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Original filename: A New Transient Frequency Acceptability Margin Based on the Frequency Trajectory.pdf
Title: A New Transient Frequency Acceptability Margin Based on the Frequency Trajectory
Author: Ancheng Xue, Jiehao Cui, Jiawei Wang, Joe H. Chow, Lei Yue and Tianshu Bi

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energies
Article

A New Transient Frequency Acceptability Margin
Based on the Frequency Trajectory
Ancheng Xue 1, *, Jiehao Cui 1 , Jiawei Wang 1 , Joe H. Chow 2 , Lei Yue 1,3 and Tianshu Bi 1
1

2
3

*

State Key Laboratory of Alternate Electrical Power System with Renewable Energy Source,
North China Electric Power University, Beijing 102206, China; jiehaocui@foxmail.com (J.C.);
15650759520@163.com (J.W.); hdyuelei@epri.sgcc.com.cn (L.Y.); tsbi@ncepu.edu.cn (T.B.)
Department of Electrical, Computer and Systems Engineering, Rensselaer Polytechnic Institute, Troy,
NY 12180, USA; chowj@rpi.edu
Research Institute of Power System, China Electric Power Research Institute, Beijing 100192, China
Correspondence: acxue@ncepu.edu.cn

Received: 9 November 2018; Accepted: 18 December 2018; Published: 21 December 2018




Abstract: When the electric power system is disturbed, the transient frequency deviation may be
large and harmful to its stable operation, especially in some small power systems. However, there is a
lack of transient frequency acceptability margin (TFAM) which could be directly used by dispatchers.
In this paper, a new TFAM is proposed based on the transient frequency acceptability index (TFAI).
First, based on the frequency trajectory and the philosophy of “different weights to the different
frequency offset levels”, a new TFAI is proposed combined with frequency thresholds and time
duration limits. The effectiveness of the TFAI is verified, and the critical acceptable disturbance is
determined by using the TFAI. Then, a new TFAM is proposed based on the critical acceptability
disturbance. The proposed TFAM can quantitatively describe the distance of the operation point
from the critical frequency acceptability point, and distinguish the transient frequency acceptability
of different disturbances. Finally, with different simulations, the effectiveness and applicability of
the proposed TFAM are verified. The TFAM can be used for disturbances with single-parameter and
multiple parameters.
Keywords: transient frequency stability; transient frequency acceptability margin (TFAM); frequency
trajectory; PMU/WAMS

1. Introduction
Frequency reflects the active power balance of generator output and load in the power system.
Frequency deviation that is overly large will lead to serious consequences for the safe operation of
generators and systems as well as users. For example, equipment such as generators and auxiliary
machines will deviate from working conditions, thus reducing their efficiency, affecting the economic
operation of power plants and also of the entire power grid. If the frequency is too low, it will also
endanger the safe operation of the whole system. Therefore, it is necessary to evaluate the transient
frequency characteristics of a power system under different disturbances to determine whether the
frequency can remain stable [1–4]. At present, more and more photovoltaic and wind power sources
are being connected to power systems, which brings more frequency instability problems. Research
by [5,6] relates to this problem.
Compared with large-scale power grids, frequency instability has a greater impact on distributed
systems. For frequency stability evaluation and control of distributed systems, scholars have conducted
significant research. Ref. [7] modeled inverter-based distributed generators to research the stability
of a microgrid. Ref. [8] suggested that the impacts of different penetration levels and distributed
Energies 2019, 12, 12; doi:10.3390/en12010012

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generation (DG) technologies on frequency stability are different. To determine frequency stability,
Ref. [9] compared two stability indexes used for load shedding of the islanded distribution network
and proposed an under-frequency load shedding (UFLS) algorithm. Similarly, Ref. [10] proposed a
novel load shedding strategy by considering power stability with economy in load shedding for a
distribution system with distributed generation. Ref. [11] introduced a hierarchical frequency control
strategy to ensure the power balance and frequency stability in multiple time scales for medium-voltage
(MV) isolated microgrids.
In addition to distributed systems, transient frequency problems of large-scale power systems have
also been studied. In general, the work in the assessment for the transient frequency characteristics can
be divided into two categories. One is transient frequency stability assessment [2–10], which determines
the stability of the transient frequency. The other is frequency acceptability assessment [11–18],
which assesses whether the duration of frequency offset exceeding a specified value exceeds the
regulation requirements after the power system suffers disturbances.
In the aspect of transient frequency stability, the traditional time domain simulation method [3,4]
is more accurate, but takes longer and requires a large amount of computation, which is not conducive
to rapid application.
With the advent of phasor measurement unit (PMU) and wide area measurement system
(WAMS) [12–14], scholars have put forward online quantitative evaluation methods of steady-state
frequency stability based on PMU data [15–17]. Ref. [15] estimated the system active vacancy by using
the pre-disturbance PMU data to realize the steady-state frequency prediction. Based on Ref. [15],
according to the PMU information, Ref. [16] considers the variation of load and transmission loss
between the instantaneous after disturbance and the steady-state after disturbance. Ref. [17] used PMU
data to predict the steady frequency and then determine the stability. It is worth noting that the above
frequency stability determination methods ignore the transient process and cannot describe the impact
of the transient process. Ref. [18] proposed a new approach to dynamic stability assessment of a power
system. This approach applied the supervised concept to a clustering neural network, and directly used
the frequencies of buses. Based on a complete-linkage clustering, Ref. [19] presented a methodology
for selecting a set of critical operating points suitable for frequency stability assessments. In order to
analyze the impact of rapid changes in the measurements on instability detection, a frequency trend
vector is considered in Refs. [20,21] calculates interference power by using the real-time initial rate
of change of frequency (ROCOF) from the WAMS, and then compares this power with the power
mismatch threshold derived from the low-order system frequency response (SFR) model to judge the
severity of the interference. Similarly, [22] used the ROCOF and maximum frequency deviation to
determine the frequency stability.
To describe the impact of the transient process in transient frequency acceptability assessment,
Ref. [23] proposed to use a fixed frequency drop threshold and time duration when frequency
is abnormal to form a number of two-element tables to quantitatively assess the acceptability of
transient frequency offset. On this basis, three transient frequency offset margin indices were proposed
by [24–26], and the quantitative analysis of security of transient frequency offset was realized. The
two indices of [24,25] can be applied to the optimization of under-frequency load shedding (UFLS).
Ref. [27] proposed that these margins should be based on the center of inertia frequency. Further,
Ref. [28] considered the influence of frequency offset and proposed a transient frequency acceptability
index based on the cumulative effect. On this basis, Ref. [29] proposed a frequency margin index which
can evaluate the severity of faults.
In recognizing that the above proposed indices take the same weights for different frequency
drops and cannot distinguish the influence of different drop degrees, Ref. [30] proposed a transient
frequency acceptability evaluation index which uses different weighting factors for different drop
degrees, and applied it to the transient frequency acceptability evaluation and security control of a
power grid connected by large-scale thermal power units. Further, Ref. [31] expanded this index to the

Energies 2019, 12, 12

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frequency acceptability assessment of a large-scale power grid with wind farms. Ref. [32] improved
the index to analyze the transient frequency acceptability when the frequency drops.
It is worth noting that the index proposed by [30–32] is difficult to be used to compare frequency
deviation acceptability directly under different faults. Hence, this index is not suitable for a dispatcher
to directly judge frequency deviation acceptability.
InEnergies
recognizing
thePEER
above
problems, based on [30,32], this paper proposes an3 of
improved
2018, 11, x FOR
REVIEW
18
transient frequency acceptability index (TFAI) and, furthermore, proposes a new transient frequency
95
It ismargin
worth noting
that the
index proposed
by [30–32] is difficult
to be used
to compare
frequency
acceptability
(TFAM)
referring
to the philosophy
of security
domain
[33,34].
This margin
96
deviation acceptability directly under different faults. Hence, this index is not suitable for a
can determine the distance between the current running point of the system and the frequency
97 dispatcher to directly judge frequency deviation acceptability.
collapse point, so as to analyze the degree of transient frequency deviation and make it convenient
98
In recognizing the above problems, based on [30,32], this paper proposes an improved transient
for
dispatcher
to determine
the(TFAI)
degreeand,
of transient
frequency
deviation
acceptability.
The main
99 thefrequency
acceptability
index
furthermore,
proposes
a new transient
frequency
contributions
of
this
paper
are
as
follows:
100 acceptability margin (TFAM) referring to the philosophy of security domain [33–34]. This margin can
101 determine the distance between the current running point of the system and the frequency collapse
(1) The TFAI proposed in this paper can use frequency deviation information to quantitatively
102 point, so as to analyze the degree of transient frequency deviation and make it convenient for the
frequency
acceptability.
103 evaluate
dispatcher
to determine
the degree of transient frequency deviation acceptability. The main
(2) By
comparingofreal-time
parameters
and the parameters obtained by off-line simulation, the TFAM
104
contributions
this paper
are as follows:
proposed
in
this
paper
directly
provides
thefrequency
distancedeviation
betweeninformation
the currenttooperating
point and
105 (1) The TFAI proposed in this paper can use
quantitatively
boundary.
This facilitates the dispatcher to judge the frequency acceptability and
106 the critical
evaluate
frequency acceptability.
107 take
(2) measures.
By comparing real-time parameters and the parameters obtained by off-line simulation, the
108
TFAMproposed
proposed in
directly
providesin
the
distance
between
current the
operating
(3) The TFAM
inthis
thispaper
paper
is expressed
power
space
and the
considers
powerpoint
mismatch.
109 Compared
and thewith
critical
boundary.
This
facilitates
the
dispatcher
to
judge
the
frequency
acceptability
and
other frequency margins, the TFAM is more convenient to formulate
security
110 control
take measures.
measures.
111 (3) The TFAM proposed in this paper is expressed in power space and considers the power
(4) The TFAM
proposed
in with
this other
paperfrequency
can be used
to compare
the
transient
frequency
deviation
112
mismatch.
Compared
margins,
the TFAM is
more
convenient
to formulate
acceptability
under
different
faults.
113
security control measures.
114 The
(4) The TFAM proposed in this paper can be used to compare the transient frequency deviation
organization of the remainder of this paper is as follows. Section 2 introduces the construction
115
acceptability under different faults.
of TFAI and TFAM. In Section 3, the relevant parameters of TFAI are determined according to the
116
The organization
of the remainder
of the
thisvalidity
paper isofasTFAI
follows.
Section 2Section
introduces
the
specific situation
of small power
grid Z, and
is verified.
4 verifies
the
117 construction of TFAI and TFAM. In Section 3, the relevant parameters of TFAI are determined
effectiveness of TFAM in many cases, focusing on the power grid Z. Finally, Section 5 draws conclusions
118 according to the specific situation of small power grid Z, and the validity of TFAI is verified. Section
and provides
discussion.
119
4 verifies the effectiveness of TFAM in many cases, focusing on the power grid Z. Finally, Section 5

120 draws conclusions and provides discussion.
2. The Construction of the TFAM
121 This
2. The
Construction
of theaTFAM
section
first presents
new transient frequency acceptability index (TFAI) and, then, proposes

122
This section
first acceptability
presents a new
transient
frequency acceptability index (TFAI) and, then,
a new transient
frequency
margin
(TFAM).
123 proposes a new transient frequency acceptability margin (TFAM).
2.1. Transient Frequency Acceptability Index
124 2.1. Transient Frequency Acceptability Index
The philosophy of the TFAI lies in assigning different weights to the accumulation of frequency
125
The philosophy of the TFAI lies in assigning different weights to the accumulation of frequency
deviation; that is, as shown in Figure 1, by dividing the frequency deviation into different bands (f -f ,
126 deviation; that is, as shown in Figure 1, by dividing the frequency deviation into different bands (f1- 1 2
f 2 -f 3 , lower
andf3f and
thanf3′f ),30 and
), and
applying
different
weights
integrating
10 -f 2f01′,-ff 2′2,0 -f
303′,, higher
127
f2, f2-f3,than
lowerf 3than
f2′-f
higher than
applying
different
weights
whilewhile
integrating
the frequency
in theindifferent
bands.
128
the frequency
the different
bands.

129
130
131

Figure
1. Frequency
band
divisionof
of transient
transient frequency
acceptability
index
(TFAI).
Figure
1. Frequency
band
division
frequency
acceptability
index
(TFAI).

Mathematically, the TFAI F can be expressed as follows:

Energies 2019,
2018, 12,
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x FOR PEER REVIEW
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44 of
18
of 18

n

m

p

i =0

j =1

k =1

=
[ F
K can
g ( f [t ]) f [t i ] − f NasΔtfollows:
+  H k hk ( f [t i ]) f [t i ] − f N Δti ]
Mathematically, Fthe
TFAI
j j bei expressed
i

132

where

n

F=

p

m

∑ [ ∑ Kj gj ( f [ti ])| f [ti ] − fN |∆ti + ∑ Hk hk ( f [ti ])| f [ti ] − fN |∆ti ]

i =0 j =1

where

(1)

(

k =1

)

(1)

 1 f ≤ f [t ] ≤ f
j +1
i
j

g j ( f [ti ])= 
( 0 f [t ] < f or f [t ] > f
 1 i f j≤
+1
f [ti ]i ≤ f jj
j +1

g j ( f [ti ]) =
0
f [ti ] < f j+1 or f [ti ] > f j
(  1
f k' ≤ f [ti ] ≤ f k' +1


f k0 ≤ f [ti ] ≤ f k0 +1
h ( f [t ])=  1
'
'
hk ( f [tik]) =i
< ffk 0or
f [t ] > f 0
00 ff [[ttii]] <
k or f [iti ] > k +f 1k +1

(

(

(

)

)

)

133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152

and ff[t
[tii]] is
is the
the frequency
frequency at time ti on the frequency trajectory;
is the
the rated
rated frequency; fk ’ is the lower
trajectory; ffN
N is
frequency of
frequency;
fj is
thethe
upper
frequency
of
of the
the frequency
frequencyband
bandk’k’that
thatisishigher
higherthan
thanthe
therated
rated
frequency;
fj is
upper
frequency
the
frequency
band
j that
is is
below
the
of the
frequency
band
j that
below
therated
ratedfrequency;
frequency;∆tΔt
i isthe
thetime
timestep
step of
of the
the frequency response
i is
calculation;
the
rated
frequency;
HkHisk
calculation; K
Kjj is the
the weight
weight factor
factorof
ofdifferent
differentfrequency
frequencyband
bandj that
j thatisisbelow
below
the
rated
frequency;
the
weight
factor
of different
frequency
bandband
k’ thatk’isthat
higher
than the
rated
n is the number
is the
weight
factor
of different
frequency
is higher
than
thefrequency;
rated frequency;
n is the
of
sampling
points; mpoints;
is the m
number
of frequency
bands that
are that
below
rated
frequency;
p is the
number
of sampling
is the number
of frequency
bands
arethe
below
the
rated frequency;
number
of
frequency
bands
that
are
higher
than
the
rated
frequency.
p is the number of frequency bands that are higher than the rated frequency.
The TFAI
TFAI (Equation
(Equation(1))
(1))reflects
reflectsfrequency
frequency
acceptability
under
a certain
disturbance.
the
acceptability
under
a certain
disturbance.
At theAt
same
same
time,
the
weight
factors
of
the
TFAI
can
be
adjusted
according
to
the
requirements
of
the
time, the weight factors of the TFAI can be adjusted according to the requirements of the frequency
frequency
acceptability.
acceptability.
The presented
presentedTFAI
TFAIfirst
firstjudges
judgeswhether
whether
frequency
enters
a certain
frequency
band,
and
thethe
frequency
enters
a certain
frequency
band,
and then
then
assigns
different
weights
to
the
frequency
in
its
vertical
direction
(fall
depth
or
rising
height).
assigns different weights to the frequency in its vertical direction (fall depth or rising height). Thus,
Thus,
the TFAI
can reflect
the influence
of different
frequency
offsets
andimprove
improvethe
theaccuracy
accuracy of
of the
the TFAI
can reflect
the influence
of different
frequency
offsets
and
frequency acceptability evaluation.
Compared to [30,31], the improvement of this paper is that for a certain sampling point, the
proposed TFAI
TFAI first
first judges
judges the frequency band of the sampling point. Then, the TFAI only considers
the weight of this frequency band, and avoids the interference of weights
weights of
of other
other frequency
frequency bands.
bands.
For example,
TFAI
is K
f ),while
whilein
in[30,31],
[30,31],the
the acceptability
acceptability
example,as
asshown
shownininFigure
Figure2,2,the
theproposed
proposed
TFAI
isnK×n ×(f (f
−f),
NN−
index is K
(fNN-f)−+f K
)+
f cr.n+−1K)2+×· ·(f·N+
K2) ×
(f 1N×−(ffNcr.2
)+
(f N − is
f cr.1
). This index
Knn ×
× (f
n-1K
×n −
(f1N ×
− f(fcr.n−1
) +⋯
−fcr.2
+K
−fcr.1
). K
This
influenced
by
N −
1 × index
is
influenced
by
weights
of
other
bands,
such
as
K
,
.
.
.
,
K
,
K
.
weights of other bands, such as Kn-1, …, K2, K1.
n −1
2
1

153
154

Figure 2.
2. Weighted
Weighted quantitative analysis of frequency.
Figure
frequency.

155
156

In addition,
thethe
proposed
TFAI
can can
analyze
the frequency
acceptability
when
In
addition,compared
comparedwith
with[32],
[32],
proposed
TFAI
analyze
the frequency
acceptability
the
frequency
rises
and
falls.
when the frequency rises and falls.

157

2.2. Transient Frequency Acceptability Margin
2.2. Transient Frequency Acceptability Margin
This section proposes a transient frequency acceptability margin from the viewpoint of the
This section proposes a transient frequency acceptability margin from the viewpoint of the
dispatchers/operators based on the TFAI presented in Section 2.1.
dispatchers/operators based on the TFAI presented in Section 2.1.
As shown in Section 2.1, the TFAI can evaluate the acceptability of the transient frequency.
However, the degree of frequency acceptability cannot be displayed directly in the power space, and

158
159
160
161

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As shown in Section 2.1, the TFAI can evaluate the acceptability of the transient frequency.
However, the degree of frequency acceptability cannot be displayed directly in the power space,
and the distance between the current operating point of the system and the acceptable margin is
implicitly contained. Therefore, it is necessary to construct a transient frequency acceptability margin
in the power space, i.e., from the viewpoint of the dispatchers/operators.
2.2.1. Power Disturbance with Single Parameter
For a specific active disturbance with parameter ∆Pdist , the TFAI F(∆Pdist ) can be obtained. If there
is a value ∆Pdist = ∆Pcr , which satisfies F(∆Pcr ) = 1, then the disturbance ∆Pcr is the disturbance which
may lead to transient frequency critical unacceptability under such disturbance.
The relationship between ∆Pcr and the actual disturbance ∆Pdist can reflect the frequency
acceptability under the disturbance, that is, if ∆Pcr > ∆Pdist , then the transient frequency is acceptable;
otherwise, the transient frequency is not acceptable. Therefore, the TFAM can be described by the
difference between ∆Pcr and ∆Pdist , i.e.,
η 0 = ∆Pcr − ∆Pdist

(2)

In order to compare the frequency acceptability of different disturbances, the normalized TFAM
under disturbance i could be defined as follows:
ηi =

∆Pcri − ∆Pdisti
× 100%
∆Pcri

(3)

where ∆Pdisti is the parameter under the disturbance i; ∆Pcri is the critical value corresponding to the
transient frequency acceptability of system under the disturbance i.
With normalization, the TFAM of the disturbance i decreases linearly with the increase of the
value of the parameter of the actual disturbance. If there is no disturbance, then ∆Pdisti = 0, and the
margin is 100%. If ∆Pdisti = ∆Pcri , the transient frequency is critical acceptable and the margin is 0.
In particular, under a single fault, when ∆Pcri is greater than the maximum disturbance quantity
∆Pmax , e.g., the maximum generator tripping quantity and load shedding quantity, it is impossible to
obtain ∆Pcri by simulation. In this case, with experience, the critical value could be set as ∆Pcri = 2∆Pmax
for the sake of unified consideration.
2.2.2. Power Disturbance with Multiple Parameters
When a disturbance involves changes of multiple parameters, the multiple n parameters can form
an n-dimensional vector (∆Pdist 1 , ∆Pdist 2 , . . . , ∆Pdist n ).
The normalized distance of the current disturbance point from any critical point k (∆Pcrk 1 , ∆Pcrk 2 ,
. . . , ∆Pcrk n ) on the critical acceptability boundary can be defined as:
s

n

i − ∆Pi
∑ ∆Pcrk
dist

ε(k) =

i =1

s

n



i =1

2
(4)

i )2
(∆Pcrk

If the transient frequency under the disturbance is unacceptable, then the TFAM under the
disturbance can be defined as:


η = −min ε(k)
(5)
k

If the transient frequency under the disturbance is acceptable, the TFAM under the disturbance
can be defined as:
η = minε(k)
(6)
k

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193
194

In the case of n = 2, the parameter of the disturbance is two-dimensional, i.e., (∆Pdist 1 , ∆Pdist 2 ).
2
2). This
This In
vector
and of
then critical
k (∆Pcrk
, ∆Pdisturbance
critical
acceptability
boundary
are
the case
= 2, thepoint
parameter
of 1the
two-dimensional,
i.e.,
(ΔPdist1, ΔP
distshown
crk ) on theis
1
2
in
Figure
vector
and3.the critical point k (ΔPcrk , ΔPcrk ) on the critical acceptability boundary are shown in Figure 3.

195
196

Figure 3. The case that a disturbance involves two parameters.
parameters.

197

Equation
Equation (4)
(4) can
can then
then be
be expressed
expressed as:
as:
q


1 1− ∆P1 1 2 2+ ∆P2 2 − ∆P22 22
∆P
dist
dist
crk
crk
Δ
P

Δ
P
+
Δ
P

Δ
P
crq
dist
crk
dist
k
ε(k) =
ε (k) =
2
2
1
2
(∆P
crk2 ) ) 2
( Δcrk
( ΔP
P 1) ) 2++(∆P

(

) (

crk

198
199
200
201

202
203
204

205
206

207
208
209
210
211
212
213

)

(7)
(7)

crk

2.2.3. Increase of the Disturbance Parameters in the Fixed Direction
2.2.3.IfIncrease
of the Disturbance
in the
Fixed
Direction
the disturbance
parameters Parameters
increase in the
fixed
direction
(assuming n parameters), the direction
of theIf disturbance
is fixed.
Assuming
the direction
is λ ∈direction
Rn and (assuming
kλk= 1, thencritical
frequency
the disturbance
parameters
increase
in the fixed
parameters),
the
n
acceptability
in this direction
could
be described
follows:is λ ∈ R and ‖λ‖= 1, the critical
direction of point
the disturbance
is fixed.
Assuming
the as
direction
frequency acceptability point in this direction could be described as follows:
1
2
n
(∆Pcrk
, ∆Pcrk
, . . . , ∆Pcrk
) = lcr · λ
(8)
1
2
n
(8)
q(ΔPcrk , ΔPcr k ,..., ΔPcrk ) = lcr ⋅ λ
2
i
where lcr is a real number equal to ∑in=1 ∆Pcrk
in Equation (4).
Because
the disturbance
disturbance
where
lcr is a the
realdirection
number of
equal
to ∑ (∆𝑃is fixed,
) inany
Equation
(4). point can be described as follows:
1
2 is fixed,nany disturbance point can be described as
Because the direction of the (disturbance
∆Pdist
, ∆Pdist
, . . . , ∆Pdist ) = l · λ
(9)
follows:
where l is a real number.
1
2
n
(9)
ΔPdist
, ΔPdist
,..., ΔPdist
) = l⋅λ
Here, the TFAM can be defined (as:
lcr − l
η=
(10)
where l is a real number.
lcr
qcan be defined as:
Here, the TFAM
2
i − ∆Pi
where lcr − l equals ∑in=1 ∆Pcrk
in Equation (4).
dist
q

lcr − l 1 2
2 2 in Equation (7), and l − l
In particular, when n = 2, λ ∈ R2 , lcr equals
∆Pcrk + ∆Pcrk
cr (10)
η
=
q


l
2
2
1 − ∆P1
2 − ∆P2
cr
equals
∆Pcrk
+ ∆Pcrk
in Equation
(7).
dist
dist
When the disturbance power point is on the unacceptable side of the critical acceptability
Δ𝑃 ) isin
Equation
(4).is η, and the further the disturbance power
where lcr−l lequals
boundary,
> lcr , and∑η <(∆𝑃
0. The−greater
l, the
smaller
pointIn
from
the critical
stability
it can be(∆𝑃
judged
by(∆𝑃
the TFAM
that the degree
) +
particular,
when
n = 2,point.
λ∈R2Here,
, lcr equals
) inηEquation
(7), andoflcrfrequency
−l equals
unacceptability
increases
gradually.
(∆𝑃 − ∆𝑃 ) + (∆𝑃 − ∆𝑃 ) in Equation (7).
When
point
is on
side of the
critical
acceptability
boundary,
When the
thedisturbance
disturbancepower
power
point
is the
on acceptable
the unacceptable
side
of the
critical acceptability
lboundary,
< lcr , and ηl >> l0.
The greater is l, the smaller is η, and the closer the disturbance power point to the
cr, and η < 0. The greater is l, the smaller is η, and the further the disturbance power

point from the critical stability point. Here, it can be judged by the TFAM η that the degree of
frequency unacceptability increases gradually.

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When the disturbance power point is on the acceptable side of the critical acceptability boundary,
7 of 18
l < lcr, and η > 0. The greater is l, the smaller is η, and the closer the disturbance power point to the
critical stability point. Here, it can be judged by the TFAM η that the degree of frequency acceptability
decreases
gradually.
critical stability
point. Here, it can be judged by the TFAM η that the degree of frequency acceptability

Energies 2019, 12, 12

decreases gradually.
3. TFAI for the Small Power Grid Z
3. TFAI for the Small Power Grid Z
This section derives the TFAI for the small power grid Z, as shown in Figure 4, which is modeled
sectionPSD-BPA.
derives the
TFAI
theof
small
gridMW.
Z, asThe
shown
in Figure
4, which
in theThis
software
The
totalfor
load
Z is power
about 700
hydraulic
power
plantisMmodeled
has the
in the software
PSD-BPA.
total
loadplants,
of Z is i.e.,
about
MW.
Thegenerators
hydraulic power
has the
largest
proportion
among The
all the
power
195700
MW
(three
with 65plant
MW),Mwhich
is
largestof
proportion
among
all the power
plants,
i.e., 195isMW
(three
generators
28.1%
the load. The
measurement
of the
frequency
located
at power
plantwith
M. 65 MW), which is
28.1% of the load. The measurement of the frequency is located at power plant M.

Figure 4.
4. The
The topology
topology of
of small
small grid
grid Z.
Z.
Figure

3.1. Frequency Bands and Time Duration Limits
3.1. Frequency Bands and Time Duration Limits
For the small power grid Z, the normal frequency fluctuation range (or acceptable frequency
For the small power grid Z, the normal frequency fluctuation range (or acceptable frequency
range) is 50 ± 0.5 Hz, so there is no weight factor in the 50 ± 0.5 Hz band (i.e., the weight is 0).
range) is 50 ± 0.5 Hz, so there is no weight factor in the 50 ± 0.5 Hz band (i.e., the weight is 0).
The setting of the under frequency load shedding (UFLS) is: first round operation frequency is
The setting of the under frequency load shedding (UFLS) is: first round operation frequency is
48.8 Hz, and delay time is 0.3 s; the special round operation frequency is 49.0 Hz, and the delay time
48.8 Hz, and delay time is 0.3 s; the special round operation frequency is 49.0 Hz, and the delay time
is 10 s. Furthermore, the action of UFLS should be avoided as far as possible to maintain frequency
is 10 s. Furthermore, the action of UFLS should be avoided as far as possible to maintain frequency
acceptability. Thus, the first acceptable frequency sag is that the duration when the frequency is below
acceptability. Thus, the first acceptable frequency sag is that the duration when the frequency is
48.8 Hz does not exceed 0.3 s. The second acceptable frequency sag is that the duration when the
below 48.8 Hz does not exceed 0.3 s. The second acceptable frequency sag is that the duration when
frequency is below 49.0 Hz does not exceed 10 s.
the frequency is below 49.0 Hz does not exceed 10 s.
Moreover, Chinese standard “DLT 1234-2013 Technique specification of power system security
Moreover, Chinese standard “DLT 1234-2013 Technique specification of power system security
and stability calculation” [35] demands that the frequency should recover to more than 49.5 Hz after
and stability calculation” [35] demands that the frequency should recover to more than 49.5 Hz after
the UFLS devices operate. Consequently, the time duration when the transient frequency is below
the UFLS devices operate. Consequently, the time duration when the transient frequency is below
49.5 Hz should not exceed 10 min.
49.5 Hz should not exceed 10 min.
Therefore, for the low frequency section in Figure 1, according to the system safety requirements,
Therefore, for the low frequency section in Figure 1, according to the system safety requirements,
the frequency trajectory can be divided into three bands. From low to high, the three bands are
the frequency trajectory can be divided into three bands. From low to high, the three bands are below
below 48.8 Hz, 48.8–49.0 Hz, and 49.0–49.5 Hz, and the corresponding weight factors are K3 , K2 ,
48.8 Hz, 48.8–49.0 Hz, and 49.0–49.5 Hz, and the corresponding weight factors are K3, K2, and K1,
and K1 , respectively.
respectively.
In the high frequency case, the first round operation frequency of the high frequency generator
In the high frequency case, the first round operation frequency of the high frequency generator
tripping of the small power grid Z is 53 Hz, and the delay time is 0.3 s. To avoid the interference of
tripping of the small power grid Z is 53 Hz, and the delay time is 0.3 s. To avoid the interference of
high frequency generator tripping on frequency acceptability, the time duration when the frequency is
high frequency generator tripping on frequency acceptability, the time duration when the frequency
beyond 53 Hz should not exceed 0.3 s. Moreover, the operation frequency of over-speed protection
is beyond 53 Hz should not exceed 0.3 s. Moreover, the operation frequency of over-speed protection
control (OPC) of the power grid Z is 51.3 Hz, and the delay time is 10 s. This requires that the time
control (OPC) of the power grid Z is 51.3 Hz, and the delay time is 10 s. This requires that the time
duration when the transient frequency is higher than 51.3 Hz should not exceed 10 s. Chinese standard
duration when the transient frequency is higher than 51.3 Hz should not exceed 10 s. Chinese
“DLT 1234-2013 Technique specification of power system security and stability calculation” [35]
standard “DLT 1234-2013 Technique specification of power system security and stability calculation”
demands that the frequency cannot be more than 51 Hz for a long period. Consequently, the time
duration when the frequency is beyond 51 Hz should not exceed 180 s.

Energies 2019, 12, 12

8 of 18

Therefore, for the high frequency section in Figure 1, the frequency trajectory can be divided into
three bands. From high to low, the three bands are above 53 Hz, 51.3–53 Hz and 51–51.3 Hz, and the
corresponding weight factors are H3 , H2 , and H1 , respectively.
3.2. Determination of Weight Factors
The setting of the weight factors makes the result of the TFAI F more than or equal to 1 when the
requirements are not satisfied. Thus, the weight factors K1 , K2 , and K3 for the low frequency section,
corresponding to different frequency bands, should satisfy the following equations:


 F = K3 × (50 − 48.8) × 0.3 = 1
F = K2 × (50 − 49.0) × 10 = 1

 F = K × (50 − 49.5) × 600 = 1
1

(11)

where the first equation in Equation (11) means that if the time duration when the frequency is lower
than 48.8 Hz exceeds 0.3 s, the TFAI F = 1. The equation corresponds to the requirement of avoiding
action of the first round of UFLS. Similarly, the second equation means that if the time duration when
the frequency is lower than 49.0 Hz exceeds 10 s, F = 1. The equation corresponds to the requirement of
avoiding action of the special round of UFLS. The third equation means that if the time duration when
the frequency is lower than 49.5 Hz exceeds 600 s, F = 1. The equation corresponds to the requirement
of frequency acceptability. The threshold-delay parameters and weight factors for the low frequency
band can be summarized as shown in Table 1.
Table 1. Low frequency threshold-delay parameters and weight factors.
Threshold-delay parameter
Weight factor
Value of weight factor

(49.5, 600)
K1
0.0033

(49, 10)
K2
0.1000

(48.8, 0.3)
K3
2.7778

For the high frequency section, the weight factors H1, H2, and H3 should satisfy the following equations:


 F = H3 × (53 − 50) × 0.3 = 1
F = H2 × (51.3 − 50) × 10 = 1

 F = H × (51 − 50) × 180 = 1
1

(12)

where the first equation in Equation (12) means that if the time duration when the frequency is higher
than 53 Hz exceeds 0.3 s, the TFAI F = 1. The second equation means that if the time duration when the
frequency is higher than 51.3 Hz exceeds 10 s, F = 1. The third equation means that if the time duration
when the frequency is higher than 51 Hz exceeds 180 s, F = 1. The threshold-delay parameters and
weight factors for the high frequency bands can be summarized as shown in Table 2.
Table 2. High frequency threshold-delay parameters and weight factors.
Threshold-delay parameter
Weight factor
Value of weight factor

(51, 180)
H1
0.0056

(51. 3,10)
H2
0.0769

(53, 0.3)
H3
1.1111

In general, the TFAI including the high and the low frequency sections can be expressed as follows.
F=

n

∑ [0.0033 × g1 ( f [ti ])| f [ti ] − f N |∆ti + 0.1000 × g2 ( f [ti ])| f [ti ] − f N |∆ti

i =0

+2.7778 × g3 ( f [ti ])| f [ti ] − f N |∆ti + 0.0056 × h1 ( f [ti ])| f [ti ] − f N |∆ti
+0.0769 × h2 ( f [ti ])| f [ti ] − f N |∆ti + 1.1111 × h3 ( f [ti ])| f [ti ] − f N |∆ti ]

(13)



1

i =0

i

N

i

2

i

i

i

N

i

(13)

+2.7778 × g3 ( f [ti ]) f [ti ] − fN Δti + 0.0056 × h1 ( f [ti ]) f [ti ] − fN Δti
+ 0.0769 × h2 ( f [ti ]) f [ti ] − fN Δti + 1.1111 × h3 ( f [ti ]) f [ti ] − fN Δti ]

278

where

Energies 2019, 12, 12

where

(

)

(

)

9 of 18

 1 f ≤ f [t ] ≤ f
j +1
i
j

g j ( f [ti ])= 
( 0 f [ti ] < f j +1or f [ti ] > f j
1
f j +1 ≤ f [ t i ] ≤ f j

g j ( f [ti ]) =
0
f [ti' ] < f j+1 or ' f [ti ] > f j

(  1 fk ≤ f [ti ] ≤ fk +1

hk ( f [ti ])=  1
f k0 ≤ ' f [ti ] ≤ f k0 +1 '
hk ( f [ti ]) =
0 f [ti ] > fk +0 1or f [ti ] < fk 0
0
f [ti ] > f k+1 or f [ti ] < f k

(

(

)

)

279
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282
283
284
285
286
287

and ff[t
at time
time tti;; ffN isis the
of the
the
and
[tii]] is
is the
the frequency
frequency at
the system
system rated
rated frequency;
frequency; ffkk’’ is
is the
the lower
lower frequency
frequency of
i N
frequency
band
k’
which
is
higher
than
the
rated
frequency;
f
j is the upper frequency of the frequency
frequency band k’ which is higher than the rated frequency; fj is the upper frequency of the frequency
band jj which
which is
is below
below the
the rated
rated frequency;
frequency; ∆t
Δti is
is the
the time
time step
step of
of the
the frequency
frequency response
response calculation.
calculation.
band
i
When
F

1,
the
frequency
deviation
is
unacceptable,
and
measures
need
to
be
taken.
When
<
When F ≥ 1, the frequency deviation is unacceptable, and measures need to be taken. When
F <F 1,
1,
the
frequency
deviation
is
acceptable.
the frequency deviation is acceptable.
In calculation
calculationofofthe
the
proposed
TFAI,
the weights
of the frequency
the
In
proposed
TFAI,
onlyonly
the weights
of the frequency
bands inbands
whichin
thewhich
sampling
sampling
points
are
located
are
used,
and
the
weights
of
other
frequency
bands
are
not
included.
In
points are located are used, and the weights of other frequency bands are not included. In [30,31],
[30,31],
the calculation
required
weights
of other
frequency
bands
between sampling
the frequency
the
calculation
required the
weightsthe
of other
frequency
bands
between
the frequency
point
sampling
point
and
the
rated
frequency.
and the rated frequency.

288

3.3. Determination of Sampling Interval

289
290
291
292
293
294
295

This subsection analyzes the impact of time interval on the result of TFAI and
and determines
determines the
suitable time interval.
disturbanceA
Aisisset
setasasthe
theload
loadatatbus
busZD
ZD
the
small
power
grid
Z increasing
50 MW,
If disturbance
ofof
the
small
power
grid
Z increasing
50 MW,
thenthen
the
the
frequency
trajectory
under
disturbance
A
can
be
obtained
as
shown
in
Figure
5.
frequency trajectory under disturbance A can be obtained as shown in Figure 5.
Figure 55 shows
shows that
that the
the frequency
frequency satisfies
Figure
satisfies the
the three
three requirements
requirements of
of frequency
frequency acceptability,
acceptability,
thus the
the system
system is
is transient
transient frequency
frequency acceptable
acceptable under
under disturbance
disturbance A.
Furthermore, the
TFAI with
with
thus
A. Furthermore,
the TFAI
different
sampling
intervals
can
be
obtained
as
shown
in
Table
3.
different sampling intervals can be obtained as shown in Table 3.

296
297

Figure 5. Frequency trajectories under the disturbance A.

298

Table
3. TFAI
TFAI with
with different
different sampling
sampling intervals.
intervals.
Table 3.

Sampling interval/s
Sampling interval/s0.02 0.02
TFAI

TFAI

0.9616

0.9616

0.04 0.04 0.1
6.0280 126.1728

6.0280

0.1
126.1728

As shown in Table 3, when the sampling interval is 0.02 s, the TFAI is less than 1, however,
when the sampling interval is 0.04 s and 0.1 s, the TFAI is greater than 1, which is not consistent with
the conclusion that the transient frequency under disturbance A is acceptable.
Clearly, with the increase of sampling interval, some useful information will be lost, and the
result may have a larger error. The smaller the sampling interval, the closer the value to the true value
(integration). Moreover, the time interval of PMU data could be set to 0.02 s, 0.04 s or 0.1 s. Therefore,
in the application of the proposed index, the time interval of 0.02 s is recommended.

299
300
301
302
303
304
305

309
310
311
312

As shown in Table 3, when the sampling interval is 0.02 s, the TFAI is less than 1, however, when
the sampling interval is 0.04 s and 0.1 s, the TFAI is greater than 1, which is not consistent with the
conclusion that the transient frequency under disturbance A is acceptable.
Clearly, with the increase of sampling interval, some useful information will be lost, and the
result may have a larger error. The smaller the sampling interval, the closer the value to the true value
Energies 2019, 12, 12
10 of 18
(integration). Moreover, the time interval of PMU data could be set to 0.02 s, 0.04 s or 0.1 s. Therefore,
in the application of the proposed index, the time interval of 0.02 s is recommended.
3.4. Effectiveness of the Proposed TFAI
3.1. Effectiveness of the Proposed TFAI
This subsection verifies the effectiveness of the proposed TFAI with different disturbances.
This
subsection
verifies theare
effectiveness
of the proposed TFAI with different disturbances.
The different
disturbances
listed as follows.
The different disturbances are listed as follows.
(1) Dis A: the load at bus ZD increases 50 MW.
(1) Dis A: the load at bus ZD increases 50 MW.
(2) Dis B: the load at bus ZD increases 55 MW.
(2) Dis B: the load at bus ZD increases 55 MW.
(3) Dis
DisC:
C:the
theload
loadat
atbus
busH
Hdecreases
decreases60
60MW.
MW.
(3)
(4) Dis
DisD:
D:the
theload
loadat
atbus
busH
Hdecreases
decreases130
130MW,
MW, the
the load
load at
at bus
bus DG
DG decreases
decreases 15
15 MW.
MW.
(4)

313
314

The
The frequency
frequency trajectories
trajectories under
under the
the above
above four
four disturbances
disturbances are
are shown
shown in
in Figure
Figure 66 and
and the
the
corresponding
Table 4.
4.
corresponding TFAI and evaluation results are shown in Table

315
316

Figure 6.
6. Frequency
Frequency trajectories
trajectories under
under different
different disturbances.
disturbances.
Figure

317

Table 4. The TFAI under four disturbances and transient frequency acceptability.
Table 4. The TFAI under four disturbances and transient frequency acceptability.

306
307
308

Disturbance

TFAI

Transient Frequency Acceptability

Disturbance

TFAI

Dis B

Dis A
1.5620

0.9616

Dis C

0.4025

Dis D

Dis B
1.6752

Dis A

0.9616

Transient Frequency Acceptability

Acceptable
Unacceptable
Acceptable
(The time duration of the frequency lower than 49 Hz is 13.01 s, more than 10 s)
Unacceptable
Acceptable
Unacceptable
1.5620
(The time duration
of the frequency lower
(The time duration of the frequency higher than 51.3 Hz is 11.3 s, more than 10 s)

than 49 Hz is 13.01 s, more than 10 s)

318
319
320
321
322

Dis C
0.4025
Acceptable
Figure 6 shows that the system is transient frequency acceptable under Dis A and C. Table 4
Unacceptable
shows that TFAI for Dis A and C are 0.9616 and 0.4025 respectively,
which are both less than 1 and
indicate that the frequency
deviation
Dis D
1.6752is acceptable.
(The time duration of the frequency higher
Furthermore, Figure 6 shows that for Dis B,than
the 51.3
timeHz
duration
thethan
system
frequency is below
is 11.3when
s, more
10 s)
49 Hz is 13.01 s, greater than the permissible time of 10 s, thus the frequency deviation is unacceptable.
TableFigure
4 shows
that TFAI
B is 1.5620,
largerfrequency
than 1, which
indicates
thatDis
the A
system
frequency
6 shows
thatfor
theDis
system
is transient
acceptable
under
and C.
Table 4
deviation
is
unacceptable.
shows that TFAI for Dis A and C are 0.9616 and 0.4025 respectively, which are both less than 1 and
Thethat
results
Dis D are
similar to
those for Dis B.
indicate
the for
frequency
deviation
is acceptable.
Thus,
the
above
simulation
results
show
thatB,the
TFAI inwhen
Equation
(13) could
effectively
Furthermore, Figure 6 shows that for Dis
theproposed
time duration
the system
frequency
is
quantitatively
frequency
below
49 Hz isanalyze
13.01 s,the
greater
than acceptability.
the permissible time of 10 s, thus the frequency deviation is

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Energies 2018, 11, x FOR PEER REVIEW

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324
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11 of 18
11 of 18

unacceptable. Table 4 shows that TFAI for Dis B is 1.5620, larger than 1, which indicates that the
unacceptable. Table 4 shows that TFAI for Dis B is 1.5620, larger than 1, which indicates that the
system frequency deviation is unacceptable.
system
frequency
deviation
is similar
unacceptable.
The
results for
Dis D are
to those for Dis B.
The
results
Energies
2019,
12, 12 for Dis D are similar to those for Dis B.
of 18
Thus, the above simulation results show that the proposed TFAI in Equation (13) 11
could
Thus, the above simulation results show that the proposed TFAI in Equation (13) could
effectively quantitatively analyze the frequency acceptability.
effectively quantitatively analyze the frequency acceptability.
4. Validation for the TFAM
4. Validation for the TFAM
4. Validation for the TFAM
4.1. Disturbance with Single Parameter
4.1. Disturbance with Single Parameter
4.1.
Disturbance
with SingleReduction
Parameter in Power Plant M
4.1.1.
Case 1: Generation
4.1.1. Case 1: Generation Reduction in Power Plant M
generation
reduction
in power
4.1.1. Considering
Case 1: Generation
Reduction
in Power
Plantplant
M M, then the TFAI under different amounts
Considering
reduction
inshown
power as
plant
M, then
TFAI
of
of generation
cangeneration
be obtained,
which is
Curve
1 in the
Figure
7. under
Curvedifferent
1 showsamounts
that if the
Considering
generation
reduction
powerasplant
M, then
the TFAI
of
can be
obtained,
which
is in
shown
insystem
Figure
7. under
Curvedifferent
1 critical
showsamounts
that if the
generation reduction
is 46.70 MW,
then
the
TFAI isCurve
1, and 1the
frequency
is
acceptable,
generation can be obtained, which is shown as Curve 1 in Figure 7. Curve 1 shows that if the
generation
reduction
is 46.70 MW, then
the TFAI
is 1, andto
the
system frequency
is criticalas
acceptable,
i.e.,
∆Pcr = 46.70
MW. Furthermore,
the TFAM
η according
Equation
(3) can be obtained
shown in
generation reduction is 46.70 MW, then the TFAI is 1, and the system frequency is critical acceptable,
i.e., ΔP2cr of
= 46.70
MW.
Curve
Figure
7. Furthermore, the TFAM η according to Equation (3) can be obtained as shown
i.e., ΔPcr = 46.70 MW. Furthermore, the TFAM η according to Equation (3) can be obtained as shown
in Curve 2 of Figure 7.
in Curve 2 of Figure 7.

336
336
337
337
338
338
339
339
340
340
341
341
342
342
343
343
344
344
345
345
346
346
347
347

Curve
compared to
to TFAI,
TFAI, the
Curve 22 in
in Figure
Figure 77 shows
shows that,
that, compared
the proposed
proposed TFAM
TFAM can
can show
show the
the degree
degree of
of
Curveacceptability
2 in Figure 7 more
showsdirectly,
that, compared
to TFAI,
theintuitive
proposed
TFAM of
can
show
the degree
of
frequency
and
provide
a
more
measure
the
distance
from
the
frequency acceptability more directly, and provide a more intuitive measure of the distance from the
frequency
acceptability more
directly, and
provide a more
intuitiveconvenient
measure of the
distance from the
current
current operation
operation point
point to
to the
the frequency
frequency collapse
collapse point.
point. It
It is
is more
more convenient for
for the
the dispatcher.
dispatcher.
current operation point to the frequency collapse point. It is more convenient for the dispatcher.
4.1.2.
Load Shedding
Shedding at
at Bus
Bus H
4.1.2. Case
Case 2:
2: Load
H
4.1.2. Case 2: Load Shedding at Bus H
Considering
load shedding
sheddingat
atbus
busH,
H,the
theTFAI
TFAIunder
underdifferent
different
amounts
load
shedding
Considering load
amounts
ofof
load
shedding
cancan
be
Considering
load
shedding
at
bus
H,
the
TFAI
under
different
amounts
of
load
shedding
can
be
be
obtained,
which
is
shown
as
Curve
1
in
Figure
8.
Curve
1
shows
that
if
the
amount
of
load
obtained, which is shown as Curve 1 in Figure 8. Curve 1 shows that if the amount of load shedding
obtained,
which
is shown
Curve
in Figure
8. Curve
1 shows
that if the
amountacceptable.
of load shedding
shedding
is 100
then as
the
TFAI 1is
1, and
the frequency
system
frequency
is critical
is 100 MW,
thenMW,
the TFAI
is about
1, about
and the
system
is critical
acceptable.
That is,That
ΔPcris,
=
is
100
MW,
then Furthermore,
the TFAI is about
1, andaccording
the system
frequency(3)iscan
critical
acceptable.
That is,
ΔP
cr =
∆P
=
100
MW.
the
TFAM
to
Equation
be
obtained
as
shown
in
Curve
cr
100 MW. Furthermore, the TFAM according to Equation (3) can be obtained as shown in Curve 2 of
100
Furthermore, the TFAM according to Equation (3) can be obtained as shown in Curve 2 of
2Figure
ofMW.
Figure
8. 8.
Figure 8.

348
348
349
349

Figure 8. TFAI and TFAM corresponding to different load shedding quantities.
Figure
Figure8.8.TFAI
TFAIand
andTFAM
TFAM corresponding
corresponding to
to different
different load shedding quantities.

Figure 7.
TFAI and transient frequency acceptability margin (TFAM) corresponding to different
7. TFAI
Figure
7.
TFAI
and transient frequency acceptability margin (TFAM) corresponding to different
generation reductions.
reductions.
generation reductions.

Energies 2018, 11, x FOR PEER REVIEW
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Energies 2019, 12, 12

350
350
351
351
352
352

12 of 18
12 of 18
12 of 18

353
353
354
354
355
355
356
356
357
357
358
358
359
359
360
360

As shown
shown in
in Figure
Figure 8,
8, similarly
similarly to
to Figure
Figure 7,
7, the
the TFAM
TFAM can
can show
show the
the degree
degree of
of the
the frequency
frequency
As
acceptability
more
intuitively,
and
provide
a
more
intuitive
measure
of
the
distance
from
the
current
acceptability
more
intuitively,
and provide
a more
intuitive
measure
of the
from
current
As shown
in Figure
8, similarly
to Figure
7, the
TFAM
can show
thedistance
degree of
thethe
frequency
operation point
point to
to the frequency
frequency collapse
collapse point.
point. Hence,
Hence, it
it is
is more
more convenient
convenient for
for dispatchers.
dispatchers.
operation
acceptability
more the
intuitively, and
provide a more
intuitive
measure
of the distance
from the current
operation point to the frequency collapse point. Hence, it is more convenient for dispatchers.
4.1.3. Case
Case 3:
3: Generation
Generation Reduction
Reduction in
in Power
Power Plant
Plant P
P
4.1.3.
4.1.3.Considering
Case 3: Generation
Reduction
in Power
Plant
P P, then the TFAI under different amounts of
generation
reduction
in power
power
plant
Considering generation
reduction
in
plant
P, then the TFAI under different amounts of
generation
reduction
can
be
obtained,
which
is
shown
as
Curve
inTFAI
Figure
9. different amounts of
Considering
generation
reduction
in
power
plant
then the
under
generation reduction can be obtained, which is shown asP,Curve
11 in
Figure
9.
Curve reduction
shows that
that
when
the generation
generation
reduction
isCurve
34 MW,
MW,
which
is the
the
maximum generation
generation
generation
canwhen
be obtained,
which isreduction
shown asis
1 in
Figure
9. maximum
Curve
11 shows
the
34
which
is
reduction,
the
TFAI
is
about
0.039,
and
the
system
frequency
deviation
is
still
acceptable.
According
Curvethe
1 shows
when
the generation
reduction
is 34 MW,
whichisisstill
theacceptable.
maximum generation
reduction,
TFAI that
is about
0.039,
and the system
frequency
deviation
According
to Section
Section 2.2.1,
2.2.1,
the is
critical
disturbance
power
canfrequency
be set
set as
as ΔP
ΔP
cr = 2ΔPmax = 68 MW.
reduction,
the TFAI
aboutdisturbance
0.039, and the
system
deviation
According to
to
the
critical
power
can
be
cr = 2ΔPis
maxstill
= 68acceptable.
MW.
Furthermore,
the
TFAM
η
according
to
Equation
(3)
can
be
obtained
as
shown as
as Curve
Curve 22 of
of
Section
2.2.1, the critical
disturbance
powertocan
be set as(3)
∆Pcan
2∆P
MW.
Furthermore,
the TFAM
η according
Equation
obtained
shown
cr = be
max = 68 as
Figure
9.
Furthermore,
the TFAM η according to Equation (3) can be obtained as shown as Curve 2 of Figure 9.
Figure
9.

361
361
362
362

Figure
9.
TFAI
and
TFAM
corresponding
to different generation reduction
reduction of
of power
power plant
plant P.
P.
Figure9.
9.TFAI
TFAI and
and TFAM
TFAM corresponding
corresponding to
Figure
different generation
P.

363
363
364
364
365
365

Figure
shows that,
that, similarly
similarlyto
toFigures
Figure 777 and
and
Figure
8, the
the proposed
proposed
TFAM
is more
morecan
linear,
can
Figure 999 shows
and Figure
8,
the proposed
TFAM isTFAM
more is
linear,
provide
Figure
shows
that,
similarly
to
Figure
8,
linear,
can
more intuitive
intuitive
measure
of the
the distance
distance
from
theoperation
current operation
operation
point
to the
the frequency
frequency
aprovide
more intuitive
measuremeasure
of the distance
from thefrom
current
point to the
frequency
collapse
provide
aa more
of
the
current
point
to
collapse
point,
and
is
more
convenient
for
the
dispatchers.
point,
and
is more
for the dispatchers.
collapse
point,
andconvenient
is more convenient
for the dispatchers.

366
366
367
367
368
368
369
369
370
370
371
371

4.2. Disturbance
Disturbance with
with Multiple
Multiple Parameters
Parameters
4.2.
Consider the
the disturbance
disturbance involving
involvingsimultaneous
simultaneousgeneration
generationreductions
reductionsatat
atpower
powerplants
plants
M
and
MM
and
P.
Consider
the
disturbance
involving
simultaneous
generation
reductions
power
plants
and
P. With
With
different
simulations,
the critical
critical
disturbances
corresponding
to the
the
critical frequency
transient
With
different
simulations,
the critical
disturbances
corresponding
to the critical
transient
P.
different
simulations,
the
disturbances
corresponding
to
critical
transient
frequency
acceptability
(when
F
=
1)
can
be
obtained,
as
shown
by
the
solid
line
in
Figure
10.
In
Figure
acceptability
(when
F
=
1)
can
be
obtained,
as
shown
by
the
solid
line
in
Figure
10.
In
Figure
10,
frequency acceptability (when F = 1) can be obtained, as shown by the solid line in Figure 10. In Figure
10,
the
area
to
the
lower
left
of
the
solid
line
is
the
region
where
the
transient
frequency
is
acceptable,
the the
areaarea
to the
lower
leftleft
of of
thethe
solid
line
acceptable,
10,
to the
lower
solid
lineisisthe
theregion
regionwhere
wherethe
thetransient
transient frequency
frequency is acceptable,
right
is
the
region
where
the
transient
frequency
is
unacceptable.
while
the
upper
right
is
the
region
where
the
transient
frequency
is
unacceptable.
while the upper right is the region where the transient frequency is unacceptable.

372
372
373
373

Figure 10.
10. Critical
Critical acceptable
acceptable boundary
boundary with
with respect
respect to
to generation
generation reduction
reduction of two power plants
plants..
Figure
Figure 10. Critical acceptable boundary with respect to generation reduction of two power plants .

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Further, ∆Pdist M and ∆Pdist P indicate the generation reduction of the power plants M and P,
respectively. If the direction of the disturbance power is fixed (∆Pdist M /∆Pdist P = const), the TFAM
of the arbitrary disturbance power point can be determined by the method of Section 2.2.3. If the
direction of the disturbance power is not fixed, the TFAM of any disturbance power point can be
determined by the method of Section 2.2.2.
As shown in Figure 10, six disturbance power points are selected randomly, and the TFAM of
these points can be obtained. The results for the random disturbances A–F are shown in Figure 10 and
Table 5.
Table 5. Generation reduction of two power plants and corresponding TFAM
Point Nomenclature

Generation Reduction of
Power Plant M ∆Pdist M /MW

Generation Reduction of
Power Plant P ∆Pdist P /MW

TFAM η/%

A
B
C
D
E
F

40
30
29.25
20
10
5

20
15
7.718
6
8
3

−46.59
−14.34
4.56
31.18
53.75
79.21

As shown in Figure 10, for the acceptable cases, the distances of the points to the critical acceptable
boundary are ranked C < D < E < F. Meanwhile, the TFAM shows that C < D < E < F. Consequently,
for the acceptable case, the farther the point is from the boundary, the greater the TFAM. In particular,
C is close to the boundary, so the TFAM of C is close to 0.
As shown in Figure 10, for the unacceptable cases, the distances of the points to the critical
acceptable boundary are ranked B < A. Meanwhile, the TFAM shows that A < B. Consequently, for the
unacceptable case, the farther the point is from the boundary, the smaller the TFAM.
The above situations indicate the consistency of the TFAM and Figure 10. Similarly to Section 3.1,
the TFAM can directly show the degree of the frequency acceptability, indicating the distance from the
current operation point to the frequency collapse point, and is more convenient for the dispatchers.
4.3. Comparison of TFAM and Other Indexes
To determine the transient frequency stability, a transient frequency deviation index (TFDI) is proposed
in [27]. The TFDI only considers the case of a frequency drop. The expression of TFDI is as follows:
ε=
where
Sd = min

Sd
( f N − f cr )tcr
Z ts +tcr
ts

(14)

( f − f cr )dt

and: f N is the nominal frequency of the system; f cr is the limit of frequency deviation; tcr is the
maximum acceptable duration for frequency deviation forgoing beyond f cr ; ts is the observing window
starting time; f is the frequency response.
When ε > 0, the frequency is stable; when ε < 0, the frequency is unstable; and, when ε = 0,
the frequency is critical stable.
Ref. [26] also proposed a new index for frequency deviation security assessment. This index also
only considers the case of a frequency drop. According to whether the frequency response curve
intersects with f cr , this index can be divided into two cases. One is if the frequency response curve
intersects with f cr . The expression is as follows:
γ=

tcr − tb
tcr

(15)

408
with fcr. The expression is as follows:
408 intersects
intersects with fcr. The expression is as follows:

t −t
tcrt

γ γ==crtcr −btb
Energies 2019, 12, 12

cr

(15)
(15)
14 of 18

409
tb is the continuous duration time when frequency is less than fcr.
409 where
where tb is the continuous duration time when frequency is less than fcr.
410
For Equation (15), when 0 ≤ γ≤ 1, tb ≤ tcr, the frequency deviation is secure. When γ < 0, tb > tcr,
410
For Equation (15), when 0 ≤ γ≤ 1, tb ≤ tcr, the frequency deviation is secure. When γ < 0, tb > tcr,
where
t is the continuous duration
time when frequency is less than f cr .
411
and
the frequency
is unsecure.
411 and the bfrequencydeviation
deviation is unsecure.
For
Equation
(15),
when
0

γ

1, response
t ≤ tcr , the
frequency
deviation
secure.
When
γ < 0, tb >istcr ,
412
The other case is that the frequency
curve
does not
intersectiswith
fcr. The
expression
412
The other case is that the frequencybresponse curve does not intersect with fcr. The expression
is
and the frequency deviation is unsecure.
413
413 asasfollows:
follows:

414
414
415
415
416
416
417
417
418
418
419
419
420
420
421
421
422
422

The other case is that the frequency response curve does not intersect with f cr . The expression is
f f − −f crf
as follows:
γ γ= = fmin
11
min
(16)
−f f cr cr+ +
(16)

γ = fmin
+
1
(16)
Nf − crf
f NN− f cr cr
lowest
point
of the
frequency
response
curve.
where
fminf is the
where
the
lowest
point
the
frequency
response
curve.
min
lowest
point
ofofthe
frequency
response
curve.
minisisthe
For
Equation
(16),
γ
>
1
is
confirmed,
and
the
frequency
deviation
must
bebe
secure.
For Equation
Equation (16),
(16), γγ >>11isisconfirmed,
confirmed, and
and the
the frequency
frequency
deviation
must
secure.
For
deviation
must
be secure.
Since
both
indexes
ε
and
γ
are
based
on
a
frequency
threshold,
we
need
to
specifyaathreshold
thresholdfor
Since
both
indexes
ε
and
γ
are
based
on
a
frequency
threshold,
we
need
to
specify
Since
indexes ε and γ are based on a frequency threshold, we need to specify a threshold
forthese
thesetwo
twoindexes,
indexes,and
andthen
thencompare
comparethem
themwith
withthe
theTFAM.
TFAM.
for these
two indexes,
and then
compare them
with the
TFAM.
Firstly,
the
frequency
threshold
of
ε
is
set
to
48.8
Hz,
and the
maximum
time allowed
to be below
Firstly, the
thefrequency
frequencythreshold
thresholdofof
is set
to 48.8
the maximum
time allowed
to be
Firstly,
ε isε set
to 48.8
Hz, Hz,
and and
the maximum
time allowed
to be below
thebelow
threshold
is
set
to
0.3
s.
By
changing
the
output
loss
of
power
plant
M,
the
TFAM
and
ε
the
threshold
is
set
to
0.3
s.
By
changing
the
output
loss
of
power
plant
M,
the
TFAM
the threshold is set to 0.3 s. By changing the output loss of power plant M, the TFAM and ε
corresponding
to each to
output
canloss
be measured.
These results
are shown
in Figure
11. The11.
and ε corresponding
each loss
output
be measured.
are shown
in Figure
corresponding
to each output
loss can
becan
measured.
These These
resultsresults
are shown
in Figure
11. The
frequency
response
curves
when
the
output
loss
of
power
plant
M
is
45
MW
and
50
MW
are
shownare
The frequency
response
curves
the output
of power
plant
is 45
MW
frequency
response
curves
whenwhen
the output
loss ofloss
power
plant M
is 45MMW
and
50 and
MW50
areMW
shown
in shown
Figure in
12.Figure 12.
in Figure 12.

423
423
424
424

Figure
11.11.
Generation
reduction
of power
plant
MM
and
thethe
corresponding
TFAM
and ε.
Figure
Generation
reduction
ofpower
power
plant
and
corresponding
TFAM
Figure
11. Generation
reduction
of
plant
M and
the corresponding
TFAM
and ε.

425
425
Figure 12. Frequency response curves when the output loss of power plant M is 45 MW and 50 MW.

When the output loss of power plant M is 45 MW, the frequency response curve is shown as
Curve A of Figure 12. At this time, the frequency meets the requirements of three thresholds, so the
frequency deviation is acceptable. According to Curve 1 and Curve 2 in Figure 11, when the output

Energies 2018, 11, x FOR PEER REVIEW

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426

Figure 12. Frequency response curves when the output loss of power plant M is 45 MW and 50 MW.

427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443

When the output loss of power plant M is 45 MW, the frequency response curve is shown
as
Energies 2019, 12, 12
15 of 18
Curve A of Figure 12. At this time, the frequency meets the requirements of three thresholds, so the
frequency deviation is acceptable. According to Curve 1 and Curve 2 in Figure 11, when the output
lossof
ofpower
powerplant
plantM
Misis45
45MW,
MW,both
bothεε and
and TFAM
TFAM are
are greater
greater than
than 0,
0, and
and the
the judgment
judgment of
of these
thesetwo
two
loss
indexes
is
correct.
indexes is correct.
Similarly, when
when the
the output
output loss
loss of
of power
power plant
plant M
M isis 50
50 MW,
MW, the
the frequency
frequency response
response curve
curve is
is
Similarly,
shown as
as Curve
Curve BB in
in Figure
Figure 12.
12. The
The duration
durationwhen
whenthe
thefrequency
frequencyisis lower
lower than
than 48.8
48.8 Hz
Hz isis greater
greater
shown
than 0.3
0.3 s, so the frequency
Curve
2 in
Figure
11,
than
frequency deviation
deviationisisunacceptable.
unacceptable.According
AccordingtotoCurve
Curve1 and
1 and
Curve
2 in
Figure
when
the the
output
loss loss
of power
plantplant
M is 50MW,
both ε both
and TFAM
less than
0, and
the0,judgment
11,
when
output
of power
M is 50MW,
ε and are
TFAM
are less
than
and the
of these two
indexes
correct. is correct.
judgment
of these
twoisindexes
In addition,
addition, because
threshold
of 48.8
Hz,Hz,
andand
TFAM
needs
to consider
three
In
because εεonly
onlyfocuses
focusesononthe
the
threshold
of 48.8
TFAM
needs
to consider
thresholds,
the zero-crossing
positions
of theof
two
in Figure
11 are11
different.
three
thresholds,
the zero-crossing
positions
thecurves
two curves
in Figure
are different.
From the
the above
above analysis,
analysis, itit can
can be
be seen
seen that
that both
both εε and
and TFAM
TFAM can
can correctly
correctly determine
determine the
the
From
acceptability of
of frequency
frequency deviation.
deviation.
acceptability
Second,the
thefrequency
frequencythreshold
threshold
is to
set49toHz,
49 and
Hz, the
andmaximum
the maximum
time allowed
to be
Second,
of of
γ isγset
time allowed
to be below
below
the
threshold
is
set
to
10
s.
By
changing
the
output
loss
of
power
plant
M,
the
TFAM
and
the threshold is set to 10 s. By changing the output loss of power plant M, the TFAM and γγ
corresponding to
to each
each output
output loss
loss can
can be
be measured.
measured. These
These results
results are
are shown in Figure 13.
corresponding

444
445

Figure13.
13.Generation
Generationreduction
reductionof
ofpower
powerplant
plantM
Mand
andcorresponding
corresponding TFAM
TFAM and
and γ.
γ.
Figure

446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462

When the
the output
outputloss
loss of
of power
power plant
plantM
M isis 45
45 MW,
MW, the
the frequency
frequency response
response curve
curve is
is shown
shown as
as
When
CurveA
A of
of Figure
Figure 12.
12. At
Atthis
thistime,
time,the
the frequency
frequency meets
meetsthe
therequirements
requirementsof
of three
three thresholds,
thresholds, so
so the
the
Curve
frequency
deviation
is
acceptable.
According
to
Curve
1
and
Curve
2
in
Figure
13,
when
the
output
frequency deviation is acceptable. According to Curve 1 and Curve 2 in Figure 13, when the output
lossof
ofpower
powerplant
plantM
Misis45
45MW,
MW, both
both γγand
andTFAM
TFAM are
are greater
greater than
than 0,
0, and
and the
the judgment
judgment of
of these
thesetwo
two
loss
indexes is
is correct.
correct.
indexes
Similarly,
when the
the output
output loss
loss of
of power
power plant
plant M
M isis 50
50 MW,
MW, the
the frequency
frequency response
response curve
curve is
is
Similarly, when
shownas
asCurve
CurveBBin
inFigure
Figure 12.
12. The
Theduration
durationwhen
whenthe
thefrequency
frequencyisislower
lowerthan
than49
49Hz
Hzisisgreater
greaterthan
than
shown
10
s,
so
the
frequency
deviation
is
unacceptable.
According
to
Curve
1
and
Curve
2
in
Figure
13,
10 s, so the frequency deviation is unacceptable. According to Curve 1 and Curve 2 in Figure 13, when
when
the
output
loss
of
power
plant
M
is
50
MW,
both
γ
and
TFAM
are
less
than
0,
and
the
judgment
the output loss of power plant M is 50 MW, both γ and TFAM are less than 0, and the judgment of
of these
indexes
is correct.
these
twotwo
indexes
is correct.
In
addition,
because
onlyfocuses
focuseson
onthe
thethreshold
thresholdof
of49
49Hz,
Hz,and
andTFAM
TFAMneeds
needsto
toconsider
considerthree
three
In addition, because γγonly
thresholds,
the
zero-crossing
positions
of
the
two
curves
in
Figure
13
are
different.
thresholds, the zero-crossing positions of the two curves in Figure 13 are different.
From the
the above
above analysis,
analysis, itit can
can be
be seen
seen that
that both
both γγ and
and TFAM
TFAM can
can correctly
correctly determine
determine the
the
From
acceptability
of
frequency
deviation.
acceptability of frequency deviation.
Compared with
thethe
TFAM
have
better
linearity,
and can
a more aintuitive
measure
Compared
with γ,
γ,ε εand
and
TFAM
have
better
linearity,
andprovide
can provide
more intuitive
of
the
distance
from
the
current
operation
point
to
the
frequency
collapse
point.
It
is
more
convenient
measure of the distance from the current operation point to the frequency collapse point. It is more
for dispatchers
to control the
frequency
using these
two indexes.
with ε, thewith
TFAM
convenient
for dispatchers
to control
the by
frequency
by using
these twoCompared
indexes. Compared
ε,
proposed in this paper can meet the requirements of multiple frequency thresholds and have a wider
range of applications.

Energies 2019, 12, 12

16 of 18

5. Conclusions and Discussions
This paper proposes a new transient frequency acceptability margin (TFAM) based on TFAI.
First, a TFAI is proposed based on the frequency trajectory and the principle that different weights
correspond to different frequency deviations. The TFAI considers the frequency thresholds and time
duration limits. Then, the effectiveness of the TFAI is verified by simulation. Further, the critical
disturbance power is determined by using the TFAI. The TFAM is proposed to measure the distance of
the actual disturbance power to the critical disturbance power. It must be noticed that the TFAM is
expressed in power space and considers the power mismatch. Consequently, compared with other
frequency margins, the TFAM is more convenient to formulate security control measures. The TFAM
can also be used to compare the transient frequency deviation acceptability under different faults.
The TFAM is convenient for dispatchers to judge the transient frequency acceptability and implement
security measures in time.
There are still some problems in the TFAI and TFAM proposed in this paper. When the frequency
oscillates violently, the area of integration is very small, so the TFAI is likely to fail to determine the
frequency acceptability. Moreover, the critical boundary for TFAM needs to be obtained by off-line
calculation. For large-scale systems, off-line calculation of the critical boundary under different
operation modes and faults is time consuming. Therefore, the calculation method of the TFAI and
TFAM needs further study.
Author Contributions: A.X. conceived and designed the experiments and analyzed the data; J.C. and J.W.
performed the experiments in the PSD-BPA and analyzed the data; L.Y. provided the power network data; J.C.
and T.B. discussed about the work and provided advices on improvement.
Acknowledgments: The authors thank the anonymous referees for their helpful comments and suggestions.
This work was supported in part by the National Natural Science Foundation of China under Grants
(51477050, 51627811) in part by the 111 Project (B08013) and the Fundamental Research Funds for the Central
Universities(2018ZD01).
Conflicts of Interest: The authors declare no conflict of interest.

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