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Power System Frequency Measurement for Frequency Relaying

Spark Xue, Bogdan Kasztenny, Ilia Voloh and Dapo Oyenuga

Abstract— Frequency protection is an important part of the

art of power system relaying. On one hand it covers rotat-

ing machines and other frequency-sensitive apparatus from

potential damage or extensive wear. On the other hand it is

a part of load shedding schemes protecting the system. Unlike

many other protection signals, power system frequency is not

an instantaneous value. Moreover, there is no unambiguous

deﬁnition of power system frequency assuming system tran-

sients and multi-machine systems. This paper discusses the

power system frequency deﬁnition, signal models for frequency

measurement, frequency measurement algorithms and funda-

mentals of frequency relaying. First, the concept of power

system frequency is discussed and clariﬁed in the context of a

large scale system. The mathematical expressions of the system

frequency and instantaneous frequency are presented on the

basis of different signal models. A summary of the requirements

for frequency measurement in applications such as under/over

frequency relays, synchrocheck relays, phasor measurement

units is given. Second, typical frequency measurement methods

are reviewed and the performance evaluation are discussed.

Third, simulation tests of the typical algorithms are provided

to demonstrate various aspects of the frequency measurement,

including the metering accuracy, the time response, the tracking

capability, and the performance under noisy/harmonics condi-

tions. In the end, some practical aspects in designing and testing

the frequency relays are discussed.

I. INTRODUCTION

Frequency is an important parameter in power system to

indicate the dynamic balance between power generation and

power consumption. The system frequency and its rate-of-

change are used directly for generator protection and system

protection. When there are disturbances or signiﬁcant load

variations, the under/over-frequency relays could trip the

units to avoid damage to the generators. When the system

is about to lose its stability, the under-frequency relays

can help to shed off non-critical loads so that the system

balance could be restored. Power system stability can also

be improved by installing power system stabilizers (PSS).

A PSS could use the frequency of voltage signal taken at

the generator terminal to derive the rotor speed, so that the

excitation ﬁeld and the power output of a generator can be

adjusted by a feedback control scheme. In addition to direct

usage in protection and control schemes, the function of

frequency tracking is an indispensable part of modern digital

relays because many numerical algorithms are sensitive to the

variation of fundamental frequency. For example, the digital

Fourier Transform (DFT) is widely used to compute phasors

of voltage and current signals. If the sampling frequency

is not the assumed multiple of signal frequency, leakage

Spark Xue, Bogdan Kasztenny, Ilia Voloh and Dapo Oyenuga are

with GE Multilin, 205 Anderson Road, Markham, Ontario, Canada

http://www.geindustrial.com/multilin/

error would occur in phasor estimation. Without proper

compensation, the overall performance of the protective relay

will be impacted. Highly accurate and stable frequency mea-

surement is always desirable for power system applications.

However, the dynamic characteristic of signals in power

system have brought challenge in designing a frequency

estimation algorithm that is accurate, fast and stable under all

kinds of conditions. To tackle this problem, researchers have

proposed many numerical algorithms, such as zero crossing,

DFT with compensation, phase locked loop, orthogonal de-

composition, signal demodulation, Newton method, Kalman

ﬁlter, neutral network etc. This paper provides a review on

the concept of power system frequency, the frequency mea-

surement algorithms and some fundamental aspects related

to frequency relaying. The paper is organized as follows: The

concept of frequency is discussed in section II. Section III

and IV describe the signal models and the requirement for

frequency relaying. Frequency measurement algorithms and

the performance evaluation are reviewed in section V and VI.

Section VII presents some simulation test results with respect

to four selected algorithms. The frequency relay design and

test are discussed in section VIII. Summary is given in the

end.

II. THE CO NCEPT OF FREQUE NCY

A. The general deﬁnition and instantaneous frequency

The general deﬁnition of frequency in physics is the

number of cycles or alternations per unit time of a wave or

oscillation. Assuming a signal has N cycles within a period

of ∆t, its frequency will be

f=N

∆t.(1)

From this general deﬁnition, one can derive that the signal

needs to be periodical and the frequency is not an instanta-

neous quantity. However, it is common that frequency is used

to characterize arbitrary signals including aperiodic signals.

Meanwhile, the term of instantaneous frequency is seen from

time to time in the literature. As a matter of fact, many

frequency estimation algorithms are based on the concept of

instantaneous frequency. These paradoxes can be resolved by

extending the deﬁnition of frequency.

Using Fourier transform, an arbitrary signal can be de-

composed into a weighted sum of periodic components in

the form of sine / cosine waves. Let the signal be s(t)in

time domain, its frequency domain correspondence is

S(f) = Z−∞

+∞

s(t)e−j2πf tdt. (2)

where a particular S(f0)gives the amplitude of the com-

ponent that has a frequency f0. If the signal is strictly

periodic, it has one fundamental frequency. If the signal

is aperiodic, it has multiple frequencies or even inﬁnite

number of frequencies. The frequency of each sinusoidal

component follows the general deﬁnition. This way, the

frequency deﬁnition is extended for aperiodic signals.

For a sinusoid signal in the form of s(t) = Asin(ϕ), it

can be viewed as the projection of a rotating phasor to the

imaginary axis of a complex plane. The angular speed of

the phasor is dϕ

dt . As a rotating phasor, the recurrence of the

signal value means that ϕis increased by 2π. Because of this,

the phasor repeats ∆ϕ

2πtimes during ∆tand the frequency

is 1

2π

∆ϕ

∆taccording to the frequency deﬁnition in Eq. (1).

Taking the limit ∆t→0, the instantaneous frequency is

deﬁned as

f(t) = 1

2π

dϕ

dt .

The above two extensions of frequency deﬁnition have

played important roles in the area of signal processing. The

Fourier transform in Eq. (2) is meaningful for stationary

signals that the spectrum are constant in a window of time.

For a non-stationary signal that the spectrum are time-

varying, the instantaneous frequency can be used to charac-

terize it. However, the concept of instantaneous frequency is

controversial and application related. For example, a complex

signal s(t)with following form

s(t) = A(t)ejϕ(t)

has both amplitude and phase that are time-varying. When

the signal is to be reconstructed from the sample values, it

could be written either in amplitude-modulation form

s(t) = A(t)ej2πf t,(3)

or phase-modulation form

s(t) = A0eφ(t),(4)

where A(t)is a time-varying and A0is a constant. The

instantaneous frequencies corresponding to Eq. (3) and Eq.

(4) would be completely different. It shows that the instan-

taneous frequency needs to be deﬁned in the context of a

speciﬁc application.

B. Frequencies for power system

In power system, the voltage or current signals for fre-

quency measurement are originated from the synchronous

machines (generators) whose rotating speed are proportional

to the frequency of the generated voltage. The mechanical

frequency of a generator is its rotor speed

fm=1

2π

dθm

dt .

where θmis the spatial angle of the rotor. The frequency of

the generated voltage is

fe=1

2π

dθe

dt .

where θeis the electrical angle that is proportional to θm

of a n-pole machine. From these equations, the frequency

has clear physical meaning for a stand-alone generator.

Both general deﬁnition of frequency and the extension of

instantaneous frequency ﬁt well in this case.

It is natural to extend the instantaneous frequency notation

of the generator internal voltage to any nodes in the system.

Using rotating phasor ~u(i, t)to represent node voltage, the

instantaneous frequency of the ith node in the system can

be deﬁned as the phasor rotating speed,

f(i, t) = 1

2π

d

dt tan−1µIm(~u(i, t))

Re(~u(i, t)) ¶(5)

where Im(~u(i,t))

Re(~u(i,t)) is the phasor rotating angle of the voltage

signal on the complex plane. The frequency for current

signal has the same expression. In power system, it is more

meaningful to use a single quantity to represent the three

phase signals. [12] proposed to use a composite space phasor

derived from αβ transform to represent the three-phase

signal. The composite phasor is actually the scaled positive

sequence component.

~up= 1/√3(u1(t) + αu2(t) + α2u3(t))

where α= exp(j2π/3). The frequency is still deﬁned

as the rotating speed of phasor ~upas in Eq. (5). Using

positive sequence component, not only all three phases can

be handled at the same time, the impact of 3rd harmonics and

dc component to the frequency measurement is also reduced.

For frequency relaying in most cases, the system frequency

is the target as it is used to reﬂect the power balance of the

system or a region. Since the frequency is obtained from

each individual node, a question arises: can the measured

frequency be taken as system frequency? The answer is yes

and no, depending on the system condition, the application

and the frequency estimation method.

In a power system, if the power generation and con-

sumption are perfectly balanced and all the generators are

in synchronism, the frequency of any node can be taken

as system frequency. However, a power network is such

a dynamic system that unbalance between generation and

load always exists. Especially, when there is a disturbance

such as fault on a critical transmission line or loss of a

large generating unit, the balance between the generation and

the load would be temporarily disturbed. Consequently, the

power balance at each individual generating unit would be

different. From [46], the swing equation of the ith generator

in a multi-machine system is

Mi

d∆ωi

dt =Pmi −Pei −Pdi.

where Miis the inertia coefﬁcient of the ith machine, Pmi,

Pei and Pdi are the mechanical power, electrical power and

damping power respectively. This equation tells that rotor is

accelerated or decelerated by the power unbalance (Pmi −

Pei) and the power Pdi absorbed by the damping forces.

The corresponding frequency would differ from generator

to generator. During the electromechanical dynamics in the

system, a generator that is close to the disturbance will have

an instantaneous rotor speed variation in response to the

disturbance. But for the generators far away, the rotor speed

and mechanical power output would not change at the ﬁrst

instant. The frequency difference will cause electromechan-

ical wave propagation in the network to produce different

frequency dynamics at different nodes in the system. From

the simulation test in [63], the speed of the frequency wave

propagation is 400-600 miles/sec for a 1800MW loss in

Eastern US system.

Therefore, the node frequency is a local quantity that may

not fully represent the system frequency, which is a global

value that can be deﬁned as the weighted average of the

node frequencies or the equivalent frequency at the center of

inertia [70],

fs=PN

i=1 Hifi

PN

i=1 Hi

(6)

where Hiis the inertia constant of the ith generator or

the equivalent generator of a region. The averaging process

in this equation should be carried out over all locations

for a ﬁxed time window to yield the system frequency.

Nowadays this has become possible by utilizing a group

of GPS-synchronized PMUs that are connected through

high speed communication network. However, for practical

reason, the system frequency is usually approximated by the

time averaging of the frequency of an individual node,

fs≈fi=1

t−t0Zt

t0

f(i, t)dt. (7)

From Eq. (6) and Eq. (7), the system frequency is not an

instantaneous value. However, the concept of instantaneous

frequency can still be used in some frequency estimation

algorithms. The average value of the estimated frequency

can be used to approximate system frequency. This leads to

another issue: the frequency results from different intelligent

electronic devices (IEDs) could be different. Some IEDs

are based on the periodic characteristic of the signal, some

are based on the concept of instantaneous frequency. Dif-

ferent algorithms also have different accuracy and different

response to harmonics, noise, time-varying amplitude, etc.

Therefore, the measured frequency at a node in the system

should be called as apparent frequency, which is a reﬂection

of the actual node frequency in the IEDs. For most IEDs,

it is usually a window of sample values that are used to

compute the frequency and the results are usually smoothed

by moving average ﬁlters. This way, the apparent frequency

would be close to node frequency and system frequency.

In brief, the node frequency, generator frequency, system

frequency and apparent frequency are different quantities,

even though their value could be very close particularly

under very slow system disturbances and in steady states.

To understand the difference would be helpful in design and

test of frequency-related applications in power system.

III. SIG NAL MOD ELS F OR FR EQU ENCY MEASURE MEN T

The modeling of signals is the ﬁrst step to the frequency

measurement problem. As a mathematical description of

signal, a model would establish the relationship between

the unknown parameters and the observed sample values.

The signal models that are commonly used for frequency

measurement are summarized in this section.

A. Basic signal model

The most widely used signal model in power system is a

voltage signal expressed by

v(t) = Acos(ωt +ϕ).(8)

where Ais the amplitude, ωis the angular frequency and

ϕis the phase angle. For a stationary signal, the frequency

is simply ω/2π. For a non-stationary signal, the frequency

and phase angle can not be considered separately from this

model. Some algorithms would take ωand ϕas two variables

but estimate them simultaneously; Some would use a ﬁxed

value for ϕand leave only ωas the only variable within the

cosine function; Some algorithms would take ωas nominal

frequency and compute the frequency deviation from the

phase angle variation

f=f0+1

2π

dϕ

dt ,(9)

which is in line with the expression of instantaneous fre-

quency.

B. Signals models with harmonics, noise and decaying DC

component

Inevitably, the voltage or current signal in a power system

could be contaminated by harmonics, noise and dc compo-

nent. Some frequency measurement algorithms assume those

should be handled by separate ﬁlters. Some just include them

in the signal model

v(t) = A0e−t/τ +

M

X

k=1

Ak(t) sin(ω0kt +ϕk(t)) + ε,

where A0is the initial amplitude of the dc component that

has time constant τ,Ak(t)represents the amplitude of the

k-th harmonic, εis noise and Mgives the maximum order

of the harmonics. From this model, the frequency deviation

can be regarded as phase angle change of fundamental

component

f=f0+1

2π

dϕ1(t)

dt ,

where ϕ1(t)represents the phase angle of fundamental

component that is slightly different from Eq. (9).

C. Complex signal model from Clarke transformation

In a power system, it is meaningful to measure the

frequency for all three phases simultaneously. Instead of

combining the measuring results from three single-phase

voltages, the Clarke transformation is used to modify the

three-phase system to a two-phase orthogonal system,

·vα

vβ¸=2

3·1−1

2−1

2

0√3

2−√3

2¸

va

vb

vc

.

From the above equation, either vαcan be used alone for

frequency measurement, or a composite signal v=vα+jvβ

can be used. Using Clarke transformation, not only the three-

phase signals could be considered at the same time, the com-

posite signal in complex form could also be useful in some

algorithms such as [2], [6], [48], [57]. The model is also

less susceptible to harmonics and noise. The disadvantage is

that system unbalance could bring errors to the transformed

signal.

D. Signal model using positive sequence component

In [11], [47], [57], the positive sequence voltage is used

for frequency estimation. The positive sequence component

has the same advantage as the composite signal from αβ

transform. Actually, they are equivalent under system bal-

ance condition. There are various solutions to compute the

sequence components from sample values. Since the positive

sequence voltage V1is a space vector rotating with angular

speed 2πf , the frequency can be computed by

f=1

2π

d

dtarg(V1).

IV. FRE QUE NCY RE LAYIN G AND T HE REQ UIREMENTS

The main applications of frequency relaying include un-

der / over frequency relays for generator protection or

load shedding schemes, the voltage / frequency (V/Hz)

relays for generator/transformer overexcitation protection,

synchrocheck relays, synchrophasors and any phasor-based

relays that incorporating frequency tracking mechanism for

accurate phasor estimation.

When there is an excess of load over the available gen-

eration in the system, the frequency drops as the generators

would slow down in attempt to carry more load. If the un-

derfrequency or overfrequency condition lasts long enough,

the resulted thermal stress and vibration could damage the

generators. If the load / generation unbalance is severe, the

generator shall be tripped by its unit protection, which could

consequently worsen the system unbalance condition and

lead to a cascading effect of power loss and system collapse.

On one hand, the underfrequency relays can be used to trip

the generators when the system frequency is close to the

withstanding limits of the units. On the other hand, the under-

frequency relays can be used to automatically shed some

pre-determined load so that the load / generation balance

could be restored. Such load shedding action must be taken

promptly so that the remainder of the system could recover

without sacriﬁcing the essential load. Most importantly,

the action shall be fast enough to prevent the cascading

of generation loss into a major system outage. For this

type of applications, the frequency relays should have high

accuracy because as little as 0.01Hz of frequency deviation

could represent tens of megawatts in power unbalance. It

is generally required that a frequency relay has a 1mHz

resolution. Meanwhile, the frequency measurement must be

stable and robust under various conditions.

When a generator unit is under AVR control at reduced

frequency during unit start-up and shutdown, or under over-

voltage conditions, the magnetic core of the generator or

transformer could saturate and consequent excessive eddy

currents could damage the insulation of the generator /

transformer. To prevent this, relays based on volts / hertz

measurement can be deployed to detect this over-excitation

condition. The accuracy requirement is the same as the

underfrequency relay.

The synchrocheck relays are used to supervise the connec-

tion of two parts of a system through the closure of a circuit

breaker. The difference of frequency, phase angle and voltage

need to be within the setting range to prevent power swings

or excessive mechanical torques on the rotating machines.

In general, a setting of around 0.05Hz is sufﬁcient and the

frequency resolution of a synchro-check relay could be in

the range of 10mHz.

For microprocessor-based relays, the frequency tracking

mechanism is critical to phasor estimation. Frequency track-

ing usually indicates that a digital relay can adjust its

sampling frequency according to the signal frequency, in

order to reduce the phasor estimation error. Most digital

relays use phasor estimation as the foundation of protection

functions since phasors can help to transform the differential

equations of electrical circuits into simple algebra equations.

Though the expression of a phasor is independent of fre-

quency, different signal frequency could result in different

phasors. Without frequency tracking, the performance of

the protection functions will be impaired under off-nominal

frequency conditions. For phasor estimation, the frequency

tracking shall be as fast as possible to follow the frequency

variations, under the condition that stability of the frequency

measurement is maintained. In addition, the range of fre-

quency tracking for generator protection needs to be wide

enough to cope with the generator starting-up and shutting-

down.

The frequency and frequency rate-of-change are also inte-

grated part of synchrophasor units for wide area protection

and control. In 2003, an internet-based frequency monitoring

network (FNET) was set up in U.S. to make synchronized

measurement of frequency for a wide area power network

[37], [71]. From the synchronized data collected by FNET,

signiﬁcant system events such as generator tripping can be

located by event localization algorithms [15] that are based

on the traveling speed of the frequency perturbation wave

and the distance between observations points. For this type

of application, the accuracy of frequency metering should be

as high as possible since minor frequency error could mean

hundreds of miles difference in fault localization.

In general, the frequency measurement should have

enough accuracy and good speed. A ±1mHz accuracy is

deemed good enough for most frequency relaying applica-

tions. However, the ±1mHz accuracy is only valid when the

frequency has slow changes. Though fast frequency tracking

could mean less dynamic error, the accuracy and the speed

requirement are mutually exclusive at a certain point. More

error could be produced in pursuit of fast frequency tracking,

especially when the system or signals are under adverse

conditions. In power system, the voltage or current signal for

frequency estimation could be contaminated by harmonics,

random noise, CT saturation, CVT transients, switching

operation, disturbance, electromagnetic interference, etc. It

is imperative that a frequency relay shall not give erroneous

results to cause false relay operation. To summarize, the

following criteria needs to be satisﬁed for frequency relaying,

•The measured frequency or frequency rate-of-change

should be the true reﬂections of the power system state;

•The accuracy of frequency measurement should be

good enough under system steady state and dynamic

conditions;

•The frequency tracking should be fast enough to follow

the actual frequency change, in order to satisfy the need

of intended application;

•The frequency tracking for generator protection should

be wide enough to handle generator start-up and shut-

down process;

•The frequency measurement should be stable and robust

when the signal is distorted.

V. FR EQU ENC Y MEAS UREMENT ALGORITHMS

In the past, a solid state frequency relay can use pulse

counting between zero-crossings of the signal to measure the

frequency. The accuracy could be as high as ±1∼2mHz

[43] under good signal conditions, but the relay is susceptible

to harmonics, noise, dc components, etc. Nowadays, with

the prevalence of microprocessor-based relays and cheaper

computational power, many numerical methods for frequency

measurement were applied or proposed, including:

•Modiﬁed zero-crossing methods [4], [3], [44], [52], [53]

•DFT with compensation [23], [28], [65], [68]

•Orthogonal decomposition [40], [55], [59]

•Signal demodulation [2], [11]

•Phase locked loop [12], [16], [27]

•Least square optimization [7], [34], [49], [62]

•Artiﬁcial intelligence [8], [13], [30], [32], [45], [58]

•Wavelet transform [9], [36], [35], [31], [64]

•Quadratic forms [29], [30]

•Prony method [38], [42]

•Taylor approximation [51]

•Numerical analysis [67]

Some of these methods are brieﬂy reviewed in this section.

A. Zero-Crossing

The zero-crossing (ZC) is the mostly adopted method

because of its simplicity. From the frequency deﬁnition, the

frequency of a periodic signal can be measured from the

zero-crossings and the time intervals between them. A solid

state frequency relay could detect the zero-crossings by using

voltage comparators and a reference signal. In a software im-

plementation, the zero-crossing can be detected by checking

00.005 0.01 0.015 0.02 0.025

-1.5

-1.0

-0.5

0

0.5

1.0

1.5

Zero crossings

Samples

t1 t2 t3

Time (s)

Fig. 1. Zero-crossing detection for frequency measurement

the signs of adjacent sample values. The duration between

two zero-crossings could be obtained from the sample counts

and the sampling interval. Fig. 1 shows the zero-crossing

detection using digital method. The accuracy of ZC could be

inﬂuenced by zero-crossing localization, quantization error,

harmonics, noise and signal distortion. The quantization error

is negligible if high sampling rate and high precision A/D

converter are used. A lowpass ﬁlter can be applied to reduce

the harmonics and noise in the signal. The random error

caused by zero-crossing localization on time axis could be

signiﬁcant if the sampling frequency is not high enough.

[4] proposed to use polynomial curve to ﬁt the neighboring

samples of the zero-crossing. The roots of the polynomial

can be solved by least error squares (LES) method and one

of the roots is taken as the precise zero-crossing on time axis.

The disadvantage of this method is the high computational

cost for curve ﬁtting and polynomial solving. In practice, the

linear interpolation is used mostly, as illustrated in Fig. 1. To

improve the accuracy of ZC, a post-ﬁlter such as a moving

average ﬁlter is usually applied.

The slow response to frequency change is another issue

for ZC since the measured frequency can be updated after at

least half a cycle. In practice, it takes a few cycles to obtain

good accuracy. Including the delay brought by the pre-ﬁlters

and post-ﬁlters, the total latency could be signiﬁcant. A level

crossing method was proposed in [44] to supplement the

ZC by multiple computations of the periods between non-

zero voltage level crossings. It makes use of all the sample

values to improve the dynamic response of the algorithm.

But the method is susceptible to amplitude variations and

signal distortion. In [1], a three-point method is used to sup-

plement the ZC. The frequency can be quickly derived from

three consecutive samples. However, the method is highly

susceptible to noise, harmonics and amplitude variations.

In brief, a zero-crossing method has its advantage of sim-

plicity. But it needs to be supplemented with other techniques

to obtain good accuracy and good dynamic response. In some

cases, the overall algorithm becomes so complicated that the

simplicity of the zero-crossing method has been lost.

B. Digital Fourier Transform

The digital Fourier Transform (DFT) is widely used for

voltage and current phasors calculation. For a discrete signal

v(k), if the DFT data window contains exactly one cycle of

samples, the phasor of fundamental frequency is given by,

Vk=√2

N

N−1

X

n=0

v(k+n−N+ 1)e−j2πn/N (10)

where Nis the number of samples and the subscript k

represents the last sample index in the data window. The

resulted phasor rotates on the complex plane with an angular

speed determined by signal frequency, which can be taken

as instantaneous frequency

f=1

2π

arg[Vk+1]−arg[Vk]

∆t.

where arg[Vk] = tan−1{Im[V k]/Re[V k]}. The phasor

estimation and frequency estimation are highly correlated

to each other. If the design assumes that sampling rate is

an integer multiple of signal frequency, DFT will produce

leakage error on both phasor and frequency measurement

for the signal with off-nominal frequency. Using DFT, a N-

point data sequence in time domain will produce N discrete

frequency bins in frequency domain. If the signal frequency

is not overlapping any of these frequency bins, the ’energy’

from the samples will leak to the neighboring bins. The

closest frequency bin that is used to approximate the signal

frequency will get the most ’energy’. Hence, the leakage

error is introduced into the estimated phasor and frequency.

Fig. 2-(a),(b) present the frequency domains of a 60Hz signal

and a 59Hz signal as the DFT results under the sampling

rate 3840Hz. The corresponding phasors out of DFT are

shown in Fig. 2-(c),(d). Without leakage compensation, the

magnitude and angle for the 59Hz signal oscillate and deviate

from the actual values. In contrast, the magnitude and angle

for the 60Hz signal are straight lines. Almost all DFT-

based frequency measurement algorithms are focusing on

how to reduce or eliminate leakage error. There are four

main approaches:

1) The length of data window is ﬁxed, the sampling

frequency is updated by the estimated signal frequency

[5];

0 20 40 60 80 100 120 140 160 180

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Magnitude

frequency (Hz)

0 20 40 60 80 100 120 140 160 180

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Magnitude

frequency (Hz)

0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16

0.98

0.985

0.99

0.995

1

1.005

1.01

1.015

1.02

Time (s)

Magnitude (pu)

0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16

1

0. 8

0. 6

0. 4

0. 2

0

0.2

Time (s)

Angle (rad)

(a) (b)

(d)(c)

Fig. 2. DFT leakage error illustration (Sampling rate = 3840 Hz). a) DFT

results of a 60Hz signal; b) DFT results of a 59Hz signal; c) Magnitudes

of the phasors for 60Hz / 59Hz signals; d) Angles of the phasors for 60Hz

/ 59Hz signals

2) The sampling frequency is ﬁxed, the length of data

window is updated by the estimated signal frequency

[14], [23];

3) The length of data window is ﬁxed, the data are re-

sampled to ensure one cycle of data in the window

[28];

4) Both the sampling frequency and the length of data

window are ﬁxed, the leakage error is compensated

[4], [47], [65], [69].

The ﬁrst three approaches are based on the fact that leakage

error can be canceled out if the sampling frequency is an

integer multiple of signal frequency, or equivalently, the

DFT data window contains exactly n(n= 1,2, ..)cycles

of samples. Under this condition, the signal frequency will

overlap one of the frequency bins in frequency domain so

that no leakage would occur.

In [5], the variable-rate measurement is proposed for

frequency measurement. A feedback loop is applied to adjust

the sampling frequency until the derived frequency is locked

with the actual signal frequency. Similar to a phase locked

loop, this type of methods can achieve high accuracy since

the feedback loop can force the error towards zero. However,

the feedback could slow down the frequency tracking speed

for a real-time application. With proper hardware and soft-

ware co-design, this method is suitable for on-line frequency

measurement.

In [23], the DFT data window has a variable length

according to the estimated frequency, so that a cycle of

samples could be included in the data window. Since the

sampling frequency is ﬁxed while the signal frequency is

uncertain, it is not guaranteed that the updated data window

would contain one cycle of samples exactly. Therefore, the

leakage error cannot be eliminated by this method. [23]

proposed to use the line-to-line voltage or positive sequence

voltage to reduce the inﬂuence of harmonics and to use a

moving average ﬁlter to smooth the estimation results. This

method is easy to implement and the measurement range

is wide, which is good for generator protection. However,

the accuracy is limited because of the incomplete leakage

compensation.

In [28], the hardware samples are re-calculated into soft-

ware samples so that the data window will always include

a ﬁxed amount of samples for one cycle of signal exactly.

A feedback loop is used to adjust the re-sampling by the

estimated frequency until the error is lower than a threshold.

This method is more accurate than [23] and simpler than

[5]. Again, the feedback loop needs careful design for good

dynamic response in real-time applications.

In [65], [68], a number of successive phasors out of DFT

are utilized to cancel the leakage error without changing the

sampling rate and the data window length. The method in

[68] does not make any approximations to cancel out the

leakage error so that high accuracy can be achieved. The

details of this algorithm is given in the Appendix. From

the simulation tests, this method can achieve both high

accuracy and good dynamic response, but it is susceptible

to harmonics, noise and dc component.

In summary, DFT can be used to estimate fundamental

frequency, phasor and harmonics simultaneously, which is

its advantage over the other single-objective algorithms.

However, the leakage effect could have signiﬁcant impact

on the phasor and frequency estimation. The DFT methods

need to be supplemented by compensation techniques for

good accuracy. Comparing various compensation techniques,

the algorithms in [5], [28], [68] are recommended for phasor

and frequency estimation.

C. Signal Decomposition

Like DFT, this group of methods will decompose the

input signal into sub-components so that the problem is

transformed and useful information can be retrieved. The ap-

proaches in [40], [55], [59] would decompose the input signal

into two orthogonal components to derive the frequency after

some mathematical manipulations.

In [40], the input signal is decomposed by a sine ﬁlter and

a cosine ﬁlter,

v1(t) = Asin(2πf t +ϕ),

v2(t) = Acos(2πf t +ϕ).

After taking the time derivatives of these two signals, the

frequency is computed by

f=v2(t)v0

1(t)−v1(t)v0

2(t)

2π(v2

1(t) + v2

2(t)) .(11)

Eq. (11) is accurate hypothetically. However, error could

stem from the signal decomposition and the approximation

of the derivatives. From the frequency response of the sine

/ cosine ﬁlters in Fig. 3, the ﬁlter gains are the same only

at nominal frequency. Due to different ﬁlter gains at off-

nominal frequencies, error will be introduced for frequency

estimation. In [40], a feedback loop is designed to adjust the

ﬁlter gains. After adjustment, good accuracy can be achieved

but only in a narrow range around nominal frequency.

Instead of using sine and cosine ﬁlters, [55] proposed to

use ﬁnite impulse response (FIR) ﬁlters designed by optimal

methods. Different coefﬁcients are used for different off-

nominal frequencies. The coefﬁcients with 1Hz step are

calculated off-line and stored in a look-up table. For other

frequencies, interpolation is performed on-line to adjust the

coefﬁcients. A feedback loop is applied to select the ﬁlters

0 100 200 300 400 500 600

0

5

10

15

20

25

30

35

f (Hz)

Filter Gain

Frequency response of sine filter

Frequency response of cosine filter

Fig. 3. Frequency response of a sine ﬁlter and a cosine ﬁlter

from the measured frequency. The accuracy is improved by

the feedback adjustment. Meanwhile, the harmonics can be

suppressed by the FIR ﬁlters. However, the convergence may

be slow for a real-time application because of the feedback

loop.

Without using feedback loop and orthogonal ﬁlters, [54]

uses a group of FIR ﬁlters to derive the frequency. After pre-

ﬁltering, the input signal is decomposed by an all-pass ﬁlter

and a low pass ﬁlter. The decomposed signals will then pass

through two groups of cascading FIR ﬁlters. The frequency

is then derived from the outputs of the two paths, during

which the error brought by ﬁlter gains are canceled out.

Compared with [40] and [55], there is no error compensation

by a feedback loop and the ﬁlters are ﬁxed so that no extra

storage of coefﬁcients is needed. However, the group delay of

the FIR ﬁlters will slow down the dynamic response, and the

frequency output is highly sensitive to harmonics and noises

so that pre-ﬁlter design is critical to the overall performance

of this method.

In [59], the impact of different ﬁlter gains are canceled

out by a sequence of decomposed signals. After ﬁltering, a

new signal is produced by combining the sub-components.

The historical values of this new signal are utilized to cancel

the impact of ﬁlter gains. The details of this algorithm are

given in the Appendix. Since the inﬂuence of unequal ﬁlter

gains are completely canceled out, high accuracy can be

achieved with this method. It is also simpler and faster in

comparison with other decomposition algorithms. However,

as it is based on the assumption that signal amplitude is

stable for a window of data, the time-varying amplitude of

a non-stationary signal could have impact on its accuracy.

D. Signal Demodulation

Instead of decomposing the input signal, a demodulation

method starts from synthesizing a new signal. Fig. 4 illus-

trates the process of computing the frequency deviation by

signal demodulation method (SDM). After the input signal

is modulated by the reference signal that has nominal fre-

quency, the resulted signal vpcontains a low frequency com-

ponent and a near-double frequency component. Through the

low-pass ﬁlter, the low frequency component vcis retrieved

and the frequency deviation is calculated as the rotating

speed of vc. More details of this method are given in Ap-

pendix. The advantage of SDM is its simplicity and potential

for high accuracy. However, the stopband attenuation of the

low-pass ﬁlter must be high enough to remove the near-

double frequency component. A compromise between the

ﬁlter attenuation and ﬁlter delay must be made for the accu-

racy and the dynamic response of frequency measurement.

Lowp ass

filter

Input si gnal i

e

Compu te frequn cy deviati on

as rotatin g speed of

-j2 0

πω

υυpυc

υc

Fig. 4. signal demodulation method

E. Phase Locked Loop

Phase

detector

Lowpass

filter

Vol tage contol led

oscillator (VCO)

Input

Reference

υpυc

υi

υ0

Fig. 5. phase locked loop

A phase-locked loop (PLL) is a feedback system that

responds to the frequency / phase change of the input signal

by raising or lowering the frequency of a voltage controlled

oscillator (VCO) until its frequency / phase matches the

input signal. A typical PLL is composed by three parts as

illustrated in Fig. 5. From the phase detector, a new signal

xpis produced from the input signal viand the reference

signal v0. The synthesized signal vpcontains a low-frequency

component corresponding to the frequency deviation. Passing

a lowpass ﬁlter, the low frequency component vcis retrieved

and used as error signal to drive the voltage-controlled oscil-

lator (VCO). The oscillation frequency of VCO is adjusted

and the VCO output v0feeds back to the phase detector. The

frequency difference of viand v0will be smaller and smaller

after each feedback until it is zero, which is the locked state

of a PLL.

It is noted that a PLL for frequency measurement is

quite similar to the signal demodulation method. Both use

lowpass ﬁlters to demodulate the synthesized signal to get

the frequency deviation. However, a PLL is characterized

as a feedback system that frequency difference would be

gradually reduced towards zero, which implied that a PLL

could achieve very high accuracy on frequency measurement

at the price of some time delay. Another advantage is that

PLL is insensitive to harmonics and noise because of the

lowpass ﬁlter and the feedback loop.

The critical part of a PLL design for frequency measure-

ment is the phase detector. In [17], the transformed αβ signal

is used as input of the phase detector. In [16], a proportional-

integral (PI) controller is used to improve the performance

and stability of the feedback system. In [27], [18], the phase

detector is consists of a in-phase component and a quadrature

component to estimate the time derivative of the phase angle

directly so that the nonlinear dependency of the error signal

to the phase difference is avoided. With this design, the range

of frequency measurement is wide and the convergence is

claimed to be within a few cycles.

Because of its accuracy and robustness, a PLL can be

applied in a line differential protection scheme for accurate

data synchronization. Combined with the GPS time, the sys-

tem frequency is also utilized to synchronize the data packet

that are exchanging continuously among the relays. For this

type of applications, fast frequency tracking is not desired.

Instead, the accurate and stable frequency measurement will

help the data alignment for relays at different locations [41].

F. Non-linear iterative methods

A number of non-linear iterative methods were proposed

for accurate frequency estimation, including: least error

squares (LES) methods [19], [49], [50], [61], least mean

squares (LMS) methods [25], [48], Newton methods [60],

[62], Kalman ﬁlters [7], [10], [26], [20], [21], [22], steep

descent method [34], etc. A common feature of these meth-

ods is to iteratively minimize the error between the model

estimations and the sample values, so that parameters or

states of the model could be derived.

1) Least Error Squares Method: [49] proposed the LES

technique to estimate the frequency in a wide range. Using

three-term Taylor expansion of Eq. (8) in the neighborhood

of nominal frequency, the voltage signal is turned into a

polynomial,

v(t1) = a11x1+a12 x2+a13x3+a14 x4+a15 x5+a16x6(12)

where v(t1)is the sample value at time t1, the coefﬁcients

a11..a16 are known functions of t1, the parameters x1..x6

are unknowns to be solved. E.g., x1=Acos ϕand x2=

(∆f)Acos ϕ. Using m > 6samples, a linear system with

n= 6 unknowns is set up and the unknowns can be

resolved by LES method. The frequency is obtained by

f=f0+x2/x1or f0+x4/x3.In this algorithm, the

accuracy of frequency estimation is affected by the simpliﬁed

signal model, the size of data window for LES, the sampling

frequency and the truncation of the Taylor expansion. In

addition, the matrix inversion that is used in every block

calculation could bring numerical error in a real-time appli-

cation. In order to improve the estimation accuracy of LES

method, some error correction techniques [50], [61] were

proposed. These techniques would increase the complexity

of the algorithm while the accuracy may still be a problem.

2) Newton Method: The Newton method in [60] takes

the dc component, the frequency, the amplitude and the

phase angle as unknown model parameters and estimate

them simultaneously through an iterative process that aims

at minimizing the error between the sample values and the

model estimations. The updating step is derived from Taylor

expansion and the steepest decent principle. The details

of a Gauss-Newton algorithm are given in the Appendix.

Using Newton methods, good accuracy can be achieved with

moderate number of iterations. Meanwhile, the phasor is

also obtained simultaneously. However, the algorithm may

not converge if the initial estimation of the parameters are

far from the actual values. The dynamic variations of both

amplitude and frequency could also delay the convergence.

To overcome these problems, the auxiliary methods such as

ZC and DFT could be applied to initialize the frequency and

amplitude and to supervise the convergence, as presented

in the Appendix. Using supervised Gauss-Newton (SGN)

method, not only the performance is improved, the frequency

estimation is more robust under adverse signal conditions.

3) Least Mean Square Method: LMS is another type of

iterative algorithm that uses an gradient factor to update

the model parameters. The product of the input and the

estimation error is used to approximate the gradient factor

for each iteration. In [48], the complex signal out of Clarke

transform is used to estimate the frequency. The relationship

between the current estimation ˆvkand previous estimation

ˆvk−1is expressed as

ˆvk= ejω∆Tˆvk−1=wk−1ˆvk−1(13)

The variable wkcan be updated by

wk+1 =wk+µekˆv∗

k−1.(14)

where ek=vk−ˆvkis the error between sample value and

the estimation, ˆv∗

k−1is the complex conjugate and µis the

tuning parameter. When the error ekis small enough, the

frequency is derived from the variable wk,

f(t) = 1

2π∆Tsin−1Im(wk).

Using LMS, both the accuracy and the frequency tracking

speed can be satisfactory. In addition, as a curve ﬁtting

approach, the algorithm is insensitive to noise. However, the

parameter µin Eq. (14) needs to be adjusted to accelerate

the convergence. The main issue of the method in [48] is that

complex model has to be used. As mentioned before, when

there is three-phase unbalance, the complex signal out of

Clarke Transform could cause error in frequency and phasor

estimation.

4) Kalman Filters: Established on stochastic theory and

state variable theory, a Kalman ﬁlter predicts the state and

error covariance one step ahead from the historical obser-

vations, then the state estimation and error covariance are

updated with the new observations. To apply Kalman ﬁlter

for frequency estimation, the critical step is to establish a

state difference equation and a measurement equation to

relate the states and observations. The general expressions

of these two equations are

xk=Axk−1+wk−1,

zk=Hxk+vk,

where xkis a vector of state variables, Arepresents the state

transition matrix, zkis the vector of current observations (the

sample values), His a relation matrix, wkand vkrepresent

the process noise and measurement noise respectively. The

state vector is recursively estimated by a Kalman ﬁlter

equation

ˆ

xk=ˆ

xk−1+Kk(zk−Hˆ

xk−1),

where Kkis the Kalman gain that can be derived from a

set of established procedures as described in [24], [66]. If

the process model is non-linear, the extended Kalman Filter

(EKF) that includes extra steps to linearize the models can

be applied.

In [7], the state difference equation is established on the

basis of a complex model,

x1k

x2k

x3k

=

1 0 0

0x1(k−1) 0

0 0 1

x1(k−1)

x1(k−1)

x2(k−1)

x3(k−1)

,

which includes three state variables

x1=ejωTs, x2=Aej ωkTs+j ϕ, x3=Ae−jωkTs−jϕ .

The measurement equation is established as

zk=£0 0.5−0.5¤

x1k

x2k

x3k

+vk.

Using EKF procedures, the state variables can be estimated

and updated. The frequency is derived from the state variable

x1.

A Kalman ﬁlter has the advantage of quick dynamic

response and it can effectively suppress the white noise.

However, the speed of convergence is up to the initial

value of state variables, error covariance matrixes and noise

covariance, which are set according to signal statistics.

The accuracy is also inﬂuenced by the linearization and

the simpliﬁcation of the noise model. The computational

expense of a Kalman ﬁlter is also considerable for a real-

time application.

VI. PE R FO RMA NC E EVALUATION

To evaluate the performance of a frequency relay or

an frequency estimation method, three aspects should be

considered: the accuracy, the estimation latency and the

robustness. The maximum error, the average error and the

estimation delay could be used as the performance indexes

for a frequency relay. The robustness is reﬂected by these

indexes under adverse conditions. Some frequency relays

claim ±1mHz resolution, which shall not be taken as the

performance index. A frequency estimation method could

be extremely accurate when the input signal is stable and

clean, but highly inaccurate when the signal is distorted or

contaminated by harmonics and noise. The accuracy should

be obtained under adverse signal conditions to reﬂect the

robustness of the relay. This also applies to the frequency

estimation latency. Most frequency estimation methods use a

window of data to derive the frequency, which would cause

estimation delay when the frequency is time-varying. It is

desirable that the latency should be as small as possible,

under the restraints of accuracy requirement and robustness

requirement. Because of the latency, the maximum error

could be high while the average error is a better index

to evaluate the relay or the algorithm. The average error

should be taken with a reasonable range according to the

requirement of a speciﬁc application. For a underfrequency

relay, a period of 10 cycles is sufﬁcient to calculate the

average error, as a time delay is usually set for the relay

to make secure operation.

For evaluation purpose, some benchmark test signals can

be used to get the maximum error and average error on the

frequency measurement. The following conditions for setting

up benchmark signals are proposed for the evaluation.

1) The frequency tracking range is 20 −65Hz;

2) To simulate power swing, the signal frequency is

modulated by a 1Hz swing, and the signal amplitude

is modulated by a 1.5Hz swing;

3) The signal is contaminated by 3rd, 5th, 7th harmonics,

the percentage is 5% each;

4) The signal contains dc component, the time constant

could be set as 0.5;

5) The signal contains random noise with signal-to-noise

ratio (SNR) 40dB;

6) The signal contains impulsive noise;

7) To simulate subsynchronous resonance, the signal con-

tains 25Hz low frequency component;

Using individual condition or combined conditions, a number

of analytical signals can be produced to test different aspects

of the frequency relay. In addition to the analytical signals,

the voltage signal or current signals obtained from transient

simulation programs (such as ATP, SIMULINK, RTDS, etc.)

could be used to test the performance of a frequency relay. A

good relay should have consistent performance indexes for

various test signals.

VII. THE SIMULATION TEST

In this section, four frequency estimation algorithms are

compared by simulation tests to demonstrate the advantage

and disadvantage of these algorithms. They are: 1. The

zero-crossing (ZC) method with linear interpolation; 2. The

smart DFT (SDFT) method from [68]; 3. The decomposition

method (SDC) from [59]; 4. The signal demodulation (SDM)

method [11]. The details of these four algorithms are given

in the Appendix. For the SDM, a 6-order Chebyshev type II

ﬁlter is used to achieve more than 100 dB attenuation for high

frequency component with reasonable ﬁlter delay. MATLAB

is used to implement the algorithms and to generate test

signals to disclose different aspects of each algorithm. For all

the discrete test signals, the sampling rate is ﬁxed at 3840Hz.

To make fair comparison, the additional pre-ﬁlters or post-

ﬁlters are not used.

A. Stationary Signal with Off-nominal Frequencies

In power system, a voltage signal under system steady

state is close to a stationary signal that frequency and

amplitude are constants. Using the stationary signals with

off-nominal frequencies, the basic performance of each algo-

rithm can be disclosed. A number of test signals are produced

by

v(t) = sin(2πf t + 0.3),(15)

where f= 61.5Hz, 59.3Hz, 58.1Hz, 45.2Hz , 20.3Hz. Ex-

cluding the initial response, the maximum error of each

algorithm is given in Table I. The tests demonstrate that all

the selected algorithms have the potential to achieve high ac-

curacy for frequency relaying. The ZC, SDFT and SDC have

wide range for frequency metering without compromising the

accuracy. The range for SDM is limited by the lowpass ﬁlter

characteristic. The maximum errors for SDFT and SDC are

almost zero, because the leakage error for SDFT and the

ﬁlter error for SDC are completely canceled out. The error

of ZC is mainly from the zero-crossing detection, which is

improved by linear interpolation. The error from SDM is less

than 1mHz if the frequency deviation is less than 10Hz.

TABLE I

MAX IM UM E RRO RS F ROM S TATIO NA RY SI GNA L TE ST

fZC SDFT SDC SDM

61.5Hz 0.1mHz 5E-9mHz 2E-10mHz 0.72mHz

59.3Hz 0.13mHz 5E-9mHz 2E-10mHz 0.49mHz

58.1Hz 0.04mHz 7E-9mHz 1.4E-10mHz 0.35mHz

45.2Hz 0.04mHz 7E-9mHz 3.5E-10mHz 2.22mHz

20.3Hz 2E-3mHz 4.2E-8mHz 2.2E-9mHz 0.60Hz

B. Tracking the frequency change

During normal operation of the power system, the fre-

quency follows the load / generation variations and ﬂuctuates

around nominal value in a range of about ±0.02Hz. When

there is major deﬁcit of active power in the system, the

frequency would drop at a rate determined by the power

unbalance and the system spinning reserve. For system

protection and unit protection, it is desirable that frequency

change can be detected promptly and accurately. To demon-

strate the frequency tracking capability, a few test signals

with time-varying frequencies are produced and tested.

1) Signal with time-varying frequency: The same equation

as (15) is used to produce the test signals with variable

frequency. To simulate the voltage signal under load / gener-

ation unbalance as the consequence of major generating unit

loss, and to reﬂect the oscillating characteristic of frequency

change, the frequency is expressed by

f(t) =57 + 2(1 + 0.4e−tcos(1.5t−0.1))+

0.2e−7t/10 cos(12t).

Fig. 6 present the frequency tracking results of the four

algorithms. The dash line in each ﬁgure represents the actual

time-varying frequency and the solid line gives the frequency

tracking results. The ZC, SDFT and SDC give better dynamic

accuracy than SDM. In this test, ZC uses half a cycle to

update the frequency so that the estimation delay is small.

For an actual application, more cycles are needed for ZC.

The latency of SDM is from the lowpass ﬁlter delay, which

0 0.5 1 1.5 2

58.8

59

59.2

59.4

59.6

59.8

60

60.2

60.4

Time (s)

Frequency (Hz)

0 0.5 1 1.5 2

58.8

59

59.2

59.4

59.6

59.8

60

60.2

60.4

Time (s)

Frequency (Hz)

0 0.5 1 1.5 2

58.8

59

59.2

59.4

59.6

59.8

60

60.2

60.4

Time (s)

Frequency (Hz)

0 0.5 1 1.5 2

58.8

59

59.2

59.4

59.6

59.8

60

60.2

60.4

Time (s)

Frequency (Hz)

(a) (b)

(d)

(c)

Fig. 6. Track the frequency that drops dynamically, using: (a) ZC; (b)

SDF; (c) SDC; (d) SDM.

also has obvious effect on the initializing stage when signal is

applied. The maximum dynamic error caused by estimation

latency is less than 0.08Hz for SDM and less than 0.04HzHz

for ZC, SDFT and SDC. This test case demonstrates that

all these algorithms are performing well with time-varying

frequency.

2) Both frequency and amplitude are time-varying: In

addition to the frequency dynamics, the dynamics of signal

amplitude could also have impact on a frequency estimation

algorithm. In power system, the voltage signal is generally

used for frequency estimation since it is more stable than

the current signal. But there are cases that only current

signals are available, such as line differential relays or bus

differential relays in some substations. What’s more, when

the system is experiencing asynchronous oscillations, the

voltage signal and current signal would oscillate in both

frequency and amplitude. To verify the performance of fre-

quency estimation algorithms under power swing conditions,

a signal that is time-varying in both frequency and amplitude

is produced per following equations

f(t) = 59.5 + sin(2πt), A(t) = √2+0.3 cos(3πt),

where the amplitude is modulated by a 1.5Hz swing and

the frequency is modulated by a swing of 1.0Hz. The

simulation results of the four algorithms are shown in Fig.

7. In principle, ZC is not affected by the variation of

signal amplitude. For SDFT, the impact of the time-varying

amplitude is minor since three successive phasors used for

each round of estimation will have similar magnitudes under

high sampling rate. Though SDM still has obvious latency

in frequency tracking, it is not caused by the amplitude

variations. More dynamic error occurs for SDC because the

algorithm assumes that sample values are all the same in the

data window.

3) The frequency step test: The step response test is

useful to disclose the characteristics of a signal process

algorithm on how it responds to signal changes in time

domain. However, the step test for power system frequency

0 0.5 1 1.5 2

58

58.5

59

59.5

60

60.5

61

Time (s)

Frequency (Hz)

0 0.5 1 1.5 2

58

58.5

59

59.5

60

60.5

61

Time (s)

Frequency (Hz)

0 0.5 1 1.5 2

58

58.5

59

59.5

60

60.5

61

Time (s)

Frequency (Hz)

0 0.5 1 1.5 2

58

58.5

59

59.5

60

60.5

61

Time (s)

Frequency (Hz)

(a)

(d)

(c)

(b)

Fig. 7. Track the frequency when both amplitude and frequency are time-

varying, using: (a) ZC; (b) SDF; (c) SDC; (d) SDM.

0.4 0.45 0.5 0.55 0.6 0.65

59.4

59.5

59.6

59.7

59.8

59.9

60

Time (s)

Frequency (Hz)

0.4 0.45 0.5 0.55 0.6 0.65 0.7

59.4

59.5

59.6

59.7

59.8

59.9

60

Time (s)

(b)

Frequency (Hz)

0.4 0.45 0.5 0.55 0.6 0.65

59.4

59.5

59.6

59.7

59.8

59.9

60

Time (s)

Frequency (Hz)

0.4 0.45 0.5 0.55 0.6 0.65 0.7

59.4

59.5

59.6

59.7

59.8

59.9

60

Time (s)

Frequency (Hz)

(d)(c)

(a)

Fig. 8. Track the frequency that has step change, using: (a) ZC; (b) SDF;

(c) SDC; (d) SDM.

estimation has no correspondence in real life. Due to the

mass inertia of the rotating machines in the system, it is

impossible for the system frequency to have any signiﬁcant

step change. Therefore, an analytical signal with 0.5Hz step

change is more than enough to test the frequency estimation

algorithms. Fig. 8 gives the measured frequency from four

algorithms in response to the MATLAB test signal that

changes from 60Hz to 59.5Hz within one sampling interval.

The transition period for the ZC, SDF and SDC are within

2 cycles while it takes about 10 cycles for SDM to settle

down. The slow response of SDM demonstrate the impact

of the lowpass ﬁlter that SDM is relying on.

C. Signal containing harmonics, noise and dc component

For a power system signal, the odd number harmonics

such as 3rd, 5th, 7th, .. harmonics are most likely to occur

due to widely-used power electronics and nonlinear load. For

system with series-compensated capacitors, the signal may

contain low frequency component due to subsynchronous

resonance. What’s more, the signal could be contaminated by

noise originated from system faults, switching operations, or

the electronic circuits. To compare the selected algorithms, a

number of discrete signals are produced to simulate extreme

conditions in a power system.

1) Signal containing 3rd, 5th, 7th harmonics: A signal

with 3rd, 5th, 7th harmonics are produced per following

equation,

v(t) =√2 sin(2πf t + 0.3) + 0.05√2 sin(6πf t)

+ 0.05√2 sin(10πf t) + 0.05√2 sin(14πf t)

where f= 59.5. In the equation, the magnitude of fun-

damental frequency component is 1.0 p.u.. The percentage

of 3rd, 5th, 7th harmonics are 5% each. The test results

are shown in Fig. 9. The performance of SDM is the best

among the four, simply because of the lowpass ﬁlter. ZC

is less susceptible to harmonics as the zero-crossings on

the time axis are mainly determined by the fundamental

frequency component, even though the signal is distorted by

harmonics. SDFT is susceptible to harmonics because the

leakage error of DFT from high frequency components are

not compensated. One solution is to add harmonics in the

0.2 0.22 0.24 0.26 0.28 0.3

58.5

59

59.5

60

60.5

Time (s)

Frequency (Hz)

0.2 0.22 0.24 0.26 0.28 0.3

58.5

59

59.5

60

60.5

Time (s)

Frequency (Hz)

0.2 0.22 0.24 0.26 0.28 0.3

58.5

59

59.5

60

60.5

Time (s)

Frequency (Hz)

0.2 0.22 0.24 0.26 0.28 0.3

58.5

59

59.5

60

60.5

Time (s)

Frequency (Hz)

(a)

(d)

(c)

(b)

Fig. 9. Frequency measurement results from signal containing 3rd, 5th,

7th harmonics, using: (a) ZC; (b) SDF; (c) SDC; (d) SDM.

signal model [69] and to compensate the leakage error with

the same technique as for fundamental frequency. Another

solution is to use a lowpass ﬁlter to pre-process the signal

and/or to use a moving average ﬁlter to smooth the results.

SDC is better than SDFT, but harmonics still make signiﬁcant

difference for it because the signal model and the frequency

equation in SDC are all based on the fundamental frequency

component.

2) Signal containing low frequency component: In power

system, the interaction between the turbine-generators and

series capacitor banks or static VAR control system could

cause subsynchronous resonance that introduces low fre-

quency component into the voltage signal for frequency

measurement. The low frequency component ranged from

10Hz to 45Hz could last long enough to cause problem to

a frequency relay. To investigate the response of different

algorithms, a signal with 10% of 25Hz components is

produced per following equation.

v(t) = √2 sin(2πf t + 0.3) + 0.1√2 sin( 25

30πft)

Part of the test signal is shown in Fig. 10 and the test

results are shown in Fig. 11. It turns out that subsynchronous

resonance could have serious impact on all the frequency

estimation algorithms. SDM is performing the best relatively

while the errors from ZC, SDFT and SDC are unacceptable.

One solution is to detect the low frequency component by

DFT and use a notch ﬁlter to remove it. However, the DFT

window has to be long enough to spot the low frequency

component. Another solution is to design a frequency secu-

rity logic to ignore large frequency variations within a few

cycles.

3) Signal containing dc component: After a system dis-

turbance or a switching operation, the voltage signal for

frequency measurement could contain dc component that

decays exponentially. A test signal is produced per following

equation,

v(t) = 0.5e−t/0.3+√2 sin(2πf t +π/6).

0.20 0.22 0.24 0.26 0.28 0.30 0.32 0.34 0.36 0.38 0.40

-1.5

-1.0

-0.5

0

0.5

1.0

1.5

Time (s)

Voltage (p.u.)

Fig. 10. Part of the test signal containing low frequency component.

0.2 0.25 0.3 0.35 0.4 0.45 0.5

54

56

58

60

62

64

66

68

Time (s)

Frequency (Hz)

0.2 0.25 0.3 0.35 0.4 0.45 0.5

54

56

58

60

62

64

66

68

Time (s)

Frequency (Hz)

0.2 0.25 0.3 0.35 0.4 0.45 0.5

54

56

58

60

62

64

66

68

Time (s)

Frequency (Hz)

0.2 0.25 0.3 0.35 0.4 0.45 0.5

54

56

58

60

62

64

66

68

Time (s)

Frequency (Hz)

(a)

(d)

(c)

(b)

Fig. 11. Frequency measurement results from signal containing low

frequency component, using: (a) ZC; (b) SDF; (c) SDC; (d) SDM.

The test results shown in Fig. 12 indicate that DC component

can have signiﬁcant impact on ZC, since the time interval of

zero-crossings are signiﬁcantly changed by dc component.

The SDM, SDFT and SDC all use the signal waveform to

derive the frequency so that the impact of dc component

is less. In most applications, a bandpass ﬁlter is applied to

remove the dc component at the price of extra delay.

0.2 0.4 0.6 0.8 1

57

58

59

60

61

62

63

Time (s)

Frequency (Hz)

0.2 0.4 0.6 0.8 1

57

58

59

60

61

62

63

Time (s)

Frequency (Hz)

0.2 0.4 0.6 0.8 1

57

58

59

60

61

62

63

Time (s)

Frequency (Hz)

0.2 0.4 0.6 0.8 1

57

58

59

60

61

62

63

Time (s)

Frequency (Hz)

(a)

(d)

(c)

(b)

Fig. 12. Frequency measurement results from signal with dc component,

using: (a) ZC; (b) SDF; (c) SDC; (d) SDM.

0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45

Time (s)

Fig. 13. Signal with impulsive noise

0 0.2 0.4 0.6 0.8 1

58

58.5

59

59.5

60

60.5

61

Time (s)

Frequency (Hz)

0 0.2 0.4 0.6 0.8 1

58

58.5

59

59.5

60

60.5

61

Time (s)

Frequency (Hz)

0 0.2 0.4 0.6 0.8 1

58

58.5

59

59.5

60

60.5

61

Time (s)

Frequency (Hz)

0 0.2 0.4 0.6 0.8 1

58

58.5

59

59.5

60

60.5

61

Time (s)

Frequency (Hz)

(a)

(d)

(c)

(b)

Fig. 14. Frequency measurement results from signal with impulsive noise,

using: (a) ZC; (b) SDF; (c) SDC; (d) SDM.

4) Signal containing impulsive noise: The voltage sig-

nal for frequency measurement could be contaminated by

impulsive noise or white noise. By changing the randomly-

selected sample values, a test signal with impulsive noise

is produced to test the frequency measurement algorithms.

Fig. 13 presents a portion of the signal. The frequency

measurement results of the four algorithms are shown in Fig.

14. The impulsive noise will have adverse impact to all the

four algorithms. SDM is comparatively better but the results

are still unacceptable. For ZC, the impulsive noise only make

inﬂuence when it is around zero-crossings. To resolve the

problem, either the impulsive noise shall be removed at the

pre-processing stage, or the singular frequency estimates can

be discarded at the post-processing stage.

5) Signal containing white noise: White noise could be

introduced by the electromagnetic interference or the dete-

rioration of the electronic components. A test signal with

white noise is produced by

v(t) = Asin(2πf t + 0.3) + ε

The parameter εrepresents the noise that can be produced by

a random function on MATLAB. The signal to noise ratio is

SNR = 20 log(1/0.01) = 40dB. The test results are shown

in Fig. 15. Again, SDM demonstrate its strong anti-noise ca-

pability, which attributes to the high stopband attenuation of

the lowpass ﬁlter for SDM. ZC is relatively less susceptible

to white noise. For SDFT and SDC, additional ﬁlters must

be applied to reduce the error caused by random noise.

0 0.2 0.4 0.6 0.8 1

58

58.5

59

59.5

60

60.5

61

Time (s)

Frequency (Hz)

0 0.2 0.4 0.6 0.8 1

58

58.5

59

59.5

60

60.5

61

Time (s)

Frequency (Hz)

0 0.2 0.4 0.6 0.8 1

58

58.5

59

59.5

60

60.5

61

Time (s)

Frequency (Hz)

0 0.2 0.4 0.6 0.8 1

58

58.5

59

59.5

60

60.5

61

Time (s)

Frequency (Hz)

(a)

(d)

(c)

(b)

Fig. 15. Frequency measurement results from signal with white noise,

using: (a) ZC; (b) SDF; (c) SDC; (d) SDM.

D. Signal from power system simulation

The above test cases are based on analytical signals to

disclose different aspects of the selected algorithms for

frequency estimation. To simulate a real system, a test signal

is generated from a simulation model as shown in Fig. 16.

In this two-source model, one of the sources is a 200MVA

synchronous machine that is controlled by hydraulic turbine,

governor and excitation system. The other source is a sim-

pliﬁed voltage source with short circuit capacity 1500MVA.

The system is initialized to start in a steady-state with the

generator supplying 200MW of active power to the load.

After 0.55s, the breaker that connects the main load to

the system is tripped. Because of the sudden loss of load

and the inertia of the prime mover, the generator internal

voltage starts to oscillate until the generator control system

damps the oscillations. The frequency tracking results from

the four algorithms are shown in Fig. 17. The rotor speed

is shown in (a) and the frequency tracking results from the

four algorithms are plotted in (b)-(d). Before the breaker is

tripped, the voltage signal is stable and the frequency output

of each algorithm is exactly 60.0Hz. After the breaker is

tripped, both the voltage signal and the current signal start to

oscillate. Since the voltage is measured at generator terminal,

the frequency change should reﬂect the change of the rotor

speed. From Fig. 17, all the four algorithm can tracking

the frequency change that is in line with the rotor speed.

G

Hydrau lic

Turbine

Excitat ion

System

Volt age

Meter

Volt age Source

1500MVA, 23 0kV

Transformer

210MVA

13.8kV/2 30kV

CB

Load

Synchron uous

Machin e

200MVA/1 3.8kV

Fig. 16. A two-source system model.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

0.98

1

1.02

Time (s)

Speed (p.u.)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

59

60

61

Time (s)

Frequency (Hz)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

59

60

61

Time (s)

Frequency (Hz)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

59

60

61

Time (s)

Frequency (Hz)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

59

60

61

Time (s)

Frequency (Hz)

(a)

(a)

(e)

(d)

(c)

(b)

Fig. 17. Frequency measurement results from simulation model test. (a)

Generator rotor speed; (b) Tracking using ZC; (c) Tracking using SDFT;

(d) Tracking using SDC; (e) Tracking using SDM.

There is a frequency jump for each frequency estimation

method after the breaker is tripped. This drastic frequency

variation is caused by the phase abnormity of the signal at the

moment of switching operation and should not be accounted

as the actual node frequency change. Comparing the four

algorithms, SDM gives the most stable and smooth frequency

tracking results. However, its recovery from the abnormal

frequency jump is also the slowest. The results from SDFT

is the worst simply because it is highly susceptible to the

harmonics and noise contained in the signal.

VIII . FRE QUE NC Y REL AY DE SI GN AN D TES T

From the simulation tests, a frequency estimation algo-

rithm alone is not enough to meet the practical requirements

for frequency relaying. In order to obtain stable and accurate

frequency measurement, it is necessary to add digital ﬁlters

and security conditions to process the signal and the fre-

quency estimates. Consequently, latency will be introduced

to the frequency measurement because of ﬁltering delay

and estimation delay. A critical aspect for frequency relay

design is to achieve the balance between the accuracy and the

group delay, under the condition of robustness. This section

discusses a few practical issues about the design and test for

frequency relaying.

A. The ﬁltering and post-processing

For a digital frequency relay, the analog anti-alias ﬁlter is

usually applied to remove the out-of-band components before

the A/D conversion (ADC). After ADC, it is necessary to

add a digital band-pass ﬁlter to remove the harmonics and

dc component. For a 60Hz system, the limiting frequencies

of the ﬁlter passband can be 20 −65Hz. The stopband

attenuation should be as high as possible. However, the ﬁlter

delay could also become signiﬁcant for excessive stopband

attenuation. To compromise with the ﬁlter delay, the average

stopband attenuation can be speciﬁed at 20 −40dB, which

means that dc component and harmonics are suppressed

to 1−10% in average. If the ﬁlter stopband has valleys

corresponding to high attenuation, it is also desirable that

the valleys shall be close to the harmonics.

A band-pass ﬁlter can effectively remove dc component

and harmonics, and helps to reduce white noise to a cer-

tain degree. But it cannot handle the impulsive noise. One

solution for impulsive noise is to use security conditions

at the post-processing stage. Another solution is to use an

impulsive noise detector at the pre-processing stage. The de-

tector can determine the impulsive noise as singular sample

values according to the adjacent samples. Once detected, the

contaminated sample can be replaced by a value that is close

to the adjacent samples.

After the frequency estimation, a post-ﬁlter would be

helpful to get better accuracy, especially when the measured

frequency has minor oscillations around the actual frequency.

In [39], a 80-coefﬁcient Hamming type FIR ﬁlter is applied

for post-ﬁltering. In [33], a binomial ﬁlter is applied. In many

cases, a simple moving average ﬁlter is sufﬁcient to improve

the measurement accuracy. As a matter of fact, a moving

average ﬁlter is the optimal ﬁlter that can reduce random

white noise while keeping sharp step response [56]. However,

the length of the moving average ﬁlter needs to be carefully

selected to achieve the balance between the accuracy and

dynamic response of the overall process.

If the signal is distorted under conditions such as CVT

transients, CT saturation, system disturbance, switching op-

erations or subsynchronous resonance, erroneous frequency

estimates may still exist after the pre-ﬁltering and post-

ﬁltering because the ﬁlters may not handle all the signal

abnormalities. Hence, it is important to have some security

conditions to validate the frequency estimation. For example,

the difference between two consecutive estimates should be

small enough to accept the new estimates; the change of

a few consecutive estimates should be consistent, etc. These

conditions are based on the fact that power system frequency

cannot have drastic change during a few sampling intervals.

The security check should also reject the estimates for the

ﬁrst a few cycles after the input signal is applied to the relay,

because a numerical algorithm needs a few cycles of data to

stabilize the estimates.

B. The df/dt measurement

The frequency rate-of-change (df/dt) is a second criteria in

a load shedding scheme or remedy action scheme to super-

vise or accelerate the load shedding. After the frequency is

estimated, the frequency rate-of-change is simply computed

by the frequency difference and the sampling interval ∆t,

df

dt =f(t)−f(t−1)

∆t.(16)

This equation could amplify the error or the high frequency

component that are contained in the estimated frequency.

Hence a lowpass ﬁlter and/or a moving average are necessary

to ﬁlter the df/dt outputs. After the ﬁltering, some security

conditions similar to those for frequency estimation shall be

used to remove abnormal df/dt values.

C. Some test results of a frequency relay

Two test cases are presented in this section to show

the frequency tracking of an actual relay that is based on

zero-crossing principle. The relay is able to achieve 1mHz

accuracy for steady state signals and can track the frequency

in a large range. To verify its performance under dynamic

conditions, the ﬁrst test signal is produced per following

equation,

v(t) =20e−t/0.5+A(t) sin(2πf (t)t+ 0.3)+

0.05A(t) sin(6πf (t)t)+0.05A(t) sin(10πf (t)t)+

0.05A(t) sin(14πf (t)t) + εw+εp,where

f(t) =60.0 + sin(2πt), A(t) = 40 + 10 cos(3πt).

From this equation, the signal contains dc component

(20e−t/0.5), harmonics, white noise εwand impulsive noise

εp. The harmonics are 3rd, 5th, 7th at 5% each and the

SNR of the white noise is 40dB. In addition, both the

frequency f(t)and amplitude A(t)are time-varying. This

signal represents the extreme condition that is designed to

challenge the relay performance. MATLAB is used to create

the signal and to save it in a comtrade ﬁle, which is then

played back to the relay by a real time digital simulator

(RTDS). From the test results in Fig. 18., the impact of dc

component, harmonics and noise are reﬂected by the delay

and error in frequency tracking. Compared with the previous

simulation test results, the relay has no drastic frequency

change or abnormal frequency values during the tracking

process, which attributes to the ﬁlters and the security check

conditions for frequency estimation.

The second signal is produced by the power system

simulation model as shown in Fig. 16. Both the frequency

and amplitude of the signals oscillate due to the power

swing. Compared with the simulation results in Fig. 17,

the test results in Fig. 19 are more stable. At the moment

when the breaker is tripped, there is no sudden jump in

tracking frequency and the frequency is stable throughout the

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

59

59.2

59.4

59.6

59.8

60

60.2

60.4

60.6

60.8

61

time(s)

Frequency (Hz)

Fig. 18. Frequency measurement from an actual relay in response to

analytical signal

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

59.7

59.8

59.9

60

60.1

60.2

time(s)

Frequency (Hz)

Fig. 19. Frequency measurement from an actual relay in response to power

system simulation signal

power swing. It shows that the relay is able to handle phase

abnormities, frequency / amplitude oscillations, harmonics

and noise, which satisfy the robustness requirement.

D. The test recommendations

To test a frequency relay, the main purpose is to verify

the accuracy of frequency measurement and the operating

time. The accuracy and operating time under different signal

conditions should be tested to verify the robustness of the

relay. The test signals need to be practical to reﬂect the power

system operation conditions. For example, a step test that has

over 0.5Hz step change may not be appropriate since the step

change would not happen to the actual system frequency due

to the inertia of rotating machines. The following test items

are recommended,

1) Inject signal with different off-nominal frequencies.

This test is to verify the steady state accuracy and the

measurement range. The accuracy should be consistent

for different off-nominal frequencies;

2) Inject signal with ramping-down frequency, the ramp-

ing step could be 0.1Hz. If possible, the injected signal,

the measured frequency and the frequency rate-of-

change shall be recorded in comtrade ﬁles for further

analysis;

3) Use the same ramping test as above to check the

operating time of the frequency relay when the signal

frequency is under/over the setting. The ramping step

can be varied in a reasonable range (such as 0.1Hz -

0.3Hz) to check the relay response;

4) Inject the relay with contaminated signal. The 3rd, 5rd,

7th harmonics that are 5% each can be added onto the

main signal. The dc component and random noise shall

also be added if possible; Otherwise, the playback test

can be used instead;

5) If the test equipment supports playback test, a num-

ber of comtrade ﬁles with simulated test signals can

be created and played back to the frequency relay.

MATLAB or MATHCAD could be used to produce

analytical signals per equations in this paper. The

transient simulation software such as EMTP can also

be used to produced signals by using power system

models. The tracking frequency and frequency rate-of-

change can be recorded in comtrade ﬁles to deduce the

maximum error, average error and the response time.

These tests should be designed to verify the robustness

of the frequency relay under adverse conditions.

During the tests, it should be noted that a frequency

relay needs a few cycles to stabilize at the beginning of the

injection test. Hence it is recommended to give the relay at

least ten cycles of stable signal before any signal variations.

The frequency measurement error during this initialization

process should not be accounted.

IX. SUMMARY

As a fundamental characteristic of considered signal, the

frequency and its measurement are important not only to

under-frequency relays but also to other protective relays.

Many numerical algorithms were discussed to pursue both

high accuracy and fast response on frequency measure-

ment. However, it should be emphasized that power system

frequency is not an instantaneous value, even though the

concept of instantaneous frequency could be utilized in some

algorithms. Due to the mass inertia of rotating machines, the

system frequency cannot have step change or fast change.

It is justiﬁable to use a window of data to compute the

average frequency to approximate the system frequency. A

few cycles delay is not only allowed but necessary for robust

frequency relaying. The accuracy of frequency measurement

only makes sense when the signal conditions are speciﬁed. If

the signal is contaminated or distorted, it is more important to

maintain a stable frequency measurement instead of pursuing

fast response. To design a frequency relay, the digital ﬁlters

and security check conditions should be applied to avoid

abnormal frequency output under various conditions, so that

the frequency relay or other protective relays can securely

protect the machines and the system.

X. APPENDIX

A. Zero-crossing with linear interpolation

For zero-crossing method implemented numerically, it is

important to detect the zero-crossings accurately on the time

axis and the linear interpolation is usually applied. A zero-

crossing is found as between two neighboring samples with

different signs. The crossing point of the line that connects

the two samples and the time axis is taken as the zero-

crossing point. The line is expressed as

p(x) = (x−m+ 1)vm−(x−m)vm−1.

where vm−1and vmare sample values, xand mare sample

indexes. Let p(x) = 0, the zero-crossing in term of sample

index is obtained by

x=m−vm

vm−vm−1

.

Though it is still called sample index, xis usually a fractional

value between m−1and m. Let Tsrepresent the sample

interval, the zero-crossing moment on a time axis is t=

xTs. The frequency is calculated from two consecutive zero-

crossings at t1and t2,

f=1

2(t1−t2)=1

2Ts(x1−x2).(17)

B. Yang&Liu’s smart DFT method (SDFT)

In [68], the phasor and frequency are derived from a com-

pensated DFT method that leakage error can be completely

canceled out. The signal model

v(t) = Vcos(2π(f0+ ∆f)t+ϕ)

is ﬁrstly written as

v(t) = vej2π(f0+∆f)t+v∗e−j2π(f0+∆f)t

2(18)

where v=V ejϕ is the phasor , f0is the nominal frequency

and ∆frepresents the frequency deviation. Applying DFT

to v(t), the estimation of fundamental frequency component

is

ˆvr=2

N

N−1

X

k=0

v(k+r)e−j2πk

N(19)

where the subscript ris the index of the ﬁrst sample in the

DFT window. Discretizing Eq. (18) by t=k

f0Nand bring it

into Eq. (19), the estimated phasor is expressed by

ˆvr=Ar+Br(20)

where

Ar=v

N

sin Nθ1

2

sin θ1

2

exp ·jπ

f0N(∆f(2r+N−1) + 2f0r)¸,

Br=v∗

N

sin Nθ2

2

sin θ2

2

exp ·−jπ

f0N(∆f(2r+N−1) + 2f0(r+N−1)¸,

θ1=2π∆f

f0N, θ2=

2π³2 + ∆f

f0´

N.

Applying DFT onto two consecutive data windows, the

following relationship holds,

Ar+1 =Ara, Br+1 =Bra−1(21)

where

a= exp ·j(2π(f0+ ∆f)

f0N¸.(22)

From Eq. (20) and Eq. (21), there are

ˆvr+1 =Ar∗a+Br∗a−1(23)

ˆvr+2 =Ar∗a2+Br∗a−2.(24)

Combining Eq. (20)-(24), the Arand Brcan be eliminated

and a 2nd-order polynomial is formed as

ˆvr+1 ∗a2−(ˆvr+ ˆvr+2)∗a+ ˆvr+1 = 0.

The root is

a=

(ˆv+ ˆvr+2 )±q(ˆv+ ˆvr+2 )2−4ˆv2

r+1

2ˆvr+1

.

From Eq. (21), the frequency is calculated by

f=f0+ ∆f=f0N

2πcos−1(Re(a)).

This method does not make any approximation during DFT.

The leakage error is canceled out from three consecutive

phasors. Hence it is highly accurate for stable signal and

can follow the frequency change with good speed.

C. Szafran’s signal decomposition method (SDC)

In [59], the orthogonal decomposition method is modiﬁed

to eliminate the error caused by unequal ﬁlter gains at off-

nominal frequencies. The signal is ﬁrstly decomposed by

a pair of orthogonal FIR ﬁlters, such as a sine ﬁlter and a

cosine ﬁlters. For an input sinusoidal signal in discrete form,

the outputs yc(n)and ys(n)of the ﬁlters are

yc(n) = |Fc(ω)|Acos(nωTs+ϕ+α(ω)),

ys(n) = |Fs(ω)|Asin(nωTs+ϕ+α(ω)),

where |Fc(ω)|and |Fs(ω)|are the ﬁlter gains at frequency ω.

A new signal gk(ω)can be composed by using the historical

output signals yc(n−k)and ys(n−k),

gk(ω) = ys(n)yc(n−k)−yc(n)ys(n−k),(25)

Again, by utilizing the historical signal g2k(ω), an expression

independent of signal magnitude can be obtained,

g2k(ω)

gk(ω)= 2 cos(kωTs).(26)

Bring Eq. (25) into Eq. (26), the frequency is calculated by

f=1

2πkTs

cos−1µ1

2

ys(n)yc(n−2k)−yc(n)ys(n−2k)

ys(n)yc(n−k)−yc(n)ys(n−k)¶.

(27)

Since the error incurred by orthogonal ﬁlters are canceled

out in Eq. (26), high accuracy can be achieved.

D. Signal demodulation Method (SDM)

Using the simple signal model in Eq. (8), a new signal

Y(t)is generated by multiplying v(t)with a reference signal

that has nominal frequency w0,

Y(t) = v(t)e−jω0t

=A

2ej((ω−ω0)t+ϕ)+A

2e−j((ω+ω0)t+ϕ).(28)

The signal Y(t)has a low frequency component and a near-

double frequency (ω+ω0) component. A low-pass ﬁlter can

be applied to ﬁlter the near-double frequency component

so that the remaining signal y(t)contains the frequency

deviation information. Using discrete form of y(t), another

complex signal U(k)is produced by

U(k) = y(k)y∗(k−1),

where y∗(k−1) is the conjugate of y(k−1). The frequency

is derived by

ˆ

f(k) = f0+fs

2πtan−1Im(U(k))

Re(U(k)) .

E. The supervised Gauss-Newton method

The supervised Gauss-Newton (SGN) method is based on

[60] with two additional auxiliary algorithms, ZC and DFT,

to supervise the Gauss-Newton process. The ZC and DFT are

used to roughly estimate the frequency and signal amplitude

so that the parameters can be properly initialized in each

iteration.

The Gauss-Newton process is an iterative method to esti-

mate the model parameters by minimizing the error between

the estimation and the observation in a least square sense.

Let the parameter vector be xand the objective function

be f(x). Starting from a initial point x0, if the descending

condition |f(xk+1)|<|f(xk)|could be enforced, a series of

vectors x1,x2, ... could be iteratively calculated and xwill

ﬁnally converge to x∗, a minimizer of the objective function

f(x). In SGN, the parameter vector xis selected per basic

signal model in Eq. (8),

x= [A(t)ω(t)ϕ(t)]T.(29)

The process of parameters updating is expressed by

x=x+∆x.(30)

The objective function f(x)is deﬁned as the error between

the estimated signal value vest(x)from Eq. (8) and the

sample value vobv,

f(x) = vest(x)−vobv .

The Gauss-Newton updating step is expressed by

∆x = (JTJ)−1JTf(x).(31)

where J(x)is the Jacobian matrix containing partial deriva-

tives for the estimated parameters. The Eq. (31) will maintain

the decent direction to minimize the objective function

|f(x)|. In order to apply the algorithm in a real-time ap-

plication, it is necessary to stop the iterations when the

error function f(x)is close to zero, or when the gradient

function JTf(x)is close to zero. It can be proven that Gauss-

Newton method can only achieve linear convergence when

xis far from the minimizer x∗while quadratic convergence

is possible when xis close to x∗. If the initial xhas

signiﬁcant difference from x∗, the convergence may not

even be achieved. In order to get proper initial parameters

and fast convergence, two auxiliary algorithms can be used

to provide initial parameters and to supervise each Gauss-

Newton updating step. The recursive DFT method is used

to estimate the signal amplitude and the zero-crossing (ZC)

method is used to estimate the frequency. The results of DFT

and ZC will be close to the actual value but not necessarily

be accurate. With this combined approach, fast convergence

of Gauss-Newton method can be achieved.

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