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Source Location of Forced Oscillations based on

Bus Frequency Measurements

´

Alvaro Ortega

Instituto de Investigaci´

on Tecnol´

ogica, ICAI

Universidad Pontiﬁcia Comillas

Madrid, Spain

aortega@comillas.edu

ORCID iD: 0000-0001-5749-0678

Federico Milano

School of Electrical and Electronic Engineering

University College Dublin

Dublin, Ireland

federico.milano@ucd.ie

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

system.

Index Terms—Forced oscillation (FO), frequency estimation,

phasor measurement unit (PMU), discrete sine transform (DST).

I. INTRODUCTION

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.

II. METHODOLOGY

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

voltages:

¯

ı(t) = ¯

Ybus ¯

v(t),(1)

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:

d

dt ¯

v(t)≈j∆ω(t)◦¯

v(t),(3)

where ◦indicates the element-by-element product and

∆ω=ω−ωs1nis 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

dt¯

ı(t)≈jBbus∆ω(t)◦¯

v(t),(4)

and, dividing element-by-element by j¯

v:

α(t) = d

dt¯

ı(t)÷[j¯

v(t)] ≈Bbus∆ω(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:

BG∆ω(t)−BBG∆ωr(t) = Bbus ∆ω(t).(6)

where ∆ωhas the same meaning as in (1);

∆ωr=ωr−ωs1mis 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) = BG∆ω(t)−BBG∆ωr(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

=Bbusa◦sin(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 Ωi, then (9)

becomes:

α(t) = Bbus X

i∈Ωi

ai◦sin(ωit+φi).(10)

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 Bbus∆ω. 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

N

N−1

X

n=0

αi[n∆t] sin(k ωon∆t),

k= 0,1,2, . . . , N −1,

(11)

where ∆tand Nare the sampling time and the number of

samples of the time series, respectively; ωo= 2π/Towith

To=N∆t; and αiis the sampled signal at bus i, given by:

αi[n∆t] = Bbus,i ω[n∆t],(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.

III. EXA MP LES

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,

respectively.

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]

−0.1

−0.05

0

0.05

0.1

αi[pu(rad Ω−1s−1)]

Bus 1

Bus 2

Bus 3

Bus 11

Bus 14

0.46 0.48 0.5 0.52 0.54

Frequency [Hz]

0

0.004

0.008

0.012

0.016

ˆαi[pu(rad Ω−1s−1)]

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:

hˆαii=ˆαi{ωFO}

max( ˆ

α{ωFO}).(13)

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.

IV. CAS E STUDY

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 #

0

0.25

0.5

0.75

1

hˆαii

1 2 3 4 5 6 8 9 10 11 12 13 14

Bus #

0

0.25

0.5

0.75

1

hˆαii

1 2 3 4 5 6 8 9 10 11 12 13 14

Bus #

0

0.25

0.5

0.75

1

hˆαii

1 2 3 4 5 6 8 9 10 11 12 13 14

Bus #

0

0.25

0.5

0.75

1

hˆαii

1 2 3 4 5 6 8 9 10 11 12 13 14

Bus #

0

0.25

0.5

0.75

1

hˆαii

1 2 3 4 5 6 8 9 10 11 12 13 14

Bus #

0

0.25

0.5

0.75

1

hˆαii

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 #

0

0.25

0.5

0.75

1

hˆαii

1 2 3 4 5 6 8 9 10 11 12 13 14

Bus #

0

0.25

0.5

0.75

1

hˆαii

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 #

0

0.25

0.5

0.75

1

hˆαii

1 2 3 4 5 6 8 9 10 11 12 13 14

Bus #

0

0.25

0.5

0.75

1

hˆαii

1 2 3 4 5 6 8 9 10 11 12 13 14

Bus #

0

0.25

0.5

0.75

1

hˆαii

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 #

0

0.25

0.5

0.75

1

hˆαii

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.

3478

12

15

16

18

20

21

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

Bus #

0

0.25

0.5

0.75

1

hˆαii

3478

12

15

16

18

20

21

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

Bus #

0

0.25

0.5

0.75

1

hˆαii

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.

1

16

327

358

392

458

507

537

588

665

684

717

771

777

866

949

1114

1143

1256

1384

Bus #

0

0.25

0.5

0.75

1

hˆαii

1

16

327

358

392

458

507

537

588

665

684

717

771

777

866

949

1114

1143

1256

1384

Bus #

0

0.25

0.5

0.75

1

hˆαii

1

16

327

358

392

458

507

537

588

665

684

717

771

777

866

949

1114

1143

1256

1384

Bus #

0

0.25

0.5

0.75

1

hˆαii

1

16

327

358

392

458

507

537

588

665

684

717

771

777

866

949

1114

1143

1256

1384

Bus #

0

0.25

0.5

0.75

1

hˆαii

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.

V. CONCLUSIONS

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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