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Probabilistic framework for transient stability assessmentof power systems with high penetration of renewablegenerationDOI:10.1109/TPWRS.2016.2630799

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Citation for published version (APA):Papadopoulos, P., & Milanovic, J. V. (2017). Probabilistic framework for transient stability assessment of powersystems with high penetration of renewable generation. IEEE Transactions on Power Systems, 32(4), 3078-3088.https://doi.org/10.1109/TPWRS.2016.2630799

Published in:IEEE Transactions on Power Systems

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Download date:01. Apr. 2020

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Abstract—This paper introduces a probabilistic framework for

transient stability assessment (TSA) of power systems with high

penetration of renewable generation. The critical generators and

areas of the system are identified using a method based on

hierarchical clustering. Furthermore, statistical analysis of

several transient stability indices is performed to assess their

suitability for TSA of reduced inertia systems. The proposed

framework facilitates robust assessment of transient stability of

uncertain power systems with reduced inertia.

Index Terms—clustering, probabilistic transient stability

assessment, renewable generation, uncertainties.

I. INTRODUCTION

URING the past several years there has been a substantial

increase in connections of intermittent and stochastic,

power electronics interfaced Renewable Energy Sources

(RES) to power systems. The inherent uncertainties of RES

operation coupled with electricity market and load (including

conventional and new types of load, e.g., electric vehicles)

driven uncertainties lead to an overall increase in both, model

and operating condition based uncertainties that are becoming

one of the key attributes of modern power systems. The

transient stability assessment (TSA) of such systems using

conventional deterministic tools and methodologies is rapidly

becoming unsuitable and new probabilistic assessment

methods are needed, and are being developed.

Probabilistic TSA has been proven to be an appropriate way

to study the impact of uncertain parameters on transient

stability [1]-[6]. Moreover, the results from probabilistic TSA

can be related to risk assessment which is important for

system operators as economic and technical reasons may drive

the systems to operate closer to the stability limit [6]. In [1]

and [2] the conditional probability approach is used, while [3]

and [4] follow the Monte Carlo approach. Uncertainties

related to transient stability of conventional power systems,

such as system loading, fault location/duration, etc., have been

most commonly considered in the past and the probability of

system stable or unstable transient behaviour following a

This work was supported by the EPSRC project ACCEPT under EPSRC-

India collaboration scheme (grant number: EP/K036173/1).

The authors are with the School of Electrical and Electronic Engineering,

The University of Manchester, Manchester, M60 1QD, U.K. (e-mail: milanovic@manchester.ac.uk, panagiotis.papadopoulos@manchester.ac.uk).

disturbance has been assessed [1]-[4]. In [1] probabilistic

transient stability indices for each line are also developed.

However, the dynamic behaviour of each specific generator is

not assessed.

Direct methods such as [7], [8] have been also used for

probabilistic transient stability assessment. In [7] the Critical

Clearing Time (CCT) for the most critical buses of the system

is calculated. In [8], the sensitivity of the stability boundary

against various parameters (e.g. power output of generators) is

calculated. While direct methods can provide additional

information that can help identify weak areas of the system,

they are generally not considered as accurate as methods based

on time domain simulations [9]. More importantly, it is not

easy to include RES with their associated controllers [10].

It has been acknowledged that the increasing participation

of RES in power generation mix of the system could

significantly affect its transient behaviour following the

disturbance, due to: i) RES exhibit different dynamic

behaviour than conventional synchronous generators; ii) the

increasing amount of connected RES results in synchronous

generators displacement either by de-loading or disconnection;

iii) changing power flows in the network depending on RES

availability, both temporal and spatial. It has not yet been

clarified, let alone quantified, though to what extent each of

these influences affects system transient behaviour [11]. Due

to spatial and temporal uncertainties associated with RES

operation, in particular, it is neither practical nor feasible to

perform TSA based on “worst case scenarios”. Probabilistic

methods are therefore used to assess various aspects of

transient behaviour of power system with RES [12]-[18].

Recently, the effect of wind generation uncertainty has been

introduced in probabilistic TSA. An approximate stability

margin including the effect of wind intermittency is calculated

in [10] but for the purpose of online probabilistic TSA for a

short time frame, i.e. 5 minutes. In [18] a probabilistic

transient stability constrained optimal power flow model is

developed using the SIngle Machine Equivalent (SIME)

method for transient stability analysis. In [16], [17] Monte

Carlo time domain simulations are used for probabilistic TSA

including wind uncertainty. In [17] the disconnection of

conventional generation due to wind generation is also

considered and one generator specific index is introduced but

only for the purpose of identifying the maximum penetration

level rather than investigating the probabilistic dynamic

Probabilistic Framework for Transient Stability

Assessment of Power Systems with High

Penetration of Renewable Generation

Panagiotis N. Papadopoulos, Member, IEEE, Jovica V. Milanović, Fellow, IEEE

D

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behaviour of each generator. In most of the above mentioned

cases the instabilities related to specific generators or groups

of generators as well as generator specific indices are not used.

The other important aspect for the overall TSA, addressed

in this paper, is the impact of network topology changes on

system transient behaviour. This issue has been addressed to a

certain extent in the past in systems with conventional

generation in [19], [20] and [5] but not in systems with RES

where this effect could be more pronounced.

This paper, proposes a comprehensive probabilistic

framework for TSA of systems with increased levels of

penetration of RES over range of operating conditions and

network topologies. The main motivation of the paper is to

propose a probabilistic methodology that can be used to reflect

changes and identify tendencies in the overall dynamic

behaviour of power systems with increased penetration of

RES and more importantly individual generators. The spatio-

temporal variation of system dynamics due to the introduction

of intermittent RES might cause local effects on individual

generators which are more pronounced considering transient

stability. The framework aims at informing decisions during

the planning phase and therefore it is based on more accurate

time domain simulations since computation time is not of

great significance. The main contributions of the paper are the

following: i) The effect of both wind generators and Photo-

Voltaic (PV) units along with the displacement of

synchronous generators and network topology changes, on

transient stability of the overall system, is investigated. ii) The

occurrence of system instability caused by specific generators

or groups of generators losing synchronism with the system is

identified using a hierarchical clustering approach. This opens

the possibility of defining risk in a more elaborate manner. For

example, instability due to one generator losing synchronism

has a smaller impact than instability due to multiple generators

losing synchronism. The identified generators or groups of

generators prone to loss of synchronism could be “acted upon”

at planning stage to improve the overall system transient

stability by applying different stabilizing measures including

the installation of devices to implement corrective control

(such as dynamic braking, Flexible AC Transmission Systems,

etc.), network reinforcements and design of appropriate

control actions (such as intentional islanding, load shedding,

generator tripping, etc.) [21]. iii) The probabilistic dynamic

behaviour of each individual generator is identified by

calculating several generator specific probabilistic indices.

This can highlight different aspects of the effect of RES and

network topology changes on the probabilistic behaviour of

individual generators. iv) Statistical analysis along with the

use of graphical methods of system wide and generator

specific transient stability indices presented in the paper can

provide input to development of special protection schemes

[22] by providing either system specific or generator specific

thresholds. This information can be used as additional measure

to distinguish between stable and unstable contingencies.

II. PROPOSED FRAMEWORK

A probabilistic framework for transient stability assessment

considering the uncertainties occurring in modern power

systems is presented in Fig. 1. The use of data mining methods

for the online identification of the dynamic behaviour of

power systems usually requires the simulation of a large

number of contingencies that are used in machine learning

training procedures. Decision Trees (DTs) are one of the tools

used for the online identification of the system behaviour after

a disturbance, triggering corrective control actions as

presented in [23]. This paper focuses on the offline procedure

that utilizes the simulation data obtained from simulated

disturbances during the training of DTs. The data from

probabilistic Monte Carlo (MC) simulations can be used to

characterize the system and individual generators’ dynamic

behaviour.

Fig. 1. Probabilistic Transient Stability Assessment framework.

A. Generation of Database of System Transient Responses

A number of Ns MC simulations are performed following

the procedure shown in Fig. 2. A power system dynamic

model suitable for stability studies is required, taking into

consideration the connected RES with the respective

controllers. Since transient stability can be also significantly

affected by pre-fault operating conditions, the uncertainties

concerning system loading and wind/PV generation for a 24

hour period are considered. Moreover, the uncertainties of the

fault location and duration are also accounted for.

Only three phase faults are considered in this paper due to

the fact that the three phase faults typically have strongest

impact on system stability. Even though this might not be true

for all systems, the proposed framework relies on time domain

dynamic simulations without any restriction on the

software/program used to perform them. Therefore, any type

of contingency that can be simulated using state of the art

software can be included in the database of transient

responses.

In a similar manner, since the proposed methodology relies

on time domain RMS simulations, a variety of models for

different power system components available in software

packages can be used. For example, in this paper constant

impedance load models are used for the cases studied but

different load models (either static or dynamic) are readily

available in most simulation software and can be easily used

when performing the RMS simulations. Furthermore, the

impact of RES connected to distribution network can be also

taken into account following approaches available in the

literature [24], [25] where aggregated dynamic models of

distribution networks with non-synchronous generation are

developed. These models can be easily implemented in power

Network planning,

protection studies, risk

assessment

Monte Carlo dynamic

simulations databse

Critical generators/

groups

System/generator

specific probabilistic

indices

Impact of RES, network

topology

Statistical

analysis

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system simulation software and can be therefore used as part

of the proposed methodology when performing the dynamic

RMS simulations.

All random variables are sampled according to appropriate

probability distributions describing the uncertain parameter

behaviour. The sampling of the respective distributions is

performed separately for each load and each RES unit in the

system to consider independent behaviour of loads and RES

units.

After the uncertainties in loading and RES contribution to

power generation have been accounted for, an Optimal Power

Flow (OPF) problem is solved to determine the output of

conventional generators. The dispatch obtained from OPF also

determines the amount of disconnection of conventional

generation and consequently system inertia reduction due to

increased RES penetration.

The rotor angles of each generator are stored as the output

of simulations and used to study the dynamic behaviour of the

system in a probabilistic manner. The whole procedure is

repeated for different network topologies and/or amount of

connected RES to study the impact of network topology

changes and RES penetration on transient stability.

The proposed framework offers full flexibility when

accounting for relevant uncertain parameters. The sampling of

uncertain factors can be done according to any probability

distribution based on historical data, prior knowledge or

forecast [26] and solving the OPF problem can include any

number of additional security constraints associated with RES,

as proposed in [27].

Fig. 2. Flowchart illustrating the proposed methodology.

B. Identifying and Clustering Critical Generators

A hierarchical clustering method is applied to determine the

groups of generators exhibiting instability in each simulated

contingency. The agglomerative (bottom up) method is

applied with a cut-off value of 360 degrees, since this is

considered to be the transient stability limit. Euclidean

distance between the data points is used as the similarity

measure and complete linkage is chosen as the linkage

criterion [23]. This results in groupings in which generators of

one group have a minimum of 360 degrees rotor angle

difference with generators of another group. Therefore,

hierarchical clustering is used to identify the unstable

generators as well as the generator grouping patterns for each

simulated contingency [23].

The percentage of cases that each generator is exhibiting

instability (i.e. the probability of instability of each generator)

is a measure of how stable each generator is. Critical

generators and generator groups can be identified in this way.

The impact of RES on transient stability can be investigated

by observing changes in the probability of instability of the

generators. The effect of added uncertainties and of the

different dynamic behaviour of RES units on system stability

is identified in this way. The impact of different network

topologies on transient stability can be also identified in a

similar manner.

C. Transient Stability Indices

Apart from ranking generators according to the number of

instabilities observed, four stability indices are also calculated.

Several indices to quantify the transient stability status of a

system are available in the literature [28]-[31]. Composite

indices based on transient energy conversion, rotor angle,

speed and acceleration have been proposed in [29], [31]. In

[30], frequency domain severity indices are introduced

utilizing PMU measurements. In this paper, measurement

based indices are chosen so that they can also be used as part

of an online TSA. Multiple indices are used, to ensure

comprehensiveness of the assessment [28] as variations of

different indices can reveal different underlying causes that

affect transient stability. System specific and generator

specific tendencies are derived from statistical analysis of the

indices. For example, for a given generator the probability to

exhibit a certain value of an index (e.g. speed deviation) is

calculated. This information can be used to define generator

specific limits for special protection schemes [22].

The four indices used are common transient stability

indices, which can also constitute the base of more complex

indices [28]-[31], and are defined by (1)-(4). All the indices

are calculated from the simulated rotor angle responses δig(t),

where i=1…Ns is the number of simulated contingencies and

g=1…Ng the number of generators. Transient Stability Index

(TSI) is an index considering the stability of the whole system

for a specific contingency [32]. Negative value of the TSI

means that the case is unstable since the difference between

rotor angles of at least two generators for the same time

instance is more than 360 degrees. The rest of the indices are

generator specific. The maximum rotor angle deviation,

maximum speed deviation and maximum acceleration of each

generator are calculated as shown in (2)-(4). In this study, the

indices are mainly used in a comparative manner between

generators as well as between different topologies to reveal

tendencies considering the transient stability of the system.

TSIi=100∙360-δmax,i

360+δmax,i

(1)

Δδigmax

=max( |δig-δig

0 | ) (2)

Power system dynamic model

Sampling of uncertainties

Run OPF

Conventional generation disconnection

Monte Carlo simulations (Ns cases)

Obtain rotor angles

Calculate Transient Stability Indices

Hierarchical clustering

Critical generators/groups

Effect of conventional generation spare

capacity

Statistical analysis

Output of Probabilistic TSA

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Δωigmax= max(|Δωig(t)|)=max( |

δig(t)-δig(t-Δt)

2πfΔt| ) (3)

aigmax=max(|aig(t)|)=max( |

ωig(t)-ωig(t-Δt)

Δt| ) (4)

where δmax,i is the maximum rotor angle deviation between any

two generators in the system for the same time instance, δig0

is

the initial rotor angle of each generator, Δωig is the speed

deviation and αig is the acceleration.

Apart from the transient stability indices, an additional

measure is used to identify the impact of conventional

generation disconnection on transient stability of the system.

When RES are connected, this will eventually lead to

conventional generation disconnection. The generator spare

capacity SCig, defined in (5), is a measure of the conventional

generator loading which is an important parameter considering

transient stability. Moreover, when spare capacity is kept low

this means that in general more conventional generation will

be disconnected leading also to lower system inertia. The

effect of conventional generation spare capacity on transient

stability is also investigated in this study.

SCig=1-PSG,ig

SSG,ig∙pfSG,ig

(5)

where PSG,ig is the power produced by each generator

(determined by OPF), SSG,ig is the apparent power of each

generator after considering any disconnection and pfSG,ig is the

nominal power factor.

D. Statistical Analysis

Statistical analysis is important in order to draw conclusions

and identify tendencies considering the dynamic behaviour of

the system, from the abundance of available simulated data.

Simple statistical measures considering central tendency,

dispersion and correlation (e.g. mean value, standard

deviation, etc.) can reveal certain tendencies but they do not

adequately provide information for the whole range of values

of the indices. Minimum/maximum value, first quartile,

median and third quartile can be visualized using boxplots to

provide an easy way to compare changes in indices (e.g. TSI)

for different network topologies and connection of RES. On

the other hand, plotting Cumulative Distribution Functions

(CDFs) of the random variables (i.e. the transient stability

indices) can reveal the associated probabilities for the whole

range of values each index takes. Moreover, non-

parametric/graphical methods, such as quantile-quantile plots

are also used in this study to highlight changes in the

probabilistic behaviour of the whole system and of specific

generators.

Quantile-quantile plots, are used to identify whether two

sampled random variables come from the same Probability

Distribution Function (PDF). In a quantile-quantile plot, the

quantiles of an observed random variable from two separate

samples are plotted against each other. If the samples fall on a

straight line (i.e. y=x) this means the two samples come from

the same distribution. Observing where the samples are placed

with respect to the straight line provides information

considering the shape of the underlying PDFs for the whole

range of values of the random variable [33]. This way, the

similarities or differences for the whole range of an index

between two different cases are investigated, highlighting the

effect of topology changes and RES penetration on transient

stability. The graphical method can be applied on any

generator specific index to investigate the behaviour of each

generator or on the TSI to study the behaviour of the whole

system. For example, if the dynamic behaviour of a specific

generator is affected by a topology change, this will be

reflected in the quantile-quantile plot of the corresponding

indices.

In general, information from statistical analysis can be used

to design special protection schemes [22], that take into

consideration additional features based on measurements, i.e.

some or all of the indices [34]. Protection device settings can

be then tuned with specific values for groups of generators

which could eventually reduce the number of false tripping

and avoid spreading of local events.

III. TEST SYSTEM AND CASE STUDIES

The test network used, is a modified version of the IEEE 68

bus, 16 machine reduced order equivalent model of the New

England Test System and the New York Power System (NETS

– NYPS). The conventional part of the test network is adopted

from [35], [36] and RES are added at the buses shown in Fig.

3. Two types of RES units are connected on each bus: Doubly

Fed Induction Generators (DFIGs), representing wind

generators and Full Converter Connected (FCC) units,

representing both wind generators and PV units. The standard

IEEE 68-bus, 16-generator test network is chosen in this paper

since its dynamic behaviour has been extensively studied in

the literature and therefore provides a good point of reference.

Moreover, the network is reasonably large to reflect reduced

order real power system models (e.g. GB equivalent network

with 12 generators [37]).

Fig. 3. Modified IEEE 68 bus test network.

A. Components modelling

The test network consists of 16 generators (G1-G16) in five

interconnected areas. NETS consists of G1 to G9, NYPS of

G10 to G13 and the three areas are represented by equivalent

generators G14, G15 and G16, respectively. Standard 6th order

models are used for all synchronous generators. G1-G16 are

equipped with either slow IEEE DC1A dc exciters or fast

G7

G6 G4

G5 G3 G2

G8 G1

G9

G16

G15

G14

G12 G13

G11

G10

NEW ENGLAND TEST SYSTEM NEW YORK POWER SYSTEM

9

29

28

26 25

8

54

1 53

47

48 40

10

31

46

49

11

3032

33

34

61

36

12

17

1343

44

39

4535 51

50

60

59

57

5552

27

37

3 2

4

57

6

22

21

23

68

24

20

19

62

6563

5864

6667 56

16

18

15

42

14

41

L44

L43

L71

L66

L69

L41

L42

L45

38

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acting static exciters type IEEE ST1A and G9 is equipped

with a Power System Stabilizer (PSS). All generators are also

equipped with generic governors, representing gas, steam and

hydro turbines.

A generic type 3 model, suitable for large scale stability

studies is used in this paper to represent DFIGs. The model

has a structure similar to the one proposed by WECC [38] and

IEC [39], as shown in Fig. 4 and is available in DIgSILENT –

PowerFactory [40]. It takes into consideration the

aerodynamic part and the shaft of the wind turbine/generator

as well as the pitch control of the blades. The rotor side

converter controller is also modeled including relevant

limitations, ramp rates and protection mechanisms, such as the

crowbar. The DFIG is represented by a typical 2nd order

model of an induction machine neglecting the stator transients

and including the mechanical equation [41]. The rotor side

converter is controlling the voltage in the rotor as in [42].

Therefore, the model represents all the relevant parts that

influence the dynamic behaviour of DFIGs.

Similarly, a type 4 wind generator model is used to

represent all FCC units. Both wind generators and PV units

can be represented by a type 4 model in stability studies, since

the converter can be considered to decouple the dynamics of

the source on the dc part. This is also suggested by the WECC

Renewable Energy Modeling Task Force [43], which develops

a PV model by slightly modifying the type 4 wind generator

model. The FCC model used in this paper and shown in Fig. 5

has a similar structure to [38], [39] and is available in the

DIgSILENT – PowerFactory software [40].

Both DFIGs and FCC units are treated as aggregate units.

Each RES unit model has a 2 MW power output and the

number of connected units is varied to determine the output of

the aggregate unit. Furthermore, all RES units are considered

to have Fault Ride Through (FRT) capabilities and remain

connected during the fault. The amount of connected RES for

each area of the system, i.e. the installed capacity of RES, is

given as a percentage of the total installed conventional

generation capacity of that area before adding any RES.

Considering the total RES installed capacity, approximately

66.67% are assumed to be DFIG wind generators and the

remaining 33.33% are FCCs. FCCs are further considered to

be 30% wind generators and 70% PV units. All dynamic

simulations are RMS simulations and are performed using

DIgSILENT – PowerFactory software [40].

Fig. 4. DFIG control structure.

Fig. 5. FCC unit control structure.

B. Modelling of uncertainties

Daily loading and PV curves are initially used, obtained

from national grid data [44] and from the literature [15],

respectively. First, the hour of the day is sampled randomly

following a uniform distribution to determine the pu values for

all the loads and all PV units according to the respective

curves. For every hour within the day, the corresponding

uncertainties are also modeled using a normal distribution for

the system load [26] and a beta distribution for the PV

generation [45]. Therefore, an extra uncertainty scaling factor

for loads and PVs is introduced which is eventually multiplied

with the corresponding value from the daily loading or PV

curve, respectively. The normal distribution for the system

loading uncertainty has mean value 1 pu and standard

deviation 3.33% and the beta distribution a and b parameters

are 13.7 and 1.3 respectively [46].

For wind generation, the mean value of the wind speed

within one day is considered constant [47], and the uncertainty

of the wind speed is modelled using a Weibull distribution

[48]. After considering the wind speed uncertainty, the power

curve of a typical wind generator is used [49] to derive the

power output. The total output of a wind farm is scaled

according to the number of wind generators. The Weibull

distribution parameters used are φ =11.1 and k=2.2 [48].

The aforementioned PDFs are sampled separately for each

load and RES unit in the system in a similar manner to [50].

However, [50] deals only with the uncertainty of loads and not

RES. This approach is followed to represent the variability of

the uncertainties in a more realistic manner.

After considering the uncertainties, OPF is solved to

determine the conventional generators dispatch PSG,ig. The

cost functions for OPF are taken from [26]. The nominal

capacity of each generator SSG,ig is then adjusted by

considering 15% spare capacity according to (5). In case the

resulting SSG,ig from (5) is larger than the initial nominal

power of the generators, it is set to that initial nominal value.

This means that there is no room for conventional generation

disconnection in this case. The disconnection of conventional

generation due to both load variations and RES penetration is

considered in the following way: Since the generators are

considered as aggregated units, reducing the nominal power, is

equivalent to a reduction in the moment of inertia of the power

plant and an increase in the generator reactance.

Only three phase self-clearing faults are considered in this

study. However, the simulation database could be extended to

include other contingencies as well. A uniform distribution is

used to model the fault location which means that the fault

may happen with equal probability at any line of the test

network and at any point along the line. A normal distribution

DFIG

Rotor side

converter control

Protection

Pitch

controlTurbine Shaft

Aerodynamic part

Electrical

Measurements

Turbine Power

Rotor Voltage

Speed

Measurement

Reference signals Grid Interface

Speed reference

Static

Voltage

Source

Converter

Controller

Grid InterfaceControlled

Voltage

Reference signals

Electrical

Measurements

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with mean value of 13 cycles and standard deviation 6.67% is

used to model the fault duration [23].

C. Monte Carlo Simulations

After considering all the uncertainties, Ns MC simulations

are performed for each case, for different network topologies

and amount of connected RES units. The number of

simulations (Ns) is chosen by keeping the error of the sample

mean up to 5%, for 99% confidence interval, considering the

TSI as the random variable. The error of the sample mean is

calculated using (6), where Φ-1 is the inverse Gaussian CDF

with a mean of zero and standard deviation one, σ2 is the

variance of the sampled random variable, δ is the confidence

level (i.e. 0.01 for this study) and XN is the sampled random

variable with N samples [51]. For this specific system (for all

case studies considered) the number of required Monte Carlo

simulations (Ns) to limit the error to 5% is 6000.

eX̅N=

Φ-1 (1-δ2)√

σ2(XN)N

XN

(6)

Five Test Cases (TCs) are presented in this section,

consisting of 6000 simulations each (i.e. 30,000 simulations in

total). In the base case (TC1) no line is disconnected and low

amount of connected RES is considered. In TC2, all the RES

units are disconnected and therefore the associated

uncertainties are also not considered. TC3 and TC4 are the

same as TC1 but lines 1 (between bus 21 and 68) and 2

(between bus 33 and 38) of NETS and NYPS are out of

service, respectively. In TC5, high amount of RES is

considered to be connected. TC2 and TC5 are used to study

the impact of RES on transient stability, while TC3 and TC4

the impact of network topology changes. The lines set as out

of service are chosen close to critical generators and groups of

generators (ones that experience the instability most

frequently) in the NETS and NYPS area, respectively. For the

given TCs the error of the sample mean varies from 3.4% for

TC5 up to 5% for TC3 for 99% confidence interval. The TCs

are summarized in Table I.

TABLE I

NUMBER OF UNSTABLE CASES AND PATTERNS OBSERVED FOR TCS

TCs RES installed capacity

(% of system nominal capacity)

Lines out of service

TC1 20% -

TC2 - -

TC3 20% Line 1

TC4 20% Line 2

TC5 60% -

All simulations are performed in DigSILENT/PowerFactory

software using a computer with an Intel Core i7 3.4 GHz

processor and 16 GB of RAM. For the TC without RES,

approximately 18 hours are required to perform 6000 Monte

Carlo simulations. For the TCs that include RES,

approximately 60 hours are required for 6000 dynamic

simulations.

IV. RESULTS OF PROBABILISTIC TRANSIENT STABILITY

ASSESSMENT

In general there are two main opposing effects which are

analyzed in the following sections. On the one hand, the

dynamic behaviour of RES might cause an improvement of

transient stability and on the other hand the disconnection of

conventional synchronous generation causes deterioration of

transient stability. Moreover, the uncertain temporal and

spatial availability of RES might cause more or less favorable

operating conditions. These spatio-temporal changes, due to

RES, in the dynamic behavior of power systems can affect

locally the transient stability of individual generators.

Therefore, investigating instabilities as well as the

probabilistic dynamic behavior of individual generators,

becomes even more significant. The results presented in this

section highlight the additional information that can be

obtained from the proposed methodology compared to

previous methods of probabilistic TSA (e.g. instabilities

related to specific generators/groups of generators, change in

probabilistic dynamic behavior of indices related to individual

generators, etc.).

A. Critical generators

The critical generators of the system are identified

according to the number of times each generator becomes

unstable. The number of cases each generator is exhibiting

instability is identified using the hierarchical clustering

approach presented in Section II B. Initially the grouping

patterns are defined and afterwards the instabilities for each

generator are observed, i.e. the number of cases each generator

belongs to any unstable group. This means that for certain

cases more than one generator might be unstable. In Fig. 6, the

results for the probability of instability for each generator are

shown, i.e. the number of instabilities divided by the total

number of simulated cases for each TC.

G9 and G11 are the most unstable generators in NETS and

NYPS area, respectively. One of the reasons is that they are

the ones having relatively smaller moment of inertia constant.

In general, generators in the NETS area exhibit more

instabilities than those in the NYPS and external areas. G9 has

the highest probability of exhibiting instability for all cases,

except from TC3 when disconnecting line 1 causes G6 and G7

to become more unstable. TC1 and TC4 do not exhibit

significant differences. However, G11 becomes slightly more

unstable because disconnecting line 2 in NYPS weakens that

area of the network in a similar manner to the one described

with TC3 and NETS area. G12-G15 do not exhibit any

instabilities in the studied cases.

Fig. 6. Unstable generators for all TCs.

0

1

2

3

4

5

6

7

8

G1 G2 G3 G4 G5 G6 G7 G8 G9 G10 G11

Pro

bab

ilit

y of

inst

abil

ity

(%)

Generator number

TC1

TC2

TC3

TC4

TC5

> ACCEPTED VERSION OF THE PAPER <

7

The probability of instability for most generators becomes

smaller as the amount of connected RES is increasing with

descending order from TC2 to TC1 and TC5, for this specific

system and studied operating conditions. (This behaviour is

further discussed in Section IV C below.)The probability of

instability of G10, however, is very slightly increasing

(approximately 0.1%) as the amount of connected RES is

increasing from TC1 to TC5. It should be mentioned here, that

for all studied TCs (TC2 without RES included), the spare

capacity is kept constant at 15% for all synchronous

generators, which means that even for TC2 there is some

disconnection of conventional generation due to different

loading level of the system. Moreover, when comparing the

same simulated contingencies (i.e. same fault

location/duration and system loading) between TCs, some of

the cases become unstable while some others become stable.

For example, when comparing TC2 to TC5 there are 106 cases

(out of the 6000) that change from unstable to stable and 92

that change from stable to unstable, i.e., net increase (14

cases) in stable cases. The overall tendency though is for the

observed instabilities to be reduced.

B. Probabilistic analysis of indices

Further analysis of the dynamic behaviour of each generator

and the whole system is carried out by calculating the four

indices defined in Section II C. Statistical analysis is applied

to the calculated values to identify underlying trends.

A boxplot with the TSI for all TCs is provided in Fig. 7.

Information such as the median, 1st/3rd quartiles, max/min

values and an estimation of the number of outliers can be

extracted from the boxplot. In this study data points are

considered as outliers when they are larger than q3+1.5(q3-q1)

or smaller than q1-1.5(q3-q1), where q1 and q3 are the 1st and

3rd quartile respectively. Between TCs the median and 1st/3rd

quartiles do not change drastically, which means that there are

small changes in the TSI for most stable cases (around 75% of

all cases). For TC2 there are no observed stable cases with TSI

values below 20 (which corresponds to 240 degrees difference

between two generators). For TC1, TC3, TC4 and TC5 (which

are the cases with RES) some outliers with values between 0

and 20 are observed. This indicates that for some cases the

system with RES can retain stability for larger excursions in

generator rotor angles. Information like this can be used to

define a threshold value for the TSI that will provide a

warning to the system operator. For this specific system, this

threshold can be considered to be approximately 20 (it is 18

for TC1, TC3 and TC4, 21 for TC2 and 24 for TC5). Positive

values lower than that for which the system is marginally

stable (large rotor angle deviations) are considered to be as the

outliers, i.e. with very low probability.

G9, even with relatively smaller acceleration than other

generators, exhibits high speed and rotor angle deviations.

This indicates possibly a weak network in the NETS area

around G9 which combined with the generator’s small inertia

leads to an increased number of instabilities. On the other

hand, G11 exhibits in general higher acceleration than the

more unstable G9 possibly due to having a smaller moment of

inertia. However, the fact that it exhibits instability less

frequently than G9 indicates a stronger network in the NYPS

area around G11. G15 exhibits the smallest deviations for all

three indices in accordance to no instabilities observed.

Fig. 7. Boxplot of the TSI for TC1-TC4.

By observing the maximum rotor angle/speed deviation and

acceleration, further information for the dynamic behaviour of

each specific generator can be extracted. In Fig. 8, CDFs for

representative generators for TC1 are provided to highlight the

probability of generators exceeding certain values of the

indices. All cases, stable and unstable, are included in the

calculation of the CDFs. Generators G1, G6, G9, G11 and

G15 are chosen as representative, by observing Fig. 6. G6, G9

and G11 are among the most unstable generators, while G1

and G15 are among the most stable.

In Fig. 8, G11 exhibits the highest maximum values for all

three indices. It however, is less probable to exhibit relatively

high values of rotor angle and speed deviation, compared to

G9. For example, the cumulative probability for G11 to

exhibit angle deviation larger than 300 degrees is around 2%,

while this probability is around 5% for G9. Similar results are

observed for speed deviation shown in Fig. 8b with the

difference that the CDF is not so steep. For acceleration

though, presented in Fig. 8c, G11 is more probable to

accelerate more than G9. This indicates that there might be

cases when certain generators can remain stable even after

exhibiting relatively high acceleration. In Fig. 8b and 8c the

maximum speed deviation and acceleration observed within

the stable cases is also marked for G9 and G11. This

information can be useful in designing specific thresholds for

individual generators as part of special protection schemes.

For example, G11 has a higher threshold for acceleration than

G9, which means that G11 exhibiting relatively higher

acceleration does not necessarily mean it will lose stability.

This kind of generator specific thresholds could be used by a

special protection scheme to avoid possible unnecessary

tripping of specific generators and vice versa.

TC1 TC2 TC3 TC4 TC5

-100

-80

-60

-40

-20

0

20

40

60

80

Test Cases

TS

I

median

q1

q3

q3-1.5(q

3-q

1)

q3+1.5(q

3-q

1)

> ACCEPTED VERSION OF THE PAPER <

8

Fig. 8. CDFs of maximum a) rotor angle deviation, b) speed deviation and c) acceleration for TC1.

C. Effect of RES and conventional generation disconnection

on transient stability

The effect of RES on transient stability is investigated in

this section using the proposed methodology. In this study, all

RES units are considered to have FRT capability. Therefore,

the control logic followed by RES units when a disturbance

happens is to increase the reactive power output to provide

support to the system. In general, this operation can improve

the transient stability of the system. However, the pre-fault

operating conditions are also affected by the uncertainties

introduced by RES, which can lead to either more or less

favorable operating conditions for specific generators.

Furthermore, the disconnection of conventional generation to

account for the connection of RES tends to have a negative

impact on transient stability, due to the reduction in inertia and

nominal capacity of the synchronous generators.

In Fig. 9a CDFs for the TSI for TC1 (low amount of

connected RES), TC2 (no RES) and TC5 (high amount of

connected RES) are presented. As explained in Section III B,

the spare capacity of conventional synchronous generation is

kept always constant at 15% (including TC2), which means

that there is some generation disconnection considered even

for this case due to load variations within the day. Relatively

high spare capacity combined with FRT capability of RES

leads to a reduction of the total number of instabilities in

system with RES. This is reflected by observing the

probability of the TSI to have a negative value which is 9.5%

for TC5, 9.7% for TC1 and 12.9% for TC2. Moreover, TSI

values above 55 tend to appear with smaller probability as the

amount of RES increases. This indicates the occurrence of

larger oscillations within the stable cases, especially for TC5.

Fig. 9. CDFs of TSI for a) TC1, TC2, TC5 and b) different spare capacity.

To provide a more detailed overview of the impact of RES

on transient stability, the most common unstable groups that

are observed in the simulated responses are presented in Table

II. The groups are identified following the procedure described

in Section II B. In addition to accounting for instability,

identifying the groups of unstable generators can provide

further information to a system operator regarding the

dynamic performance of the system. The number of grouping

patterns that appear tends to decrease as more RES are

connected in the system. Moreover, the frequency of

appearance of certain unstable groups varies, indicating that

the power system dynamic behaviour is changing when RES

are introduced. The frequency of appearance of some groups

0 200 400 600 800 1000 1200 1400 1600 1800 2000 22000

0.5

1

Maximum rotor angle deviation (degrees)

Cu

mu

l. P

rob

.

G1

G6

G9

G11

G15

0 50 100 150 200 250 3000

0.5

1

a)

0 0.02 0.04 0.06 0.08 0.1 0.12 0.140

0.5

1

Maximum speed deviation (pu)

Cu

mu

l. P

rob

.

b)

0 0.1 0.2 0.3 0.4 0.50

0.5

1

Maximum acceleration (pu)

Cu

mu

l. P

rob

.

c)

0.429

0.038

0.029

0.194

-100 -80 -60 -40 -20 0 20 40 60 800

0.2

0.4

0.6

0.8

1

TSI

Cu

mu

lati

ve

Pro

ba

bil

ity

a)

TC1

TC2

TC5

-100 -80 -60 -40 -20 0 20 40 60 800

0.2

0.4

0.6

0.8

1

TSI

Cu

mu

lati

ve

Pro

ba

bil

ity

b)

TC2 no disconnection

TC1 SC10

TC1 SC15

TC1 SC20

-10 -5 0 5 10 15 20 25 30

0.08

0.1

0.12

0.14

TC1

TC2

TC5

-10 0 10 20 300

0.05

0.1

0.15

TC2 no disconnection

TC1 SC10

TC1 SC15

TC1 SC20

> ACCEPTED VERSION OF THE PAPER <

9

increases (e.g. groups 1, 8, 10, 11), while for others it

decreases (e.g. groups 3, 7, 9, 12).

TABLE II

MOST SIGNIFICANT UNSTABLE GENERATOR GROUPS

Unstable generator groups

Frequency of appearance (%)

TC1 TC2 TC5

1 (G9) 49.38 42.96 51.39

2 (G11) 22.12 19.91 10.24

3 (G2-G9) 2.65 7.63 0.01

4 (G4-G5)/(G6-G7) 1.95 1.35 -

5 (G3) 3.01 4.04 3.39

6 (G4-G7,G9)/(G3) 0.88 - -

7 (G4-G7) 4.78 5.69 1.2

8 (G4-G5) 4.60 1.20 5.78

9 (G1-G9) 1.59 2.40 -

10 (G8) 2.12 1.80 3.19

11 (G5) 0.88 0.15 2.19

12 (G1-G10) 0.35 2.69 -

13 (G10) 0.18 2.25 1.59

14 (G1-G9)/(G10) - 2.25 -

To further investigate the effect and importance of

synchronous generation spare capacity, the simulations for

TC1 are repeated for fixed spare capacity of 10% and 20% for

the hour of the day that the maximum load occurs. Moreover,

TC2 with no synchronous generation disconnection (i.e.

keeping the initial SSG,ig values) for the same hour is also

considered. This approach for TC2 means that all generators

are only operating at lower loading (de-loaded), which is the

most favorable assumption considering transient stability.

Only 1000 cases are simulated for this comparison since the

variation of the load within the day is not considered. CDFs

for the TSI are presented in Fig. 9b, showing that there is

significant impact of the spare capacity in transient stability.

The probability of instability (i.e. TSI to exhibit negative

value) increases with reduction in spare capacity from

approximately 4.8% to 9.7% and 14.7% for spare capacity

20%, 15% and 10%, respectively. For TC2 the probability of

instability is 7% without considering any disconnection which

is between the values with spare capacity 15% and 20%.

Therefore, it is important to keep the spare capacity to a high

value (e.g. above 15% for this specific system) by giving

priority to de-loading instead of disconnecting some of the

synchronous generators to ensure the transient stability of the

system does not deteriorate.

D. Effect of network topology changes

When direct comparison between two random variables is

needed (as in the case of network topology changes), quantile-

quantile plots can comprehensively visualize the differences

for the whole range of values the variables take. The effect of

topology changes in the probabilistic behaviour of a certain

generator can be highlighted in this way. If a generator is not

affected by a topology change, it exhibits a similar

probabilistic dynamic behaviour and therefore a similar

probability distribution for the calculated indices. A

representative case for maximum rotor angle deviations

between TC1 and TC4 is provided in Fig. 10 for G11 and G6,

respectively.

G11 is affected by tripping line 2, since it is close enough,

while G6 follows almost the same distribution. The quantile-

quantile plot provides also more information about how the

generator is affected. The initial points on the left above the

dashed line are cases that G11 exhibits instability in TC4

while it is stable in TC1. For larger values of angle deviation

(above 1500 degrees) which are associated with more severe

faults, the quantiles start following the straight line y=x, which

means G11 exhibits the same behaviour for those values. The

impact of tripping line 2 for these cases is not as significant,

since the contingencies are severe enough to cause instability

in both TC1 and TC4. However, there are some values

between 800 and 1500 degrees that are unstable in both TCs

but exhibit different values which might cause changes to the

generator grouping patterns as defined by the hierarchical

clustering method used. In general, network topology changes

exhibit mainly local effects on system transient stability. The

impact might be small as observed in TC4 or major as in TC3

when a group of generators (G6, G7) becomes significantly

more unstable.

Fig. 10. Quantile-quantile plot for maximum angle deviation between TC1

and TC4 of a) G11 and b) G6.

An additional TC, namely TC6, is also studied where a line

close to generators G2 and G3 (between buses 57 and 58) is

out of service. The quantile-quantile plots of maximum rotor

angle deviation for G2 and G3 are presented in Fig. 11. Both

generators are negatively affected, with G3 being a slightly

more affected than G2. It should be mentioned that the

purpose of this study is to show that the proposed framework

can be used to highlight in a detailed manner the effect of

network topology changes on transient stability of the system

and of individual generators, rather than provide a conclusive

result considering network topology changes.

0 500 1000 1500 2000 25000

500

1000

1500

2000

2500

X Quantiles (TC1 G11 max rotor angle deviation)

Y Q

ua

nti

les

(TC

4 G

11

)

a)

0 100 200 300 400 500 600 7000

200

400

600

800

X Quantiles (TC1 G6 max rotor angle deviation)

Y Q

ua

nti

les

(TC

4 G

6)

b)

> ACCEPTED VERSION OF THE PAPER <

10

Fig. 11. Quantile-quantile plot for maximum angle deviation between TC1 and TC6 of a) G2 and b) G3.

V. CONCLUSIONS

A framework for probabilistic transient stability assessment

of power systems with high RES penetration is proposed in

this paper. The critical generators of the system, i.e. the ones

going unstable more frequently, are identified using an

approach based on hierarchical clustering. Statistical analysis

of four transient stability indices as well as observed changes

in the critical generators are used to investigate the effect of

high RES penetration and network topology changes.

RES with FRT capability can provide support during faults

and can therefore have a beneficial impact on transient

stability. However, disconnecting conventional synchronous

generators has a detrimental effect on transient stability. The

spare capacity, which is a measure of the loading of

synchronous generators, is significant in determining how

critical an operating point is. Keeping a high amount of spare

capacity of synchronous generators, e.g. more than 15% for

the specific studied system, can ensure transient stability does

not deteriorate. The proposed framework takes into

consideration both aspects to identify the overall impact on

transient stability of the system and of individual generators.

Network topology changes, as expected, could also cause

significant impact on the probability of instability of

generators. Disconnecting lines close to groups of generators

that tend to become unstable, might increase drastically the

probability of instability of the groups. While this is not a

certainty, the proposed methodology can highlight in detail the

exact impact of network topology on system and individual

generator transient stability.

In general, the proposed probabilistic framework can help in

identifying the underlying tendencies that affect transient

stability of power systems, including the probabilistic dynamic

behaviour of individual generators. The results of the analysis

can point to critical areas of the network where it is more

probable to encounter transient stability problems and help in

designing and applying measures to improve the system

transient behaviour.

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(TC

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a)

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Panagiotis N. Papadopoulos (S’05-M’14) received the Dipl. Eng. and Ph.D.

degrees from the Department of Electrical and Computer Engineering at the

Aristotle University of Thessaloniki, in 2007 and 2014, respectively. Since 2014 he has been postdoctoral Research Associate at the University of

Manchester. His special interests are in the field of power system modeling,

simulation and investigation of dynamic behaviour of power systems with increased penetration of non-synchronous generation.

Jovica V. Milanović (M'95, SM'98, F’10) received the Dipl.Ing. and M.Sc. degrees from the University of Belgrade, Belgrade, Yugoslavia, the Ph.D.

degree from the University of Newcastle, Newcastle, Australia, and the Higher

Doctorate (D.Sc. degree) from The University of Manchester, U.K., all in electrical engineering. Currently, he is a Professor of Electrical Power

Engineering, Deputy Head of School and Director of External Affairs in the

School of Electrical and Electronic Engineering at the University of Manchester, U.K., Visiting Professor at the University of Novi Sad, Serbia,

University of Belgrade, Serbia and Conjoint Professor at the University of

Newcastle, Australia.