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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: mohammadpourfard@sut.ac.ir

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

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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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