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IET Research Journals
Probabilistic Framework for Transient Stability Contingency Ranking of Power Grids with Active Distribution Networks: Application in Post Disturbance Security Assessment
ISSN 1751-8644
doi: 0000000000
www.ietdl.org
Mohsen Tajdinian1, Mehdi Allahbakhshi2, Mostafa Mohammadpourfard3∗,Behnam Mohammadi4,Yang Weng 5, Zhaoyang Dong6
1 School of Electrical and Computer Engineering, Shiraz University, Shiraz, Iran
2 School of Electrical and Computer Engineering, Shiraz University, Shiraz, Iran
3Department of Electrical and Computer Engineering, Sahand University of Technology, Tabriz, Iran
4Faculty of Electrical and Computer Engineering, University of Tabriz, Tabriz, Iran
5School of Electrical, Computer and Energy Engineering, Arizona State University, Tempe, USA
6School of Electrical Engineering and Telecommunications, University of New South Wales, NSW, Australia
* E-mail: [email protected]
Abstract: The conventional distribution power network is becoming an active grid due to widespread integration of distributed energy resources (DERs). This integration raises great concerns about the stability and security of the power system considering the growing interactions between the transmission and active distribution systems. One way to ensure the stability of the integrated power system is the prediction of critical contingencies which can jeopardize the power system security. Therefore, this paper aims to investigate transient stability contingency ranking in power grids considering the uncertainties raised by DERs of active distribution networks. To this end, a probabilistic transient stability prediction framework is provided, in which the probability density function of transient stability condition is calculated based on the normalized stability indicators. The normalized stability indicators are based on the rotor angle and rotor speed. Unlike the previously suggested transient stability indicators, the normalized stability indicators are only utilized in the case of using fault data for prediction. Based on the proposed scheme, the instability of generators is anticipated through the proposed probabilistic transient stability prediction framework. Also, the effects of the probable topology changes on the line overload and also the voltage profile are investigated. Finally, the performance of the proposed method is validated and compared with the time domain simulation under various operating conditions.
I. INTRODUCTION
HE occurrence of an unusual event in the power system
affects the stability level of the system and thereby
threatening the power system security. Large disturbance
stability or so-called transient stability (TS) is one of the
important types of power system stability. TS is defined as the
ability of a power system to maintain the synchronism between
the rotor angles of the generators when it is subject to a severe
disturbance [1]. Since power systems are operated near the
maximum capacity, the inability or failure in fast prediction of
the TS condition may lead to wide area blackouts. As a result,
situational awareness of the TS condition is crucial in the power
security assessment [2].
Increasing penetration of renewable energy sources (RESs)
has significantly affected the transient stability of the power
system with mix power generations. The presence of different
generators with different dynamic behaviors, replacing
synchronous generators by RES and stochastic behavior of RES
existence are the factors enhancing the challenge of TS in a high
penetrated RES power system. As a result, performing TS
analysis for the worst case condition considering uncertainties
due to the presence of RES seems neither practical nor feasible
[3]. References [4]- [6], discuss the effects of distributed
generations (DGs) on controlling frequency and voltage issues
in the combined transmission and distribution systems. They
emphasize the importance of considering the interactions
between transmission grids and active distribution systems in
power system operation, stability and security assessments.
Focusing on the significance of considering such interactions in
power system operation, several publications can be found that
try to address different challenges in the integrated power
systems such as power flow solution [7], optimal power flow
[8], economic dispatch [9], unit commitment [10], reliability
assessment [11], voltage stability [12] and static contingency
analysis [13]. However, transient stability contingency ranking
has not been fully discussed considering the impacts of DGs in
distribution networks on the transmission grids.
Although situational awareness in the TS assessment for a
T
2
certain contingency is very crucial, there are some critical
contingencies in the power system that, if they happen, will
jeopardize the security level of the power system. As a result,
TS prediction and contingency ranking are connected to each
other and can be analyzed together [14]. It is worth mentioning
that the contingency analysis in this paper is focused on the (N-
1) TS contingency ranking with the aim to find the impact of
TS contingency on the post disturbance condition of the power
system security.
Regarding TS assessment and contingency ranking, several
publications have been found to investigate these issues
separately or together. To have a general understanding of both
TS assessment and contingency ranking, the literature
surveying both issues is given in the following. It should be
noted that the main focus of this paper is to probabilistically
investigate TS contingency analysis.
It has been acknowledged that the previously published
papers are divided into two general groups: deterministic and
probabilistic assessment. Most of the previously published
papers have discussed TS issues in a deterministic way either
for evaluation or prediction. Deterministic methods try to find
TS condition or margin for a certain condition contingency.
Deterministic approaches are usually designed to predict TS for
online application. Conventionally, time domain simulation
(TDS) [2], and the transient energy function [15]– [22]
approaches have been used for dealing with TS issues in a
deterministic way. Modern analysis, which is based on the
machine learning algorithms and pattern recognition methods,
provides more feasibility and flexibility in TS awareness and
power system security classification [23]- [25]. In addition,
some hybrid methods are designed to predict TS condition for
online application in a deterministic way [26]- [30]. It is
obvious in the deterministic analysis of TS stochastic variation
of the load and generation is ignored. Also, the probabilistic
nature of disturbance will be missed in the calculation. In
contrast with deterministic TS assessment, there are two
probabilistic approaches dealing with the TS assessment: 1.
Analytical-based methods [31],[32], which use conditional
probability theory to evaluate the probability of generator
stability. These methods consider the uncertainty in type,
location, and duration of the fault and are implemented in both
offline and online manners. 2. Monte Carlo simulation (MCS)-
based methods [3],[33],[34]. Unlike analytical – based
methods, the MCS based methods compute the probability
density function of critical clearing time (CCT) for each
generator without sacrificing the details of power grids
network. Although the MCS based methods are more accurate
than the analytical based methods, it is obvious that the MCS
based methods are not appropriate for real-time application due
to the large computational burden. Also, the probabilistic
analysis may not help to real-time prediction. However, it can
profoundly improve the planning issues related to the transient
stability of the power grids [3],[34].
TS contingency ranking has been addressed in several
publications based on the time domain simulation and equal
area criterion [35]- [38], transient energy function [39]- [41],
the combination of PMU with fuzzy neural network [42], [43],
regression based model [44], data mining [45] and DT [46],
[47], which are focused on the online CR. As a result, the pros
and cons of these methods are almost limited by the speed of
calculation and assessment. Investigating the previously
published contingency ranking methods, it has been found that
the probabilistic contingency ranking and also post disturbance
power system security assessment based on the probable
change caused by the obtained contingency ranking have not
been adequately addressed in the literature. The probabilistic
contingency ranking may not be realized in online or real-time
calculation even in the close future. However, it can provide a
general overview for TS prediction with offline calculation and
it can be employed for online situational awareness for the post
fault power system security assessment [14], [35].
This paper provides a probabilistic framework for TS
contingency prediction and consequently CR. This framework
is designed for probabilistic identifying of severe contingencies
that result in instability of generators. Based on the latter
identification, the power system security is analyzed for
probable topology changes caused by the identified severe
contingencies. The paper has following contributions:
normalized transient stability index (NTSI) is introduced
which is based on critical rotor angle (CRA) and critical
rotor speed (CRS). Unlike several previous suggested
indices [48] which require post fault data for calculation,
the NTSI is calculated at the clearing time and as a result,
it is independent of post fault. Also, the NTSI varies
between (0, 1), so the range of NTSI variation is known.
Based on the Monte-Carlo simulation (MCS), probability
density function (PDF) of NTSI is calculated for different
influential random variables. It is noteworthy that the
effects of load and generation variation, clearing time and
different types of fault are considered in the calculation of
the PDF of NTSI.
Based on the probabilistic transient stability prediction
(PTSP) framework, the contingency ranking (CR) analysis
is conducted based on the new criterion so-called Weighted
Normalized Total Stability Index (WNTSI) which takes all
generators inertia and PDFs of NTSI into consideration.
Same as PDF of NTSI, WNTSI takes the effects of load
and generation variation, clearing time and different types
of fault into consideration. As a result, WNTSI has a
general overview due to power system stochasticity
considerations in comparing with deterministic indices
given in [35]- [48].
PTSP identifies the critical contingencies resulting in
generator instability. However, unlike previous
investigations such as [35], [40], [44], and [48] the
probable topology changes due to generators and line
outage, are employed to assess the level of line overload
and voltage deviation for the critical contingency. In other
words, the impact of transient stability contingencies on
voltage and active power security indices are investigated.
As one can see in simulation results given in section IV,
PTSP method can anticipate the instability based on the
fault clearance time data, which will lead to instability
anticipation much faster than instability of generator rotor
3
angle (generator outages may occur 1 or multiple seconds
after fault clearance). In addition, the CR analysis through
the proposed framework helps to anticipate the behavior of
the probable impact of the generators outage in the
transient stability analysis on the violation of bus voltage
and line overload constrains.
The structure of the paper is as follows: the fundamental
mathematics of the TS indicator for prediction and contingency
ranking framework is discussed in Section II. The
implementation of the proposed framework is discussed in
Section III. Simulation results and some comments on
conclusions are provided in sections IV and V, respectively.
II. THE PROPOSED CONTINGENCY RANKING FRAMEWORK
This section provides and discusses the basic mathematics of
the TS indicator for prediction. Also, some indices for CR
analysis and active/reactive line overload and allowable voltage
margin are discussed in this section. It should be noted that the
main focus of this paper is to equip a MCS framework with two
simple-calculable indices so that it quickly predicts stability,
thus contributing to situational awareness. Then, as an
objective, the proposed scheme is applied to CR analysis to
analyze the power system security condition for the probable
topology changes.
A. Calculation of Transient Stability Indicators
Surveying literature shows that several indices have been
suggested for TS and dynamic security evaluations.
Investigating the definition of the previously suggested indices,
it has been found that most of the indices are defined based on
the post fault condition of the power system. Requiring post
fault data does not question the performance of a certain index.
It means that more data are required to obtain a reliable value
of the index. Besides, most of the previous indices have not
been defined with a known variation range. The most
comprehensive list of TS and dynamic security evaluation
indices can be found in [48].
To confront with the aforementioned drawbacks, two
normalized stability indices (NSIs) are introduced as follows:
,
,
,
( )( )
1 ( )
i cl
i cl c i
c ii
i cl c i
tif t
NRASI
if t
(1)
,
,
,
( )( )
1 ( )
i cl
i cl c i
c ii
i cl c i
tif t
NRSSI
if t
(2)
max ,i i iNTSI NRASI NRSSI (3)
Where, clt denotes fault clearing time. ( )i clt and ( )i clt
represent the rotor angle and rotor speed of the ith generator
during a fault, respectively. iNRASI , iNRSSI and iNTSI are
normalized rotor angle stability, normalized rotor speed
stability and normalized total stability indices of the ith
generator, respectively.
The parameters ,c i and ,c i are critical rotor angle and rotor
speed of the ith generator, respectively. The aforementioned
critical parameters are calculated for a three-phase fault at the
ith generator bus, which is the worst case condition (WCC) that
may happen for any generator [27].
While normalizing based on the WCC may result in strict
prediction, it should be noted that two NSIs have two salient
features compared with the previously suggested indices: first,
the variation of two NSIs is between 0 and 1. Second, these two
NSIs only require ( )clt and ( )clt for each generator.
B. The Criterion for Contingency Ranking
For a set of scenarios, a point of the power network is
identified as the critical one if the fault scenario results in a large
number of unstable generators. To perform CR analysis based
on the PDF of NTSI, a CR criterion (CRC) must be defined
based on the obtained PDF of NTSI of the generators. It is
assumed that for a fault at the ith bus of the power network, the
PDF of NTSI for the jth generator is denoted by ( )i
jg X . The
following expression is defined for the ith fault location:
1
1
( )g
g
N
i
j j
ji
N
j
j
M g X
WNTSI
M
(4)
Where, jM denotes the inertia of the jth generator.
Expression (4) shows the WNTSI for the ith fault location. In
other words, WNTSI is a PDF for the TS condition of the whole
power system. A critical contingency is identified if at least
three of the following pre-assumed constrains for the ith WNTSI
are satisfied:
1 2 3
1 2
2 3
1 3
1: ( ) 0.75
2 : ( ) 0.85
3 : ( ) 0.8
i
i
i
C C C
C Ci th Crittical Contingency
C C
C C
C min WNTSI
C max WNTSI
C mean WNTSI
(5)
The limits of the constrains given in (5) are selected based
on the several fault scenarios considering different operation
conditions, fault location, fault resistance, and fault type. From
multiple scenarios, it is found that in these limitations, the
condition of the whole power system is more likely to be near
the critical one.
Critical contingencies predict the critical generators through
the proposed PTSP framework. However, the type of transient
instability may remain unrecognized. For instance, for one
contingency, a generator may experience instability in the first
swing and for another contingency, the same generator may
experience instability for multiple further swings. Fast
prediction helps the situational awareness of corrective actions
[3], [35]. However, after critical contingencies, the topology
changes may result in the changes of the overload and voltage
profile of the power network so that the power system security
4
would be threatened. To have fast situational awareness of the
overload of transmission lines and also voltage deviation of
buses, utilizing the predictive ability of the proposed PTSP
framework, the level of overload and voltage stability index for
the probably predicted topology change can be calculated as
follows: after before
k k
k before
k
P PAPOL
P
(6)
after before
i i
i before
i
V VVSI
V
(7)
Where, APOL and VSI denote active power overload for
the kth line and voltage stability index for the ith bus,
respectively [49].
1
LNj
kj k
L
APOL
TVoPIN
(8)
1
bNj
ij i
b
VSI
TVoVIN
(9)
Where jTVoPI and
jTVoVI are total variation of active power
overload and total variation of voltage stability index for the jth
contingency. bN and bN denote the line number and bus
number, respectively.
III. IMPLEMENTATION
The implementation of the proposed PTSP, CR analysis and
post disturbance security assessment is presented in Fig.1. As
previously discussed, the presented framework is designed to
deal with the TS situational awareness taking into account the
uncertainties of the power system.
There are two types of random variables, which are considered
in this investigation.
First, random variables that belong to stochasticity of the
disturbance in power system. Fault type, fault location, and
fault clearing time are the random variables that are
considered to show the stochasticity of the disturbance in
power system.
Second, random variables that belong to stochasticity of
the power system operation, and climate behaviors. Load
and generation are the random variables that are considered
to demonstrate stochasticity of the power system.
In the proposed method, in each level of power generation,
several fault scenarios may happen and for each fault scenario,
several differential-algebraic equations should be solved in time
domain simulation. In other words, considering PDFs of
influential parameters for pre-fault and during fault conditions,
the PDFs of transient stability margins through Monte Carlo
simulation is extracted.
To implement the proposed probabilistic framework, this paper
utilizes MCS, in which the stochastic behavior of the power
system is repeatedly applied to the test system. The test system
is implemented in MATLAB/Simulink. Fig.1 illustrates the
implementation stages of the proposed framework. According
to Fig.1, the procedure for performing the proposed framework
is as follows:
Stage 1 (Initialization): in stage 1, after reading system data, the
power flow is conducted on the test system. Also, the CRA and
CRS are calculated for all generators in WCC (i.e. three-phase
fault at generator terminal).
Stage 2 (Probabilistic Transient Stability Prediction): for a
certain fault location, the NTSI is calculated through applying
MCS on (3). In the MCS procedure, fault clearing time (FCT),
fault type (FT), active and reactive powers of generators and,
active and reactive powers of load are selected as random
variables (RVs).
FT is assumed to have a uniform distribution as follows [50]:
0 0.65
0.66 0.80
0.81 0.90
0.9 1
c then LG fault
c then LL faultc
c then LLG fault
c then LLLG fault
(10)
Where c is the probability of FT and (0,1)c . Also different
FTs in (10) consist of line-to-ground (LG), line-to-line (LL),
double line-to-ground (LLG) and three line-to-ground (LLLG)
respectively. The distribution function of fault clearing time
(FCT) is assumed to have a uniform distribution as follows:
: [300,340]
: [280,300]
: [250,280]
: [200,250]
LG fault Uniform
LL fault UniformFCT
LLG fault Uniform
LLLG fault Uniform
(11)
It should be noted that in (11), the numbers are in millisecond.
The active and reactive powers for generators and loads are
assumed with a normal distribution. The mean values of the
active and reactive powers are assumed equal to the given
deterministic values for power flow as described in [28]. It is
assumed that for all the active and reactive powers, standard
deviation varies between 5% to 15% of the mean value.
Stage 3 (Contingency Ranking): the procedure of CR is
performed by calculating WNTSI for all fault locations. Having
all WNTSI , the criteria given in (5) are checked for any
WNTSI . If any of WNTSI belongs to the described sets in (5),
the corresponding fault location is selected as a critical
contingency. After identifying the critical contingencies, the
contingency can be sorted to specify the criticality level of the
identified contingencies.
Stage 4 (Post Disturbance Security Assessment): each critical
contingency may lead to the topology changes that definitely
influence the power system security level. To determine the
post disturbance security level, using (6) to (9), the level of
violation in the active and reactive power flow of the line as
well as voltage magnitude are calculated. These values are also
sorted for different identified contingencies so that the impact
of each critical transient stability contingency on the post
disturbance condition is determined.
IV. SIMULATION RESULTS AND DISCUSSION
The performance of the proposed multi-objective framework is
validated by applying the framework on the IEEE 39 bus test
system. The transmission grid is included with 8 active
5
distribution networks (ADNs). According to Fig.2.b, ADNs are
constructed from multiple IEEE 33 bus test system. Each ADN
contains a certain type of distributed generation (DG) including
doubly fed induction generators (DFIGs) based wind turbine
(WT) and photovoltaic (PV) which are specified in Fig.2. a.
About 200MW DGs are installed in the test system, as shown
in Fig.2. The characteristics of the IEEE 39 bus test system,
IEEE 33 bus test system, DFIGs and PVs could be found in [3],
[26], [52]. As previously mentioned, in this paper, the
stochasticity of DGs are only considered in the power
generation PDFs. In this investigation, the Weibull, beta, and
normal PDFs are considered for power generations of WT, PV
and synchronous generators, respectively. In addition, the PDF
of load is assumed normal distribution. The characteristics of
load and generation PDFs are adopted from [3], [26], [53].
The performance of the proposed framework is verified in four
steps. In the first step, a case study is provided in which the
performance of the PTSP is illustrated, investigated and
compared with time domain simulation. In the second step, the
performance of the proposed method is compared with two
previously published methods in contingency screening
including a Lyapunov function based approach [40], and a
regression based approach [44]. In the third step, the
contingency ranking and critical contingency selection are
performed by the proposed framework. Eventually, in the
fourth step, the security assessment for the probable topology
changes is concluded from the critical contingency ranking.
Fig. 1. The procedure of implementing PTSP, CR analysis and post
disturbance security assessment
(a) (b)
Fig. 2. Single line diagram of the test system (IEEE 39 bus) with active distribution networks (ADNs), (a) integrated transmission and distribution grid,
(b) active distribution network with distributed generation (DG)
6
A. Case1) Evaluation of PTSP
As previously discussed, the proposed PTSP is provided to a
PDF that shows the stability margin of each generator. If the
PDF sample of NTSI is close to 1, then the post fault TS
condition of the generator is predicted as critical. It must be
emphasized that the NTSI is a normalized index and it is
calculated only at clearing fault point. To show the
effectiveness of the proposed PTSP, a three-phase fault, with
fault duration (FD) of 0.3 s, is applied on bus 14. The fault is
initiated at 2t s and as shown in Fig.3.a, the rotor angles of
G3,G4,G10, and WT1 become unstable. The proposed PTSP is
also applied for the selected fault location. After applying 1000
iterations with different FTs and FCTs, the PDF of NTSI is
calculated for all generators. The results of the MCS are shown
in Fig.3. b. As illustrated in Fig.3.b, the PDF of NTSI for G3,
G4, G10, and WT1 are at the right side of the vertical dash line.
It means that the NTSI samples obtained by MCS for G3, G4,
G10 and WT1 are very close to 1 and as a result, these
generators are predicted to have critical behavior for this fault
location at the post fault condition. It should be noted that the
proposed PTSP only provides an anticipation for the post fault
TS condition of the generator. However, the proposed PTSP
cannot show whether the generator definitely becomes unstable
or not. But as seen in Fig.3, the proposed PTSP only needs the
clearing fault point of rotor angle and can predict the stability
margin with good precision. As seen in Fig.3.a, the rotor angles
of G3, G4, G10 and WT1 become unstable almost after 2 or 3
seconds of fault clearance, while the proposed method
anticipates the critical generators utilizing the clearing fault
data.
B. Case2) Comparative analysis with previous published
methods
In the subsection A (Case 1), the PDFs of NTSI was calculated
and further evaluated with TDS method. This subsection
provides some case studies for performance evaluation of the
proposed method and previous published contingency
screening methods. Two methods have been selected: A
Lyapunov function based approach [40], and a regression based
approach [44]. Considering TDS method as the benchmark of
results correctness, the performance of proposed method, [40]
and [44] are evaluated for the scenarios given in Table 1. It
should be noted that each fault scenario in Table 1 is initiated
at t=2 s and the total simulation time in all scenarios is 10 s. The
rotor angles of the generators for all scenarios are given in
Fig.4. The unstable generators of all scenarios for the proposed
method, [40] and [44] are given in Table 2.
Table 1: specification of fault scenarios in Case 2
Scenario 1 2 3 4
Fault type LG LL LLG LLLG
Fault duration
(s) 0.54 0.445 0.325 0.25
Fault location
(bus number)
18 24 3 27
(a)
(b)
Fig. 3. Performance evaluation of PTSP for the fault at bus 14 (FD=0.3 s), (a) rotor angle of the generators, (b) PDF of NTSI
7
To make efficient comparison between the methods, the
number of identified unstable generators (NUG), the number of
correct identified (NCI) unstable generators, and total
computational time (TCT) from fault initiation until instability
occurrence are tabulated in Table 3.
From Tables 2 and 3, and Fig.4 it can be concluded that:
The NUG of the proposed method in all scenarios is in
compliance with TDS method, while the other methods
almost have failed to identified the number of unstable
generator. Moreover, NCI of the proposed method have
very good accuracy comparing with TDS method, while
the other methods have not been able to identified correct
unstable generators. The reason of good precision of the
proposed method is that it obtains required data from
solving structure preserving of synchronous generator, WT
and PV. On the contrary, the methods in [40] and [44] treat
distribution networks as a constant load and do not take
into account the dynamic models of WT and PV in the
transient stability evaluation.
As it can be seen in Table 3, the TCT of the proposed
method is very low comparing with TDS method. It hould
be mentioned that TCT of [40] and [44] is almost similar
to the proposed method. The only difference between
proposed method and the other methods is that the
proposed method only utilizes data from fault initiation
until fault clearance, but the nature of the other methods is
that they require post fault data for the assessment. The
Table 2: Unstable generators of all scenarios for the under investigation methods in Case 2
1 2 3 4
TDS G8, WT5 G7, WT4 G3, G4,
G10,WT1
G3, G4, G5, G8, G10,
WT1,
WT2,WT5
[40] G8, WT5 G9, WT4, WT5 G3, G4,
G6, G9, WT2
G3, G4, G6, G7, WT1
[44] G3, G7, G10 G3, WT4 G3, G5,
G10,WT1, WT4
G2, G4, G6, G10,
WT3, WT5
Proposed
method G8, WT5 G7, WT4
G3, G4,
G10,WT1
G3, G4, G5, G8, G10,
WT1, WT2, WT4
Table 3: NUG, NCI and TCT of the under investigation methods in case 2
Scenario 1 Scenario 2 Scenario 3 Scenario 4
NUG NCI TCT
(s) NUG NCI
TCT
(s) NUG NCI
TCT
(s) NUG NCI
TCT
(s)
TDS 2 2 85.4 2 2 95.4 4 4 104.3 8 8 126.4
[40] 2 2 24.2 3 1 25.1 5 2 32.2 5 3 29.2
[44] 3 0 29.7 2 1 27.3 5 3 37.1 6 3 31.4
Proposed
method 2 2 31.2 2 2 28.2 4 4 30.8 8 7 33.8
(a) (b)
(c) (d)
Fig.4: Rotor angles of the generators for scenarios of case 2, (a) LG, (b) LL, (c) LLG, (d) LLLG
8
latter simply means if we able to apply parallel computing
for reducing TCT, then according to case 1 in the previous
section, the proposed method is probably able to anticipate
the unstable generator may be multiple seconds before
rotor angle crosses to ±180º.
C. Case3) Contingency Ranking
The previous case studies showed that the proposed PSTP is
able to provide a PDF, which can properly anticipate the post
fault TS condition of each generator at the clearing fault point.
This case study, however, is designed to obtain the critical fault
location points, where the post fault TS condition of power grid
becomes critical. To find critical contingencies, according to
section III, at each bus considered as a fault location, over 1000
random scenarios are applied and the PDF of WNTSI are
calculated. As illustrated in Fig.4.a, for some fault locations, the
PDF of WNTSI are at the right side of the vertical dash line,
that is to say these PDFs have samples that are close to 1.
Applying (5) to the obtained PDFs of WNTSI, the critical fault
locations are listed in Table 1. As seen in Table 1, for almost all
identified critical fault locations, the mean, min and max values
do not violate the critical criteria of (5). The ranking of fault
location for all buses, based on the mean, min and max values,
are illustrated in Fig.5.b to Fig.5. d. It should be noted that the
number above the bar shows the fault location. From Fig.5, the
identified critical fault locations are 7, 24, 20, 19, 27, 6, 31, 4,
3 and 16 respectively.
It should be mentioned that for each fault location, the PDF of
WNTSI shows the anticipated post fault TS condition for the
whole power system. It means that some generators may not
experience instability; however, the behavior of all generators
reflected in WNTSI shows the degree of criticality. Table 4
provides a list of critical generators for the identified critical
bus. For example, for the first rank, critical fault location is
selected at bus 7. In this fault location, for 1000 MCS, it is
observed that the generators G3, G8, G10, WT1 and WT4 have
NTSI samples very close to 1 and they may experience
instability at post fault condition.
D. Case 4) Security Assessment for the Probable Topology
Changes
The first goal of the proposed PTSP is to quickly find post
disturbance TS condition of the generators with high accuracy.
However, knowing post disturbance TS condition of generators
can lead to some topology changes in the power grid. Such
aforementioned topology changes may result in the violation of
allowable voltage bus variation or active power line overloads.
As a result, it is necessary to identify what critical TS
contingency results in more violation in voltage of buses or line
overload after the probable topology changes. To find the
effects of identified critical TS contingencies on the voltage and
line overload constrains, the probable topology changes given
in Table 4 are applied on the test system and, for these ten
critical contingencies, the TVoVI and TVoPI are calculated,
ranked and illustrated in Fig.6.
From Fig.6 and comparing the values of TVoVI and TVoPI
between ten critical TS contingencies, it is concluded that the
violation in the line overload is expected to more than voltage
stability. In other words, in the values of TVoVI, there is a
significant difference between the first two contingencies and
the rest ones while such a difference is not observed between
them in the values of TVoPI.
Additionally, from Fig.6, it is concluded that the most critical
contingency in the TS condition of power system does not
necessarily lead to the most violation in the security indices
after topology changes. As seen in Table 1, buses 19 and 20 are
identified as the 4th and 3rd critical fault locations from TS point
of view, respectively. However, as seen in Fig.6.a, if the
topology changes corresponding to these contingencies happen,
the TVoVI will capture the highest value among all critical
contingencies. The high value in TVoVI means the high voltage
variations of buses after these contingencies, as compared to
before topology changes. Similar results can be found about
TVoPI. According to Table 1, buses 4 and 6 are identified as
the 8th and 6th critical fault locations from TS point of view,
respectively. However, according to Fig.6.a, TVoPI
corresponding to these contingencies obtain the highest value
among all critical contingencies. It simply means that the
corresponding topology changes to these fault locations will
lead to more line overloads than the other critical TS
contingencies.
Investigating all simulation results, it can be concluded that the
proposed NTSI is able to show the TS margin at post fault
condition using the clearing fault point, as investigated in case
1. This achievement in combination with probabilistic
simulation will provide a general view of stability margin for
each generator and the whole system. Applying the proposed
PTSP in case 3 showed that the proposed PTSP can simply and
approximately anticipate the TS margin and identify the most
critical area of the power grid. It should be noted that this
identification is accomplished using the fault clearing point and
the probable topology changes caused by generator outages
may occur 1 or multiple seconds after fault clearance.
Moreover, identified critical generators are utilized in case 4, to
show how they may affect the power system security
constrains. Also in case 4, it is shown that the critical TS
contingency may not lead to the most violation in the voltage
bus constrains or line overloads. Considering the discussions
above, the proposed PTSP can be used for probabilistic TS
assessment and contingency ranking to provide some
information about probable topology changes and power
system security constrains. It is also noteworthy that such
information can be useful in other research fields such as cyber-
physical attack detection [51], [52] and transient stability-
constrained optimal power flow [54], [55] in power system with
high-penetration of renewable energy resources.
9
(a)
(b)
(c)
(d)
Fig. 5. Application of proposed PTSP for finding critical contingencies, (a): PDF of WNTSI, (b): Ranking of TS contingencies based on mean value, (c): Ranking of TS contingencies based on min value, (d): Ranking of TS contingencies based on max value
10
(a)
(b)
Fig. 6. Application of proposed PTSP for security assessment of post fault
probable topology changes, (a): Ranking of critical TS contingencies based on the severity of bus voltage, (b): Ranking of critical TS contingencies based on
the severity of line overload
Table 4
Ranking of Critical Contingencies and Probable Generators/Line outage
(* Fault location is near bus i)
Rank Critical
Bus Critical generator
Faulted Line (between
buses i and j)
1 7 G3,G8,G10,WT1,WT4 7 and 6
2 24 G6,G7,WT2,WT3,WT5 24 and 16
3 20 G4,G5,G6,WT1, WT4
,WT3,WT5 20 and 19
4 19 G4,G5,G6,WT1, WT2 WT4
,WT3,WT5 19 and 16
5 27 G9,G10, WT1, WT2 ,WT5 27 and 26
6 6 G3,G8,WT2,WT1,WT4 6 and 5
7 31 G3, G4 ,WT3 ,WT4 31 and 6
8 4 G8,G10, WT1,WT4,WT5 4 and 5
9 3 G9, G10,WT1,WT2, WT4,WT5 3 and 2
10 16 G6, G9 ,WT3,WT5 16 and 17
V. CONCLUSION
Power system security can be threatened due to the failure in
the prediction of critical contingencies. However, the raised
uncertainties and complexities due to widespread introduction
of DERs have significantly increased the challenges of critical
contingency prediction and ensuring the stability of the
integrated power system. Therefore, in this research study, TS
contingency ranking was investigated in power grids taking the
RESs uncertainties into account. TS contingency ranking was
conducted based on the new PTSP framework. In the proposed
PTSP framework, the PDF of NTSI and WNTSI were provided,
in which the post fault TS condition can be calculated only
utilizing fault data. Based on the proposed PTSP scheme, CR
was conducted in two stages. First, the contingencies resulting
in the instability of generators are anticipated through the
proposed PTSP framework. In the second stage, based on the
predicted unstable generators, the effect of the topology
changes on the line overload and also voltage profile was
investigated. The proposed NTSI was able to show the TS
margin at the post fault condition using only clearing fault point
data. The numerical results showed that the proposed PTSP is
able to identify the most critical area of the power grid and
predict the post fault TS margin. The identification was
obtained utilizing only the fault clearing point. It should be
noted that the probable topology changes caused by generator
outages may occur one or multiple seconds after fault clearance.
As a result, the proposed PTSP model is capable of predicting
the TS margin before instability occurs. Additionally, the
application of PTSP on the probable topology changes in the
power system security constrains was investigated. It was
observed that the critical TS contingency may not result in the
most violation in the voltage bus constrains or line overloads.
According to the simulation results, the proposed PTSP can be
utilized for probabilistic TS assessment and contingency
ranking considering the impact of DGs in active distribution
networks and bulk generations in transmission grids
simultaneously. Taking this interaction into consideration will
help to find the role of active distribution networks and raised
uncertainties on the stability assessment and also security of the
whole power system. Moreover, the proposed framework can
provide insightful information about the probable topology
changes. These topology changes which could have significant
impacts on the normal operation of distribution networks, might
be used in situational awareness of power system security after
TS contingency occurrence.
Future work will focus on employing the proposed method
on finding cascading dynamic and static contingencies in smart
transmission and active distribution grids. Knowing the latter
cascading contingencies will help to enhance immunity of
cyber-physical attack detection algorithm in future smart grids.
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