremental changes
(i.e. derivatives) from an equilibrium. For example, voltage
security analysis using Thévenin equivalent [8] involves
estimating the derivative of steady-state (slow) voltage-to-
current injection changes and therefore involves smoothing/low
pass filter to remove dynamics. Along the same lines, there
have also been attempts to estimate such derivatives using a
polynomial approximation [9] and even power flow Jacobian
from ambient data [10], which is a partial derivative of real and
reactive power angles and voltages.
Given that the power system operates in ambient conditions
[4], extending standard synchrophasor analytics to ambient data
for continuous monitoring has sparked substantial interest. For
example, there has been a surge in interest in online inertia
monitoring [4], which, as previously noted, is fully reliant on a
precise estimation of the double derivative of phase angle.
Working with ambient data presents a significant challenge
because the signal to noise ratio is low. This makes it difficult
to visually fine-tune the derivative design, as is commonly done
for large signal events.
While multiple applications depend on a signal’s derivatives,
surprisingly, the problem of computing derivatives hasn't been
examined in depth in the power system literature. It has only
been investigated on a case-by-case basis in each application
scenario and therefore, there is a lack of a generic framework
for derivative design that can be applied to any type of
synchrophasor signal. Often, not much thought is given to their
actual effect on a signal’s content. At most, data is smoothed
before/after an ideal type of numerical differentiation.
Numerical differentiation, if not designed carefully, increases
higher frequencies in the estimated derivative’s spectrum,
thereby undoing the effect of smoothing if applied previously.
The current work proposes a derivative design framework for
PMU applications that allows the user to account for the
application's validity in relation to the time scale of interest, in
order to address the numerous challenges surrounding
derivative estimation, as well as a lack of a practical generic
framework. Furthermore, because the framework is wholly in
the frequency domain, it may be used with spectrum analysis
tools [11] to build derivatives for ambient data applications,
which has gone entirely unaddressed in the current literature.
This paper is organized as follows. Section II of this paper
presents the n-th derivative estimation problem, along with a
methodology for designing its computation in the frequency
domain. The results are obtained for multiple practical use cases
on synthetic as well as real PMU data from Dominion System
in Section III, while Section IV outlines future work.
II. DERIVATIVE ESTIMATE
A. Derivative Estimation Problem in Frequency Domain
The existing practices in power systems often involve using
ideal derivatives with or without a smoothing operation
(typically, a moving average). The major challenge of reliably
computing these derivatives can be understood by analyzing
their effect in the frequency domain. An ideal, order
derivative operation on a signal  results in,

(1)
where denotes the Fourier Transform of any signal.
Observe that the gain of an order derivative is proportional
to . The higher the order of the derivative estimated, the
more distorted the signal’s content will be at higher frequencies,
resulting in higher frequency noise. Thus, applying numerical
differentiation when computing derivatives can often undo the
effect of smoothing the signal, as seen in Fig. 1.
Figure 1. Effect of applying an ideal derivative with moving average
B. Ambient System Characteristics and Derivative
Requirements
A derivate estimate procedure should ensure that there is no
loss of relevant information as observed in the signal while at
the same time suppressing high frequency noise. Now, for
ambient conditions, we will show how the information content
of the signal can be understood to guide the derivative design.
Power system in ambient conditions can be approximately
modeled as a linear system driven by random perturbations. The
frequency spectrum [11] of any general scalar output
measurement in such systems can be written as,



(2)
where are the system eigen values and  are complex
constants. Note that the above can be faithfully estimated from
the measurement data and does not require a system model.
Now, the eigenvalues characterize the oscillatory behavior
(modes) of the system and therefore, the output signal spectrum
can yield insights into the time scales of the underlying
system’s dynamics as observed in the signal. Thus, it can be
used to understand what range of frequencies do not contain any
information and therefore can be suppressed in the derivative
estimate to reduce the estimate’s overall variance. This
approach will be later on demonstrated on the real-world
measurement data in the results.
Here it is important to mention that while the discussion is
focused on linear systems and small signal response of
nonlinear systems, the same design approach can be applied for
large signal response in nonlinear systems as well. However, in
that case, the frequency range to retain depends on the
underlying time scales of the system’s dynamics that are not
straightforward to obtain as in the case of linear systems.
C. Methodology for Designing a Robust Derivative
Once the spectrum is estimated for the measurement signal,
the next step is to design how the derivative of the signal is to
be estimated. The following two design criteria are desirable:
1. Present a similar behavior to an ideal derive (as shown
in (1)) in frequency range where the relevant system
dynamics are present. For e.g. < 1 Hz for
electromechanical oscillations.
2. Suppressing unwanted high frequency noise/ fast
dynamics irrelevant to the application. For e.g. suppress
frequencies above 5 Hz in the case of electromechanical
oscillations.
To meet these criteria, we expand the derivative design
approach in [12], keeping in mind that it will be applied to PMU
data. Since the derivative operator is linear in nature, it is
realized using a linear, causal FIR filter. The output when a
filter with window length and coefficients 
acts on a signal  is given by,
  


(3)
here, sampling frequency is
. Next, assume we want
to realize an  order derivative by appropriately choosing
. We start by obtaining a Taylor series expansion of the
exponential terms on the RHS at an arbitrary angular frequency
to get a polynomial representation,




(4)
Now, the first design criterion is met by matching the above
polynomial to an ideal derivative operation, as given in (1), in
the vicinity of . For this, the first Taylor coefficients
at need to be zeroed out and in addition, the coefficient
for term need to be equal to 1. This yields linear
equations in filter coefficients . For a window length of ,
the remaining degree of freedom for realizing the second
criterion is . The second design criterion is
met by zeroing out the first Taylor coefficients evaluated at
the Nyquist angular frequency . The classical two-
point derivative approach corresponds to , i.e. a
minimum window length and thus, no regards for the second
design criterion. Fig. 2 shows the transfer function magnitudes
for those filters when compared to ideal derivative, which
clearly demonstrates the improvement in high frequency
suppression properties with increasing window length . Note
that the axis on the plot is normalized frequency i.e. for higher
sampling rates, a longer will be needed to achieve the same
level of suppression in absolute frequency values. For reader’s
reference, Table I shows the derivative filter coefficients to be
used in (3) for first and second order derivatives for .
Figure 2. Robust derivatives of 1st and 2nd order
for different window sizes
TABLE I DERIVATIVE FILTER COEFFICIENTS
N
First Order








III. RESULTS
To demonstrate the virtues of the proposed framework,
studies are conducted on two popular synchrophasor analytics
problems Thevenin equivalent for steady state voltage security
analysis and frequency estimation from phase angle data.
However, before showing practical application examples,
results are also presented for a synthetic signal to analyze the
performance of the methodology under ideal conditions. This is
carried out because when using real-world measurements, the
ground truth is not known beforehand. When choosing the
appropriate window length for ambient data, the frequency
spectrum is estimated using Welch’s periodogram [11] with
Hanning Window and an FFT window length of 2 mins.
A. Synthetic Ringdown Signal
For the first test, we create a synthetic ringdown signal with
5 Hz and 2 Hz modes with a high variance additive Gaussian
Noise , sampled at 
 .
(5)
This test demonstrates that while the derivative design is
focused on suppressing higher frequencies, it provides the
additional benefit of suppressing measurement noise due to the
low pass filter type effect.
Figure 3. Ringdown Signal Derivative Estimates
The first and second order derivative estimates using the
proposed approach for an arbitrary window size are compared
against the classical, two-point derivative approach in Fig. 3.
Also, compared to the quality of the first order derivative
estimate, the second order estimates are extremely poor for the
2-point approach. This can be explained using (1), which shows
that higher-order derivatives are more prone to enhancing high
frequency noise.
B. Thevenin Equivalent
This application aims at the online monitoring of steady state
voltage stability/security by estimating Thevenin equivalent
from synchrophasor measurements. For the stability analysis of
a monitored load bus, the idea behind this approach is to
represent the remaining power grid using a constant voltage
source  behind a series impedance  satisfying,
 
(6)
Under normal operation conditions, if no major changes
occur in the power grid,  do not change rapidly and
therefore can be treated as a constant in a time window lasting
several minutes. can be estimated independent of  as [8],


(7)
That being said,  are supposed to represent the
response of the grid to slow (quasi-steady state) changes only.
However, the power system does not have a flat frequency
response. This implies that the estimate for  obtained using
the above equation is very sensitive to the dynamic nature of
the load current , which poses an issue.
Now, to mimic realistic operating conditions, we create a
synthetic data set with the load current  gradually
ramping up along with a small, additive, poorly damped
oscillation at 0.8 Hz. The response of the power grid to such a
signal is given by a frequency-dependent impedance  with a
value of 0.2 pu below 0.1 Hz (slow dynamics) and 0.5 p.u.
above it with a smooth transition.  is set to a constant 1 p.u.
Gaussian measurement noise  is added. The results
when using a standard two-point derivative () are
compared against robust derivative with  Firstly, the
spectrum for the estimated current derivatives is plotted below.
Figure 4. I and d/dt(I) Spectrum
It can be seen from the current magnitude spectrum that the
effect of slow ramping is restricted to < 0.2 Hz while the
oscillation is around 0.8 Hz. This is typical of practical systems
where the underlying dynamics are usually separable in the
frequency domain. Also, observe that the derivative estimates
are the same up to 0.3 Hz beyond, which the longer window
derivative suppresses the faster dynamics.
Figure 5 Thevenin Impedance Estimates
Next, we plot the Thevenin Equivalent estimate obtained
from (7) using the estimated derivatives. As expected, the two-
point derivative gives a wrong estimate for  due to it also
capturing the interaction of 0.8 Hz components of and ,
while increasing the window size yields a better estimate by
filtering out that portion.
C. Frequency Estimation from Phase Angle Data
By definition, frequency represents the first-order derivative
of phase angle. While the IEEE C37.118 standard on
Synchrophasor Measurements in Power Systems [13]
standardizes the phase angle calculation, it does not prescribe
how to estimate frequencies. Most PMU algorithms approach
frequency estimation by averaging the derivative calculation
over a multi-cycle window of data, to remove the effect of
measurement noise and transients [1], yielding a response
similar to Fig. 1. However, there are substantial differences in
how each manufacturer implements it resulting in different
results even under ideal conditions [2], which become
particularly prominent when studying small-signal response in
ambient data. Furthermore, frequency estimates can be plagued
with significant quantization errors, which makes them even
less trustworthy. Therefore, in a practical system with multiple
devices from different manufacturers, it is preferable to not
directly use frequency estimates from the various PMUs, which
makes a posteriori frequency estimation from phase angle an
important problem.
Figure 6. Phase Angle Spectrum
For the present study, a parcel of 20 min phase angle
measurements was taken from two synchrophasor device
manufacturers at different substations (A and B, respectively).
Fig. 6 shows the absolute phase angle spectra in the 0-5 Hz
range. The initial part, <1 Hz, has a curved spectral baseline due
to the absolute phase angle drifting from the 60 Hz power
system frequency. There are also sharp peaks at 1, 2 and 4 Hz.
Based on a prior analysis[14], these are not physical modes but
the effect of periodic phase angle corrections made internally
by the PMUs to account for clock drift. Note that only
frequencies below 5 Hz are of interest.
Substation A’s spectrum shows spectral peaks around 0.5
Hz, 1.5 Hz, and 2.5 Hz. These are a result of mechanical rotor
oscillations from a combined cycle power generation plant.
Below, we compare the frequency estimate encoded inside the
device (in blue, DFR) and reported as synchrophasor data with
different robust derivative options differing in window lengths.
The estimated gain of the device’s internal frequency
calculation stays close to an ideal derivative up to 0.4 Hz,
increases from 0.4-0.6 Hz, and beyond that, rolls off. The gain
continues to bounce beyond 1 Hz, loosely similar to that of a
moving average with an ideal derivative (see Fig. 1). It neither
annihilates the unwanted frequencies nor resembles a derivative
in the retained frequency range. A good derivative estimate will
closely match an ideal derivative-like operation for the
frequency ranges with dynamics of interest, while at the same
time annihilating everything at higher frequencies. From the
plot in Fig. 7, to account for electromechanical oscillations up
to 2.5 Hz, can be a good choice for the robust derivative
with better characteristics than the frequency computed from
the actual PMU device.
Figure 7 Frequency Estimates Substation A
Figure 8 Frequency Estimates Sub B
At Substation B, the device’s internal frequency estimate is
loosely similar to an ideal derivative up to 0.2 Hz, beyond
which it increases until 4 Hz and then rolls off gradually.
However, the quality of the derivative is poor. This can be seen
in Fig. 9 by comparing the estimates in the time domain where
even a simple two-point derivative (orange) is less noisy than
the device’s own estimate.
Figure 9 Time derivatives with N=2 & 18 compared to DFR
Finally, based on the phase angle spectrum, since there isn’t
much information beyond 1 Hz at this location, a more
aggressive approach can be taken, using a longer derivative
window of  for which case, the gain roll-off starts close
to 1 Hz, as shown in Fig. 8. However, this results in a larger
phase shift due to the causal nature of the filter, which is an
acceptable tradeoff. That being said it is important to highlight
that the requirement for a causal derivative filter is only relevant
for online applications. On the other hand, for applications
allowing for time delays such as offline analysis or most PMU
control center applications, which operate using rolling
windows of data, a noncausal derivative can be designed with
an additional constraint that minimizes the phase lag.
IV. DISCUSSION AND FUTURE WORK
In this work, a framework for designing derivatives in the
frequency domain is presented. This framework enables
designing derivatives guided by the time scale of dynamics of
interest in the underlying system while suppressing high-
frequency noise. Future work will explore the use of this
approach for state estimation for new monitoring applications.
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Monitoring long term voltage stability of the power grid is necessary to ensure the secure operation of the system. In this paper, the voltage locus of the load phasor as a function of the load increase is studied is studied from a geometric perspective. This locus is used to propose a Voltage Stability Index based on the magnitude of the measured voltage and calculated thevenin equivalent angles. A novel method to calculate only the necessary thevenin parameters is presented. The proposed Voltage Stability Index and the efficient thevenin calculation method are tested on the IEEE-30 Bus system with lagging, unity and leading power factors.
Article
Transient stability of an electrical power system refers to the ability of the system to settle at the stable equilibrium point in the post-fault system subsequent to a specific fault scenario. This stability problem can be studied either as a system stability or a structural stability problem. In this paper, the concept of Lyapunov Exponents (LEs) is used to analyze the transient stability of IEEE 3-generator 9-bus and IEEE 16 generator 69 bus benchmark systems. A spectrum of LEs is calculated from the mathematical models and the Largest Lyapunov Exponent (LLE) being negative implies the exponential stability of the post-fault power system. The system stability regions for specific fault scenarios are determined using the invariance property of the LEs from the initial conditions. Furthermore, the structural stability regions in terms of parameters of the pre-fault system dynamic equations are also determined. This study demonstrates that the concept of LEs can be used as a novel method to determine if the post-fault power system is within the stability region of attraction of the new stable equilibrium point, and hence to determine the stability region. It is also proposed that the largest average exponential rate calculated over a short time window is a sound approach for early prediction of the stability.