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Source Location of Forced Oscillations based on Bus Frequency Measurements

Source Location of Forced Oscillations based on
Bus Frequency Measurements
Alvaro Ortega
Instituto de Investigaci´
on Tecnol´
ogica, ICAI
Universidad Pontificia Comillas
Madrid, Spain
ORCID iD: 0000-0001-5749-0678
Federico Milano
School of Electrical and Electronic Engineering
University College Dublin
Dublin, Ireland
ORCID iD: 0000-0002-0049-9185
Abstract—The paper proposes a technique to locate the source
of forced oscillations in power systems. The only requirement
is the availability of bus frequency measurements as obtained,
for example, from phasor measurement units. The proposed
technique is model-agnostic, optimization-free, non-confidential,
and independent from the source and the frequency of the forced
oscillations. Accuracy and robustness with respect to noise are
demonstrated through two examples based on the IEEE 14-bus
and New England 39-bus systems, as well as a case study based
on a 1,479-bus dynamic model of the all-island Irish transmission
Index Terms—Forced oscillation (FO), frequency estimation,
phasor measurement unit (PMU), discrete sine transform (DST).
Forced oscillations (FOs) in power systems result from
external periodic perturbations as opposed to natural or free
oscillations that are originated from the coupling of power
system devices. While the latter can be generally damped
by means of control actions, FOs are usually suppressed
by disconnecting their source. It is thus crucial for system
operators to reliably and swiftly locate FOs in order to prevent
potential cascading phenomena.
References [1] and [2] provide detailed reviews of available
techniques to locate the source of FOs. All the approaches
discussed in [1], [2] and in the references therein show some
relevant drawback, such as: (i) the need of detailed information
of the models [3], [4]; (ii) unfitness for large systems [5];
(iii) applicability only to post-fault analysis studies [4]; (iv)
inability to identify the source of FOs in a wide range of
frequencies [6]; and (v) inability to cope with sources of
different nature [7]. A recent promising approach to locate FOs
is proposed in [8], where a multi-stage optimization algorithm
is proposed, which can cope with parameter uncertainties in
real-time applications. In [8], however, the authors do not
tackle the issue of the scalability of the technique proposed,
and they focus only on generators as the source of the FOs
in a rather narrow range of frequencies of the FOs, namely
from 0.5 to 0.86 Hz, while the typical range of frequencies
FOs spans a quite wider interval, namely from 0.12 to 14 Hz.
This work was supported by the European Commission, by funding
A. Ortega under Grant No. 883985 and F. Milano under Grant No. 883710.
F. Milano is also funded by Science Foundation Ireland, through project
AMPSAS, Grant No. SFI/15/IA/3074.
Several references, including recent works, have made use
of Fourier-related transforms and indices such as the Power
Spectral Density (PSD) applied to measurement signals from
Phasor Measurement Units (PMUs) for the detection of FOs,
e.g., [9]–[12]. These references are aimed at detecting the
presence of FOs, not to identify the location of their sources.
The possibility to use the PSD to estimate the location of
the FOs source is illustrated in [1] through voltage signals.
The approach above shows low accuracy when FOs are
characterized by frequencies that are relatively low or close to
system natural modes. Another approach to locate FOs based
on PMU measurement is described in [13], where the sources
are identified by means of the concept of dissipating energy
flows (DEFs). DEF-based approaches, however, require the
assumptions that FOs are originated by synchronous machines,
lossless network and constant power loads.
This paper proposes a technique to estimate the location
of the source of FOs based on a byproduct of the Frequency
Divider Formula (FDF) originally presented by the authors
in [14], and on the processing of the resulting signals through
the well-known Discrete Sine Transform (DST). Thanks to the
features of the FDF, the proposed technique is:
Model-agnostic, as one does not need to know anything
about the device that originates FOs, nor about any
other system dynamics, since only information about the
system topology is required;
Optimization-free, as the estimation of the FO source is
based on the direct processing of PMUs data;
Non-confidential, as it only requires bus frequency mea-
surements, which can be readily obtained with PMUs;
Independent from the source and frequency of the FOs,
being reliable for any source and in the typical FO
frequency range; and
Scalable, as it is suitable for large electrical networks
(thousands of buses).
The paper is organized as follows. The methodology of the
proposed approach is formulated and duly discussed in Sec-
tion II. A set of illustrative examples based on two well-known
benchmark networks is presented in Section III. Section IV
discusses a case study based on the detailed dynamic, 1,479-
bus model of the All-island Irish Transmission System. Finally,
Section V draws conclusions and future work directions.
The objective of this section is to derive an expression of
the harmonic “frequency injections”, called αin the remainder
of this work, and their link with the frequency variations ω
at the buses of the grid.
We derive these expressions in two ways. First Section
II-A shows that the time derivative of the current injection
at buses is linked to bus frequency variations. This fact
indicates that the FO harmonic sources and bus frequency
variations are, as expected, intertwined. Then, Section II-B
shows that the obtained expression is, in fact, an extension of
the FDF. This observation allows stating the equivalence of
FO harmonic injections to the variations of frequency during
an electromechanical transient. Finally, Section II-C presents
the proposed technique to locate the FO sources.
A. Link between Current Injections and Bus Frequencies
Let us consider the current injections at the network buses
as a function of the network admittance matrix and the bus
ı(t) = ¯
Ybus ¯
where ¯
ıand ¯
vare assumed to be quasi-steady state phasors.
Then, let us apply to (1) common simplifications utilized in
transient stability analysis, namely, (i) the elements of the
admittance matrix are constant, i.e., their dependency on the
frequency can be neglected, except for discrete events such as
line outages; and (ii) the resistance of the transmission lines
are negligible compared with their reactances, thus:
Ybus jIm{¯
Ybus}=jBbus .(2)
We then calculate the time derivative of (1), assuming that the
derivatives of the bus voltages can be approximated as:
dt ¯
where indicates the element-by-element product and
ω=ωωs1nis the vector of frequency variations at the n
network buses, with ωsthe fundamental (synchronous) angular
frequency of the system, e.g., 314.16 rad/s for a 50 Hz system.
The approximation of the time derivative of the bus voltages
in (3) is fully justified in the context of electromechanical
transients as thoroughly discussed in [14] but is also perfectly
consistent with the typical range of frequencies of FOs, which,
as discussed in Section I, span a range from 0.12 to 14 Hz.
With these assumptions, the time derivative of (1) can be
written as: d
and, dividing element-by-element by j¯
α(t) = d
v(t)] Bbusω(t),(5)
where ÷is the element-by-element division between two
vectors. Equation (5) is the sought expression that links the
time derivative of harmonic current injections with frequency
variations in the network.
As a consequence of the assumptions utilized to obtain (5),
the right hand side of (5) is real, hence also the elements of α
are real. Moreover, (5) has the same structure as (1), meaning
that bus frequency variations ωare “potentials”, whereas α
are “flows”. This observation is reprised in Section II-C and is
key for the proposed technique to locate the sources of FOs.
B. Equivalence of FOs and Electromechanical Transients
The expression in (5) is utilized in this paper to determine
the sources of FO. The derivation given so far, however, does
not justify the physical meaning of the vector α, which is,
in turn, that of a “current source” of frequency variations at
buses. To clarify this point, we utilize the concept of FDF.
The FDF originally proposed in [14] can be written as:
BGω(t)BBGωr(t) = Bbus ω(t).(6)
where ωhas the same meaning as in (1);
ωr=ωrωs1mis the vector of the frequency variations
of the rotor speeds of the msynchronous machines; BGis a
diagonal matrix whose i-th diagonal element is non-zero if
there is a synchronous machine connected to bus iand its
value is the inverse of the machine internal reactance; BBG
is the susceptance incidence matrix between generators and
network buses with inclusion of the internal reactances of the
synchronous machines; and Bbus is the imaginary part of the
standard network admittance matrix;
Comparing (6) and (5) leads necessarily to conclude that,
for synchronous machines, one has:
αG(t) = BGω(t)BBGωr(t),(7)
that is, the “harmonic” injection of synchronous machines
depends on the deviation of their rotor angular speeds with
respect to the reference frequency of the system.
From (5), we can now extrapolate the concept of “genera-
tors” in a wider sense, i.e., not only synchronous machines, but
actually any device that locally imposes a frequency variation,
and, in particular, an FO source. The elements of αcan
thus be interpreted as the “boundary conditions” for the bus
frequencies of the grid. The vector αitself can be thought
as the sum of two terms, the frequency variations due to
synchronous machines and those due to FOs, i.e.,
α(t) = αG(t) + αFO(t).(8)
C. Identification of the Location of FO Sources
The relevance of (5) in the context of this work is that it
states that the effect of αcan be determined by measuring bus
frequency variations and properly weighting such variations
with the elements of the matrix Bbus. This property of the
FDF has been exploited in [15] to estimate the rotor speed
of synchronous machines and in [16] and [17] to quantify
the frequency support and inertial response provided by non-
synchronous devices, respectively.
Here, we propose another application of such a property,
namely, to study the frequency spectrum of the elements of
αto determine where the FO is located. With this aim, let
us assume a stationary condition where each AC voltage of
the system oscillates with the fundamental frequency ωsand a
constant FO with angular frequency ωFO. Then, the expression
(5) becomes:
α(t) = Bbusωs1n+aFO sin(ωFOt+φFO )ωs1n
=Bbusasin(ωFO t+φ),(9)
where the vectors a(aFO) and φ(φFO) are the amplitudes
and phase shots, respectively, of the “total” (FO) harmonic at
every bus of the grid. The expression (9) shows that, while in
stationary conditions, the FDF “sees” the harmonics as time-
dependent periodic variations with respect to the fundamental
frequency ωs.
If the system includes a set of harmonics, say i, then (9)
α(t) = Bbus X
Both (6) and (10) state that the amplitude of frequency
variations in a grid follow the same behavior of voltages in
a resistive DC circuit. In a resistive circuit, the voltage at the
source is always greater than at the load. Since the network
is interconnected, all bus frequencies “see” the FO injected
at bus i, but its effect is more evident the closer the bus is
(electrically) to the location of the FO. It has to be expected,
thus, that if the i-th element of vector αFO is non-null, the
frequency variations due to the FO will impact mostly the i-th
element of Bbusω. Moreover, being (10) a linear expression,
the superposition principle holds, and hence each harmonic can
be considered to be independent from the others.
These two concepts together lead to the following propo-
sition. The i-th element of the vector of amplitudes aFO
corresponding to the i-th bus where the FO is injected into
the grid is independent from other harmonics and, being
equivalent to a voltage source in a resistive circuit, is greater
than the other elements in the same vector.
In the vast majority of cases, FOs are characterized by a
single constant fundamental frequency, ωFO as in (9). Hence, a
well-known Fourier-related transform widely used in harmonic
analysis, namely the Discrete Sine Transform (DST), is used
to process the right-hand side of (5) as follows [18]:
ˆαi{k ωo}=2
αi[nt] sin(k ωont),
k= 0,1,2, . . . , N 1,
where tand Nare the sampling time and the number of
samples of the time series, respectively; ωo= 2π/Towith
To=Nt; and αiis the sampled signal at bus i, given by:
αi[nt] = Bbus,i ω[nt],(12)
where Bbus,i is the i-th row of matrix Bbus. Note also that,
due to the sparsity of the network susceptance matrix Bbus,
the number of frequency measurements required to estimate
the i-th element of the vector αis small, as one only needs
to measure the frequency at bus i, and its adjacent buses (see
also the discussion in [15]).
Two remarks on the proposed technique are relevant. First,
the assumption of stationarity conditions is not crucial for the
validity of (5), and hence (9), but are assumed here to simplify
the Fourier analysis discussed above. If the FO has a transient
behavior as discussed in the example [13], a DFT with mobile
window as commonly implemented in PMUs can be utilized.
As a second remark, in the literature, the DFT of several
other quantities, e.g., power flows and bus voltages, have
been considered for the identification of the locations of the
sources of FOs. None of these quantities, however, is a linear
combination of the frequency variations of the buses as the
vector α. This is the feature that makes the approach proposed
in this paper particularly adequate for the identification of FO
sources. This point is thoroughly illustrated and discussed in
the following sections.
The proposed approach to estimate the location of the source
of forced oscillations in a transmission grid is first illustrated
by means of two well-known benchmark networks used for
dynamic analysis, namely the IEEE 14-bus and the New-
England 39-bus test systems, in Subsections III-A and III-B,
Uncorrelated stochastic processes of the loads are utilized to
model load volatility. Unless otherwise indicated, the Ornstein-
Uhlenbeck process (mean-reverting) [19], is considered for
all loads of the systems discussed in this section as well
as in the case study of Section IV. PMUs used to measure
the bus frequencies are modeled as Synchronous Reference
Frame Phase-Locked Loops (SRF-PLLs). For all scenarios,
time domain simulations last 600 s, with a time step of 20 ms.
The simulation tool used in the examples of this section,
as well as in the case study of Section IV is Dome [20].
The simulations were executed on a 64-bit Linux Ubuntu
18.04 operating system running on an 8-core 3.40 GHz
Intel©Core i7TM with 16 GB of RAM.
A. IEEE 14-bus system
The description and data of this benchmark system are
provided in [21].
In this example, FOs have been applied, individually and
in separate simulations, to the mechanical power of the syn-
chronous machines at buses 1 and 2, as well as to the active
power consumption of all loads, which are connected to all
buses except for 1, 7 and 8. All FOs have a frequency ωFO
of 0.5 Hz, and their amplitude is proportional to either the
capacity of the machine (2%), or to the active power load
consumption (10%).
1) Active Power-Driven Forced Oscillations: Figure 1
shows the trajectories of the elements of vector αand their
respective DSTs for the case where the FOs are injected by
the machine at bus 1. By processing αsignals by means of
the DST, one can estimate the location of the source of the
FOs by analyzing the peak values observed at the frequency
0 2.557.5 10 12.5 15 17.5 20
Time [s]
αi[pu(rad Ω1s1)]
Bus 1
Bus 2
Bus 3
Bus 11
Bus 14
0.46 0.48 0.5 0.52 0.54
Frequency [Hz]
ˆαi[pu(rad Ω1s1)]
Bus 1
Bus 2
Bus 3
Bus 11
Bus 14
Fig. 1. IEEE 14-bus system – Trajectories of the elements of vector αand
their respective DSTs when FOs are applied to the mechanical power of the
synchronous machines at bus 1.
of such oscillations. As anticipated in Section II, the highest
peak corresponds to the bus where the FOs originate. This is
also shown in Fig. 2 for the cases of the two synchronous
generators, and four loads at buses 2, 3, 11 and 14. Figure 2
shows the values ˆ
α{ωFO}of the buses to which a device
is connected (either generator, compensator or load). These
values are normalized as follows:
max( ˆ
For all cases studied, the bus with the highest peak corresponds
to the source of the FOs.
2) Voltage-Driven Forced Oscillations: To illustrate that the
nature of the source of FOs does not affect the accuracty of
the results, Fig. 3 shows the values of hˆαiiat ωFO = 0.5Hz
when the oscillations are originated by the Automatic Voltage
Regulators (AVRs) of the machines at buses 1 and 2. These os-
cillations can be captured accurately by the proposed approach.
An oscillation originated by an AVR, in fact, introduces
oscillations in all variables of a synchronous machine and,
thus, also in the rotor speed of the machine.
3) Spectrum Overlapping: The next scenario discussed in
this Section considers the case when the frequency of the
FOs overlaps with that of the electromechanical modes of the
synchronous machines. Figure 4 shows the values of hˆαiiat
ω= 1.45,1.5and 1.55 Hz when FOs (ωFO = 1.5Hz) are
applied to the active power consumption of the load at bus 11.
It can be seen that, while the electromechanical modes of the
machines span a range of frequencies around 1.5Hz, the FOs
are localized at a very narrow range around ωFO. This property
of FOs allows estimating their source even when their ωFO
overlaps with other oscillating modes of the system that can
even show higher values of hˆαii.
4) Natural Oscillations: The last scenario discussed in this
section considers the 20% overload of the IEEE 14-bus system,
with the loads modeled as constant PQ. Under this conditions,
the system can show undamped natural oscillations due to a
pair of complex conjugate eigenvalues with positive real part.
In this scenario, the contingency that triggers the limit cycle is
the outage of the line connecting buses 2 and 4 and the results
of the DST at the oscillatory frequency are shown in Fig. 5.
The synchronous generator at bus 1 is the most susceptible to
such oscillations. Indeed, the pair of complex eigenvalues that
crosses the complex plain imaginary axis is associated with
the AVR and stator voltage of the machine at bus 1.
A less expected result of simulation results is the high
peak observed at bus 4. As stated above, the load at such
a bus is modeled as a constant PQ load, thus no active
device is connected to bus 4. Therefore, it could be expected
that hˆα4i= 0, and this is indeed what it can be observed
if, e.g., only the active power injections at system buses
are monitored. However, the ability of a load to maintain a
constant power consumption requires the current to vary in
order to compensate any bus voltage variations due to, e.g.,
the natural oscillations of the limit cycle [22]. These current
oscillations, which cannot be detected if measuring the bus
active power injection, are detected by the proposed hˆαii.
Note that the load at bus 4 is the second largest load in
the system, just behind the load at bus 2 which is connected
together with a synchronous machine and thus its effect is
coupled with that of the generator. Bus 4 is also, together
with bus 5, the only bus with a load in the 69 kV region of
the IEEE 14-bus system. However, the load at bus 5 is about
an order of magnitude smaller than that at bus 4. These facts
explain the high peak observed in Fig. 5 with respect to the
other loads of the system.
B. New-England 39-bus system
The second example considers the New-England 39-bus,
10-machine test system [23].
Two simultaneous FOs are applied, namely to the me-
chanical power of the machine at bus 35 (ωFO,1= 0.5Hz),
and to the active power consumption of the load at bus
25 (ωFO,2= 10 Hz). Figure 6 shows the values of hˆαiiat
ωFO,1(top) and ωFO,2(bottom) for all buses where either a
machine or a load is connected. The proposed approach is
able to accurately estimate the source location of both FOs,
despite being of different nature (generation vs. load) and with
different ωFO.
Note that the New England system shows a uniform and
fairly symmetrical topology, whereas the IEEE 14-bus system
of the previous example has all generation in the 69 kV region,
and most of the load is located in the 13.8 kV region. Hence,
the proposed approach shows to be accurate regardless of the
topology of the grid.
To test the accuracy of the proposed approach on a large,
heterogeneous system, the All-island Irish Transmission Sys-
tem (AIITS) is considered in this case study. The model
of this grid includes 1,479 buses, 1,851 transmission lines
and transformers, 245 loads, 22 conventional power plants
1 2 3 4 5 6 8 9 10 11 12 13 14
Bus #
1 2 3 4 5 6 8 9 10 11 12 13 14
Bus #
1 2 3 4 5 6 8 9 10 11 12 13 14
Bus #
1 2 3 4 5 6 8 9 10 11 12 13 14
Bus #
1 2 3 4 5 6 8 9 10 11 12 13 14
Bus #
1 2 3 4 5 6 8 9 10 11 12 13 14
Bus #
Fig. 2. IEEE 14-bus system – From left to right and from top to bottom, elements hˆαiifor ωFO = 0.5Hz when FOs are applied to the mechanical power
of the synchronous machines at buses 1 and 2, and to the active power consumption of the loads at buses 2, 3, 11 and 14.
1 2 3 4 5 6 8 9 10 11 12 13 14
Bus #
1 2 3 4 5 6 8 9 10 11 12 13 14
Bus #
Fig. 3. IEEE 14-bus system – From top to bottom, elements hˆαiifor
ωFO = 0.5Hz when FOs are applied to the output voltage of the AVRs
the synchronous machines at buses 1 and 2.
1 2 3 4 5 6 8 9 10 11 12 13 14
Bus #
1 2 3 4 5 6 8 9 10 11 12 13 14
Bus #
1 2 3 4 5 6 8 9 10 11 12 13 14
Bus #
Fig. 4. IEEE 14-bus system – From top to bottom, elements hˆαiifor ω=
1.45,1.5and 1.55 Hz when FOs of ωFO = 1.5Hz are applied to the active
power consumption of the load at bus 11.
1 2 3 4 5 6 8 9 10 11 12 13 14
Bus #
Fig. 5. IEEE 14-bus system – Elements hˆαiiwhen a Hopf bifurcation induces
a limit cycle due to the outage of the line 2-4 at 20% system overload.
Fig. 6. New England system – Elements hˆαiifor ωFO ,1= 0.5Hz (top)
and ωFO,2= 10 Hz (bottom) when simultaneous FOs of different ωFO are
applied to the mechanical power of the synchronous machine at bus 35 and
to the active power consumption of the load at bus 25.
with AVRs and Turbine Governors (TGs), six Power System
Stabilizers (PSSs), and 176 Wind Power Plants (WPPs) [24].
In the simulations, apart from the load fluctuations described
in the introduction of Section III, wind perturbations have also
been modeled. In the long term, the stochastic process applied
to the wind follows a Weibull distribution or some other non-
symmetrical distribution. However, in the short term, one can
approximate wind fluctuations with a Gaussian process. In this
case study, the noise is modeled as a zero-average Ornstein-
Uhlenbeck process.
Fig. 7. AIITS – From top to bottom, a set of hˆαiiwhen FOs are applied to
the generation/loads connected to buses 327, 588, 717 and 777.
A selection of 20 buses of the AIITS are chosen to be
monitored during the simulations. These buses represent syn-
chronous generators, WPPs and loads distributed across the
network. FOs with ωFO = 0.5Hz are applied, individually, to
four of these buses.
Figure 7 shows the values hˆαiiat ωFO = 0.5Hz of the
20 buses when FOs are applied to the mechanical torque of
the WPPs at buses 327 and 588, to the mechanical power of
the synchronous machine at bus 717, and to the load at bus
777. The proposed approach is able to accurately estimate the
location of the FO sources despite all the external perturbations
that affect the system, namely wind and load fluctuations.
This paper proposes a technique to identify the location
of the source of active power/voltage phase angle-driven
Forced Oscillations (FOs) based on measurements of the
bus frequencies through PMUs. Simulation results based on
two benchmark networks, as well as on a large real-world
transmission system model, show the accuracy and reliability
of the proposed approach. The technique proposed in this
paper is model-independent; is suitable for any system size
and topology, and for on-line applications; and can cope with
a variety of types of FOs sources and frequencies.
Future work will further elaborate on the estimation of
the location of FOs-induced resonances in power systems in
cases where devices other than the synchronous machines are
involved. Techniques to improve the accuracy of the proposed
index when FOs overlap with resonance modes of the system
will also be studied and developed.
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The Discrete Cosine Transform (DCT) is used in many applications by the scientific, engineering and research communities and in data compression in particular. Fast algorithms and applications of the DCT Type II (DCT-II) have become the heart of many established international image/video coding standards. Since then other forms of the DCT and Discrete Sine Transform (DST) have been investigated in detail. This new edition presents the complete set of DCT and DST discrete trigonometric transforms, including their definitions, general mathematical properties, and relations to the optimal Karhunen-Lo ve transform (KLT), with the emphasis on fast algorithms (one-dimensional and two-dimensional) and integer approximations of DCTs and DSTs for their efficient implementations in the integer domain. DCTs and DSTs are real-valued transforms that map integer-valued signals to floating-point coefficients. To eliminate the floating-point operations, various methods of integer approximations have been proposed to construct and flexibly generate a family of integer DCT and DST transforms with arbitrary accuracy and performance. The integer DCTs/DSTs with low-cost and low-powered implementation can replace the corresponding real-valued transforms in wireless and satellite communication systems as well as portable computing applications. The book is essentially a detailed excursion on orthogonal/orthonormal DCT and DST matrices, their matrix factorizations and integer aproximations. It is hoped that the book will serve as a valuable reference for industry, academia and research institutes in developing integer DCTs and DSTs as well as an inspiration source for further advanced research. Key Features - Presentation of the complete set of DCTs and DSTs in context of entire class of discrete unitary sinusoidal transforms: the origin, definitions, general mathematical properties, mutual relationships and relations to the optimal Karhunen-Lo ve transform (KLT). - Unified treatment with the fast implementations of DCTs and DSTs: the fast rotation-based algorithms derived in the form of recursive sparse matrix factorizations of a transform matrix including one- and two-dimensional cases. - Detailed presentation of various methods and design approaches to integer approximation of DCTs and DSTs utilizing the basic concepts of linear algebra, matrix theory and matrix computations leading to their efficient multiplierless real-time implementations, or in general reversible integer-to-integer implementations. - Comprehensive list of additional references reflecting recent/latest developments in the efficient implementations of DCTs and DSTs mainly one-, two-, three- and multi-dimensional fast DCT/DST algorithms including the recent active research topics for the time period from 1990 up to now.
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