power system analysis on distribution network

Electrical Power and Energy Systems 33 (2011) 887–900

Contents lists available at ScienceDirect

Electrical Power and Energy Systems

journal homepage: www.elsevier .com/locate / i jepes

Fault detection in transmission networks of power systems

S. Saha a,⇑, M. Aldeen a, C.P. Tan a,b

a Department of Electrical and Electronic Engineering, The University of Melbourne, Parkville, Victoria 3010, Australiab School of Engineering, Monash University, Jalan Lagoon Selatan, 46150 Bandar Sunway, Malaysia

a r t i c l e i n f o a b s t r a c t

Article history:Received 4 November 2009Received in revised form 8 December 2010Accepted 9 December 2010Available online 18 February 2011

Keywords:SMIBSynchronous generatorFault detectionModellingSliding mode observerTransmission lines

0142-0615/$ - see front matter � 2011 Elsevier Ltd. Adoi:10.1016/j.ijepes.2010.12.026

⇑ Corresponding author. Tel.: +61 383440374.E-mail address: [email protected] (S. S

An online fault detection scheme for a sample power system is introduced in this paper. The detectionapproach is based on the use of a variable structure system called ‘‘sliding mode observer’’, where infor-mation contained in the output measurements is utilized to detect the onset of faults in the transmissionnetwork of the sample power system in real time and online. The power system comprises a generatingunit connected to an infinite bus through double line transmission network. The common case of a faultoccurring on a transmission line is illustrated through simulation studies.

� 2011 Elsevier Ltd. All rights reserved.

1. Introduction

Power systems are prone to frequent faults, which may occur inany of its components, such as generating units, transformers,transmission network and/or loads. It is well known that faultscan cause significant disruption of supply, destabilise the entiresystem and may also cause injuries to personnel. Detection offaults is therefore of a paramount importance from economic andoperational viewpoints. In addition faults should be detected asquickly as possible, in real time if possible, so that an appropriateremedial action can be promptly taken before major disruptionsto the power supply can occur.

So far such faults are detected by fault locators, which form anintegral part of ‘‘protection systems’’. However, the accuracy offault locators is known to be below desirable levels. It is alsoknown that fault locators may not able to detect faults in real time.In this paper we introduce a software based scheme where faults inthe transmission network are able to be detected in real time, usingcommonly available measurements of speed, load angle, terminalvoltage, power, etc. Only symmetrical faults are considered here,as asymmetrical faults require dynamical incorporation of sym-metrical components into the overall dynamical systems, whichis beyond the scope of this paper. However research in this direc-tion by the authors is ongoing and new results will soon be submit-ted for possible publication.

ll rights reserved.

aha).

The novelty of the proposed scheme is two folds; developmentof a fault dependent model for a sample power system and designof a real time fault detectors, referred to in the control literature as‘‘fault detection filter’’. The result of this study are among a veryfew model based results that have been reported in the open liter-ature (see for example [1,2]) and may therefore offer tangible ben-efits to the protection industry if incorporated in the design of faultlocators. It can be noted that this paper does not deal with ‘‘protec-tion systems’’ per say but with how to locate faults in real timeusing control and estimation theory. Protection systems expertsare therefore invited to make use of this approach.

Other reported contributions are reported in [16–18]. In [16],pre-fault and during fault phasor currents and voltages are mea-sured and used in a two-bus Thevenin equivalent network modelof the transmission system to locate a fault after its occurrence.In this method the dynamics of the system are ignored and eachgenerator is represented by a voltage behind transient reactance.In [17], adaptive extended Kalman filter and probabilistic neuralnetworks are combined to estimate different harmonic compo-nents in fault current signals. The harmonic components are thenfed into a forward neural network for training and identificationof high impedance faults in power distribution feeders off line. In[18], discrete wavelet transform and neural networks are used toparameterise and characterise fault signals.

Unlike the approach presented in this paper, the above ap-proaches are not suitable for real time fault detection as they arebased on acquiring fault date, which can then be used in knowl-edge based techniques for analysis and characterisation.

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Nomenclature

w flux linkage.0 superscript refers to nominal value.q subscript refers to the quadrature axis.G,Q subscript refers to damper windings on the quadrature

axis.d subscript refers to the direct axis.D subscript refers to damper winding on the direct axis.F subscript refers to field winding.ex subscript refers to exciter.N subscript refers to network.G subscript refers to governorx state vectory output vector

f fault parametersA, B, C, D, E, G, W constant parameter matricesd rotor shaft anglex rotor shaft speedxb speed base quantityv, V machine and network voltagei, I machine and network currentj�j magnitude of a vectorEFD field winding excitation voltageH synchronous machine inertia constantkd rotor damping coefficientTm, Pm mechanical torque and power

REFV

FDEAV

1R

R

K

sτ+ 1A

A

K

sτ+

1F

F

sK

sτ+

+ -

VV2

VF

|VN|

-R

+

Fig. 1. IEEE type 1 excitation system.

888 S. Saha et al. / Electrical Power and Energy Systems 33 (2011) 887–900

This paper is structured as follows: We start with the develop-ment of a comprehensive model of a power system suitable forfault studies and test its validity through simulation studies. Thenwe introduce a software based fault detection algorithm (filter)capable of detecting faults online and real time. The fault filter isthen applied to the sample power system to detect faults in thetransmission network. We finally present comprehensive simula-tion studies carried out on the power system under various scenar-ios and discuss the results in full.

2. Generating unit model

In the following we provide a detailed linearised model of agenerating unit comprising an 8th order synchronous generator,a 3rd order model of an IEEE Type 1S exciter [3], and a 3rd ordermodel of an IEEE steam-turbine speed governor [4]. These modelsare based on those presented in with some changes.

2.1. Synchronous generator

Detailed models of a synchronous generator should includeelectrical (flux-voltage equations) and mechanical (rotor shaft mo-tion) parts [5–7]. We start with modelling the flux-voltage dynam-ical relationships along the d-axis. These relationships may bemodelled by the set of linear equations listed below.

_wd ¼ xbwqð1þ _d�Þ þxbw�q_dþxbvd þxbrid ð1Þ

_wF ¼ xbvF � rF iFxb ð2Þ_wD ¼ xbvD �xbrDiD ð3Þ

In the q-axis the voltage-flux dynamics are described by the follow-ing linear equations:

_wq ¼ �xbwdð1þ _d�Þ �xbw�d_dþxbvq þxbriq ð4Þ

_wG ¼ xbvG �xbrGiG ð5Þ_wQ ¼ xbvQ �xbrQ iQ ð6Þ

The mechanical part of the machine is due to the motion of the rotorshaft and is expressed as follows:

€d ¼ �xb

2Hkd

_dþ ðTm � TeÞxb

2Hð7Þ

Te ¼ wdi�q þ w�diq � wqi�d � w�qid ð8Þ

Fig. 2. IEEE steam turbine speed governing system.

2.2. Excitation system

The synchronous machine is equipped with the following IEEEexcitation system.

The state equations for the excitation system shown in Fig. 1 arederived as follows:

_VR ¼ �1sR

VR þKR

sRjVN j ð9Þ

_VF ¼ �1sF

VF þKF

sF

_VA ð10Þ

_VA ¼ �1sA

VA þKA

sAðVREF � VR � VFÞ ð11Þ

2.3. IEEE governor model

A model of a steam turbine speed governing system used in thisstudy is shown in Fig. 2 and its model is derived below:

_PST ¼ �1

TSTPST þ

1TST

PGV ð12Þ

_PGV ¼ �1

TSMPGV þ

1TSM

PSR ð13Þ

_PSR ¼ �1

TSRPSR þ

1TSR

Pr �_d

RxB

!ð14Þ

In order to interface the governor with the synchronous generator,the governor output, which is the mechanical power, Pm, needs to

S. Saha et al. / Electrical Power and Energy Systems 33 (2011) 887–900 889

be expressed in terms of mechanical torque, Tm, to act as an input tothe generator. This is accomplished as follows. The relationship be-tween the mechanical torque and the power is given as:

Pm ¼ Tmx ¼ Tmð1þ _dÞ

Linearization of this equation and re-arranging will give

Tm ¼Pm � T�m _d

ð1þ _d�Þð15Þ

3. Transmission network model

A Single Machine Infinite Bus (SMIB) power system is shown inFig. 3, where possible three phase line-to-ground faults, denotedby f1,2, on the two lines are also shown. The location of the faultis at a distance g1,2 from the generator terminal. In this study onlysymmetrical faults are considered. Asymmetrical faults requiredynamical incorporation of symmetrical components into theoverall dynamical systems, which is beyond the scope of this pa-per. New results in this respect will soon be submitted for possiblepublication.

From Fig. 3, the individual line currents are expressed as

IN1 ¼f1

g1Y1VN þ ð1� f1ÞY1ðVN � VBÞ

IN2 ¼f2

g2Y2VN þ ð1� f2ÞY2ðVN � VBÞ

IN ¼ IN1 þ IN2

ð16Þ

From (16) it is easy to derive a generic expression for a SMIB systemthat has as many as lines as required, say k lines, as follows:

VN ¼Xk

i¼1

fi

giY i þ f1� figYi

� �" #�1

IN þXk

i¼1

ð1� fiÞYiVB

( )ð17Þ

For a two-line configuration Eq. (17) is written as

VN ¼f1

g1Y1 þ f1� f1gY1 þ

f2

g2Y2 þ f1� f2gY2

� ��1

� fIN þ ð1� f1ÞY1VB þ ð1� f2ÞY2VBg

Or in a more compact form as

VN ¼1g1� 1

� �Y1

1g2� 1

� �Y2

� �f1

f2

� �þ Y1 þ Y2

� ��1

� IN þ ðY1 þ Y2ÞVB � ½Y1 Y2 �VBf1

f2

� �� ð18Þ

Therefore Eq. (18) can simulate any possible scenario of faultsoccurring on either line and at any distance from the generator. Itmay also be expressed in a vector form as shown below:

Fig. 3. SMIB power system.

VN ¼ ½l1 l2 �f1

f2

� �þ Y12

� ��1

� IN þ ðY1 þ Y2ÞVB � ½Y1 Y2 �VBf1

f2

� �� ð19Þ

where

l1 ¼1g1� 1

� �Y1; l2 ¼

1g2� 1

� �Y2; Y12 ¼ Y1 þ Y2

Or

VN ¼ ðlf þ Y12Þ�1fIN þ Y12VB � Yf VBfg ð20Þ

where

l ¼ ½l1 l2 �; f ¼f1

f2

� �; Y12 ¼ Y1 þ Y2; Yf ¼ ½Y1 Y2 �

It is clear that Eq. (20) represents any number of faults that may oc-cur in any transmission network. It however can be simplified if weuse the fact that the two lines, 1 and 2, are the same. Then we canwrite

Y ¼ Y1 ¼ Y2 ¼ ðRþ jXÞ�1 ð21Þ

Using Eq. (21) in Eq. (20), we obtain the following fault cases.No fault case: For no fault condition, i.e. f1 = f2 = 0, Eq. (20) is re-

duced to the nominal case expressed as

VN ¼12

Y�1IN þ VB ð22Þ

Fault on line 1: In case of fault occurring on line 1, for example, i.e.f1 = 1, but line 2 is fault-free then Eq. (20) reduces to

VN ¼ 1þ 1g1

� �Y

� ��1

fIN þ YVBg ð23Þ

Fault on line 2: Similarly, in case of fault occurring on line 2, forexample, i.e. f2 = 1, but line 1 is fault-free then Eq. (20) reduces to

VN ¼ 1þ 1g2

� �Y

� ��1

fIN þ YVBg ð24Þ

Faults on line 1 and line 2: In case of fault occurring on both line 1and line 2, for example, i.e. f1 = f2 = 1 then Eq. (20) reduces to

Fig. 4. Phasor relationship between VB and VN.

0 5 10 15 20 25 30 35 40

Load Angle Response

Speed Response

0

-0.5

0.5

Terminal Voltage0 5 10 15 20 25 30 35 40

-1

-0.5

0

0.5

Accelerating Torque Response0 5 10 15 20 25 30 35 40

-1

-0.5

0

0.5

Terminal Power Response0 5 10 15 20 25 30 35 40

-4

-2

0

2

0 5 10 15 20 25 30 35 40

Time (Seconds)

-2

0

2

4

Fig. 5. Output responses to a 5% change in VREF.

0 10 20 30 40 50 60

0 2 4 6 8 10 12 14 16 18

Load

Ang

leLo

ad A

ngle

15 20 25 30 35 40 45 50 55 60

Load

Ang

le

Time (Seconds)

-40

-20

0

20

-1

-0.5

0

0.5

-40

-20

0

Fig. 6. Load angle response.



VN ¼1g1þ 1

g2

� �Y

� ��1

IN:

Finally, we may re-write Eq. (20) in a state space form by defining

bi ¼ 1þ 1gi

� �1and manipulating to obtain.

VN ¼12þ b1 � 1

2

�b2 � 1

2

�� f1

f2

� �� Y�1IN

þ 1þ ½ ðb1 � 1Þ ðb2 � 1Þ �f1

f2

� �� VB ð25Þ

Eq. (25) represents a generic fault-dependent state space expressionthat can be easily incorporated into the overall SMIB dynamicalmodel, as outlined below.

As shown in Fig. 1, the input to the exciter is the magnitude ofthe terminal voltage. In order to incorporate this into the overall

0 10 20

Spee

d

0 2 4 6 8

Spee

d

15 20 25 30 35

Spee

d

Time (s

-4

-2

0

2

-1

-0.5

0

0.5

-4

-2

0

2

Fig. 7. Speed

0 10 20 3

15 20 25 30 35Ti

Vn

-1

-0.5

0

0.5

Vn

-1

-0.5

0

0.5

Vn

-1

-0.5

0

0.50 2 4 6 8

Fig. 8. Terminal vo

generating unit model it is necessary to represent the machine ter-minal voltage in terms of the generating unit state variables. This isaccomplished next.

A phasor diagram representing this relationship is shown inFig. 4, where the load angle d is one of the state variables, definedin Section 2 above and the magnitude of VB is commonly assumedconstant, 1 pu. From the figure, the components of VB on the d- andq-axes, are expressed as

vBq ¼ jVBj cos d; vBd ¼ jVBj sin d

Thus taking the d-axis as the reference axis, we write

VB ¼ vBd þ jvBq ð26Þ

Similarly, we write

30 40 50 60

10 12 14 16 18

40 45 50 55 60

econds)

response.

0 40 50 60

40 45 50 55 60me

10 12 14 16 18

ltage response.


VN ¼ vd þ jvq ð27Þ

Equating Eq. (27) with Eq. (25), substituting for IN = id + jiq,VB = vBd + jvBq and using Y = R + jX yields

vd þ jvq ¼

12þ b1 � 1

2

�b2 � 1

2

�� f1

f2

� �� ðRþ jXÞðid þ jiqÞ

þ 1þ ½ ðb1 � 1Þ ðb2 � 1Þ �f1

f2

� �� ðvBd þ jvBqÞ

26664

37775ð28Þ

From Eq. (28), the real and imaginary parts of the terminal voltageare derived as

0 5 10 15 20

δ, load

dδ / dt, loa

Ta, accelera

PN, machine te

-0.5

0

0.5

0 5 10 15 20VN, terminal volta

-5

0

5

-5

0

5

0 5 10 15 20

0 5 10 15 20

0 5 10 15 20

-10

0

10

-10

0

10

Fig. 9. The response of the outp

vd ¼

12þ b1 � 1

2

�b2 � 1

2

�� f1

f2

� �� ðRid � XiqÞ

þ 1þ ½ ðb1 � 1Þ ðb2 � 1Þ �f1

f2

� �� jVBj sin d

26664

37775 ð29Þ

vq ¼

12þ b1 � 1

2

�b2 � 1

2

�� f1

f2

� �� ðRiq þ XidÞ

þ 1þ ½ ðb1 � 1Þ ðb2 � 1Þ �f1

f2

� �� jVBj cos d

26664

37775 ð30Þ

Linearising Eqs. (29) and (30) gives, respectively

25 30 35 40

angle

d speed

ting torque

rminal power

25 30 35 40ge of generator

25 30 35 40

25 30 35 40

25 30 35 40

uts when VREF is increased.


vd ¼

jVBj cos d�dþ 12 Rid � 1

2 Xiqþ

b1 � 12

�Ri�d � b1 � 1

2

�Xi�q

þðb1 � 1ÞjVBj sin d�

( )b2 � 1

2

�Ri�d � b2 � 1

2

�Xi�q


( )" #f1

f2

" #0BB@

1CCA

ð31Þ

vq ¼

�jVBj sin d�dþ 12 Riq þ 1

2 Xidþ

b1 � 12

�Ri�q þ b1 � 1

2

�Xi�d

þðb1 � 1ÞjVBj cos d�

( )b2 � 1

2

�Ri�q þ b2 � 1

2

�Xi�d


( )" #f1

f2

" #0BB@

1CCA

ð32Þ

z1 and its esti

z2 and its esti0 5 10 15 20




0 5 10 15 20

-4

-2

0

2

-20

-10

0

10

-10

-5

0

5

-1

0

1

2

-0.5

0

0.5

Fig. 10. The response of the outputs of the augmented system and the

Finally the magnitude of the machine terminal voltage is now ob-tained from

jVNj ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiv2

d þ v2q

qð33Þ

A linearised form of the machine terminal voltage is derived as

jDVNj ¼v�dDvd þ v�qDvqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

v�2d þ v�2q

q ð34Þ

defining ac ¼ 1=ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiv�2d þ v�2q

qand dropping D results in

mate (dotted)

mate (dotted)25 30 35 40




25 30 35 40

corresponding observer outputs (dotted) when VREF is increased.


jVN j ¼

acv�d

jVB j cos d�dþ 12 Rid � 1

2 Xiqþ

b1 � 12

�Ri�d � b1 � 1

2

�Xi�q

þðb1 � 1ÞjVB j sin d�

( )b2 � 1

2

�Ri�d � b2 � 1

2

�Xi�q


( )" #f1

f2

" #0BB@

1CCA

þacv�q

�jVB j sin d�dþ 12 Riq þ 1

2 Xidþ

b1 � 12

�Ri�q þ b1 � 1

2

�Xi�d

þðb1 � 1ÞjVB j cos d�

( )b2 � 1

2

�Ri�q þ b2 � 1

2

�Xi�d


( )" #f1

f2

" #0BB@

1CCA

0BBBBBBBBBBBB@

1CCCCCCCCCCCCA

ð35Þ

In order to express the terminal voltage in terms of the state vari-ables of the generating unit, we first need to express the currentsin terms of the state variables as outlined below:

Ix ¼ L�1xw and L ¼

�Ld Lad Lad 0 0 0

�Lad LF Lad 0 0 0

�Lad Lad LD 0 0 0

0 0 0 �Lq Laq Laq

0 0 0 �Laq LG Laq

0 0 0 �Laq Laq LQ

26666666664

37777777775

ð36Þ

where

Ix ¼ ½ id iF iD iq iG iQ �T

Hence we can write

id ¼ e1L�1xw

iq ¼ e4L�1xw

ð37Þ

where

e1 ¼ ½1 0 0 0 0 0 �; e4 ¼ ½0 0 0 1 0 0 �

As a result Eq. (35) can be expressed as

jVN j ¼

acv�d

jVB j cos d�dþ 12 Re1L�1xw � 1

2 Xe4L�1xwþ

b1 � 12

�Ri�d � b1 � 1

2

�Xi�q


( )b2 � 1

2

�Ri�d � b2 � 1

2

�Xi�q


( )" #f1

f2

" #0BBB@

1CCCA

þacv�q

�jVB j sin d�dþ 12 Re4L�1xw þ 1

2 Xe1L�1xwþ

b1 � 12

�Ri�q þ b1 � 1

2

�Xi�d


( )b2 � 1

2

�Ri�q þ b2 � 1

2

�Xi�d


( )" #f1

f2

" #0BBB@

1CCCA

0BBBBBBBBBBBBB@

1CCCCCCCCCCCCCA

ð38Þ

Eq. (38) can be expressed in a matrix-vector compact form as

jVNj ¼ ½Gw Gd � xTw xT

d

� T þ Gf f ð39Þ

0 10 20 30 40

The fault

0

0.2

0.4

0.6

0.8

1

Fig. 11. The fault (left) and its reconstru

where

Gw ¼ac

2v�dRþ v�qX�

e1L�1 þ v�qR� v�dX�

e4L�1n o

ð40Þ

Gd ¼ acv�dVB cos d� � acv�qVB sin d��

0h i

ð41Þ

Gf ¼

acv0d

b1 � 12

�Ri�d � b1 � 1

2

�Xi�q


( )þ acv0

q

b1 � 12

�Ri�q þ b1 � 1

2

�Xi�d


( )

acv0d

b2 � 12

�Ri�d � b2 � 1

2

�Xi�q


( )þ acv0

q

b2 � 12

�Ri�q þ b2 � 1

2

�Xi�d


( )2666664

3777775

T

ð42Þ

4. Derivation of state space model

4.1. State equations

A linearised model of the generating unit can now be expressed,in state space form, by combining the synchronous machine, exci-ter, and governor models. As a result the following state equation isobtained

_x ¼ Axþ Buþ Exf ð43Þ

where the state vector is defined as

x ¼ xTw xT

d xTex xT

gov� T

; xw ¼ ½wd wF wD wq wG wQ �T ;

xd ¼ ½ d _d �T ;

xex ¼ ½VR VF VA �T ; xgov ¼ ½ PST PGV PSR �T

and the input vector u is defined as.

u ¼ ½VREF PREF �T

VREF is the reference voltage and PREF is the reference power. Thematrix parameters A, B, Ex are derived in Appendix A.

4.2. Output equations

For the purposes of simulation, state estimation and controllerdesign, it is common to use the most easily available measurements,such as load angle, speed, terminal voltage, accelerating torque andterminal power. We therefore define our output equation as

y ¼ ½ d _d VN Ta PN �T ¼ Cxþ Eyf ð44Þ

The parameters C, Ey are derived in Appendix B.

0 10 20 30 40

The fault reconstruction

0

0.2

0.4

0.6

0.8

1

ction (right) when VREF is increased.


5. Fault studies

In this section, we simulate cases fault conditions for the linearpower system described above. A number of scenarios are consid-ered where fault occur on one line only and on both lines at differ-ent locations.

5.1. Case study I: fault on a single line

This study involves the following sequence of events. First werun the system unperturbed (i.e. no changes in the reference inputsand no fault). Second, at time t = 2 s, we increase the reference volt-age VREF by 5% and run the simulation for another 25 s. Third, we

0 5 10 15 2

0 5 10 15 2

0 5 10 15 2

δ, load a

-4

-2

0

2dδ / dt, lo

-4

-2

0

2

VN, terminal volta

Ta, accelera

0 5 10 15 2-5

0

5

10

0 5 10 15 2-10

-5

0

5

PN, machine te

-0.5

0

0.5

Fig. 12. The response of the outp

inject three-phase line-to-ground fault on line 2 of the transmis-sion system at time 25 s.

The output responses are shown in Fig. 5. It is clear from theindividual figures that the load angle slips back as a result of theincrease in the terminal voltage. This is due to the fact that asthe voltage increases, the terminal power increases and as the in-put power remains constant, the increase in the terminal power ismet by release of the kinetic energy stored in the rotor shaft, hencethe slipping back of the rotor angle. The slipping back of the loadangle is accompanied by a decrease in the speed, but as the unitis connected to an infinite bus, the speed recovers to that of theinfinite bus and therefore no change in the speed occurs at steadystate.

0 25 30 35 40

0 25 30 35 40

0 25 30 35 40

ngle

ad speed

ge of generator

ting torque

0 25 30 35 40

0 25 30 35 40rminal power

uts when PREF is increased.


With respect to the accelerating torque, as the speed initiallydecreases, causing a decrease in the electrical torque, the rotorshaft accelerates (Ta = Tm � Te), but when the speed recovers tozero change, the acceleration reduces to zero too, as expected.

The figures also show clearly that, due to the exciter action, theterminal voltage is increased by the required amount of 5%.

When the fault is applied, the same chain of events takes place,only in this case more profoundly.

5.2. Case study II: fault on both line

In this study, we consider an initially unperturbed system (i.e.no changes in the reference inputs and no fault). Then we affect

z1 and its estimate

0 5 10 15 20

z2 and its esti

0 5 10 15 20

0 5 10 15 20

z5 and its esti

0 5 10 15 20

z4 and its esti

-20

-10

0

10

-0.5

0

0.5

0 5 10 15 20

z3 and its esti

-4

-2

0

2

-10

-5

0

5

-1

0

1

2

Fig. 13. The response of the outputs of the augmented system and the

a 5% step increase in reference voltage VREF at 2 s. This is followedby a 3 phase balanced fault at 10 s in line 1 at 30% length of thetransmission line from the generator end and run the simulationtill 30 s. After 30 s we inject a fault in line 2 at 80% length of theline from the generator end. Output responses are given in Figs.6–8.

Each of the output responses are subdivided into three figures,top one shows the overall response, middle one shows first 17 sand the bottom one shows the response between t = 16 s andt = 60 s. The responses up to 30 s are same as the responses in studycase I.

After 30 s when fault takes place in line 2, the generator connec-tion to the infinite bus is completely severed. As a result the load

(dotted)

25 30 35 40

mate (dotted)

25 30 35 40

25 30 35 40

mate (dotted)

25 30 35 40

mate (dotted)

25 30 35 40

mate (dotted)

corresponding observer outputs (dotted) when PREF is increased.


angle continues slipping back at a constant rate and the speed set-tles to a new steady state, after the transient period. The terminalvoltage undergoes a large fall immediately after the fault, as shownin Fig. 8, but it then recovers to 5% above the initial condition dueto the action of the voltage regulator (exciter).

6. Fault detection using a sliding mode observer

Conventional fault detection algorithms are typically based onobserver theory, where the error residual between the observerand the plant is used as an indicator for faults. Example of residualgeneration algorithms incorporating unknown input observer the-ory are found in [8–10], and incorporating H1 are found in [11,12].However, recent approaches such as sliding mode observer (SMO)theory [13] have proven to be quite effective in fault detection insystems with unknown inputs such as faults.

In the following we first present a brief outline of the theory ofsliding mode observer. Readers who are not familiar with controltheory may skip this section and the next section move onto Sec-tion 8.

For convenience let us recall the SMIB state-space equations

_x ¼ Axþ Buþ Exf

y ¼ Cxþ Eyfð45Þ

where x 2 Rn¼14; y 2 Rp¼5; u 2 Rm¼2; f 2 Rq¼1. Since Ey has full col-umn rank, there exists an orthogonal matrix Tr 2 Rp�p such that

TrEy ¼0

Ey2

� �; Try ¼

y1

y2

� �TrC ¼

C1

C2

� �ð46Þ

where Ey2 is a square and invertible matrix, and C1 and C2 are gen-eral matrices with no particular structure.

Let yf 2 Rq be the output of a user defined stable filter driven byy2:

_yf ¼ �Af yf þ A2y2 ¼ �Af yf þ Af C2xþ Af Ey2f ð47Þ

Combine (45) and (47) to get the following augmented state-spacesystem of order �n :¼ nþ q

_x_yf

� �¼

A 0Af C2 �Af

� �|fflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflffl}

A

x

yf

" #x

þB

0

� �B

uþEx

Af Ey2

� �|fflfflfflfflffl{zfflfflfflfflffl}

Ex

f

y1

yf

" #z

¼C1 00 Iq

� �|fflfflfflfflfflffl{zfflfflfflfflfflffl}

C

x

yf

" # ð48Þ

0 10 20 30 40

The fault

0

0.2

0.4

0.6

0.8

1

0

0

0

0

Fig. 14. The fault (left) and its reconstru

Define z :¼ y1yf

� �; z 2 Rpþq¼6 (It is clear that z is a measurable

signal.)

Theorem 1. If the following conditions are satisfied

(i) rankðCExÞ ¼ rankðExÞ; and(ii) the zeros of ðA; Ex; CÞ (if any) are stable,

then for the state-space model described by (48) there exists a changeof coordinates such that the triple ðA; Ex; CÞ can be re-written as:

A ¼A11 A12

A21 A22

� �; Ex ¼

0M2

� �; C ¼ ½0 T �; M2 ¼

0Mo

� �ð49Þ

where A11 2 Rð�n�pÞ�ð�n�pÞ, matrix T 2 Rp�p is orthogonal, M2 2 Rp�q, and

Mo 2 Rq�q is invertible. Any unobservable modes of (A11,A21) are theinvariant zeros of A; Ex; C

� and are stable. Full proof is given in [14].

By Theorem 1, a sliding mode observer of the structure shownbelow can be designed for the system (48) to reconstruct the faultf as well as the entire state and output variables.

_w ¼ Awþ Bu� Gley þ Gnm ð50Þ

where ey ¼ Cw� z and m is a nonlinear switching term defined by

m ¼ �qey

keyk; ey–0;q 2 Rþ ð51Þ

The matrices Gl; Gn are to be designed; in particular Gn has the

structure Gn ¼ �LTT

TT

� �P�1

o , where Po 2 Rp�p is a symmetric positive

definite matrix, and Lo 2 Rð�n�pÞ�ðp�qÞ. It has been proven in [15] that a

sliding motion will take place in finite time on the surfaceSe = {ey = 0} if the following conditions hold:

Condition (i): there exists a matrix P with the structure

0

0

.2

.4

.6

.8

1

ction (ri

P ¼P11 P11L

LT P11 TT PoT þ LT P11L

� �> 0; P11 2 Rð�n�pÞ�ð�n�pÞ;

that satisfies the following inequality

PðA� GlCÞ þ ðA� GlCÞT P < 0 ð52Þ

Condition (ii): The scalar q in the switching function m satisfiesthe following inequality

q > kPoTM2k � kfk:

When sliding motion has been achieved, the fault can be recon-structed from the following signal:

10 20 30 40

The fault reconstruction

ght) when PREF is increased.


fe :¼ 0 M�1o

� P�1

o meq

where meq is a version of m required to achieve and maintain slidingmotion. The signal meq is computable online by replacing m with meq

where

meq ¼ �qey

keyk þ cð53Þ

and c is a small positive scalar that governs the degree of accuracyof meq.

7. Observer design

In this section, the theory presented in Section 6 is used todesign a SMO suitable for the power developed in Section 4. Wethen demonstrate, through extensive simulations, the designedsliding mode observer’s ability to detect faults on line and in realtime.

Using the data given in Appendix C, then for the given value ofEy, a suitable value for Tr was found to be

Tr ¼

1 0 0 0 00 1 0 0 00 0 1 0 �1:11940 0 0 1 00 0 0 0 1

26666664

37777775 ð54Þ

This transformation matrix resulted in the output equation beingtransformed to the structure shown in Eq. (46). Let us partition yand z from (45) and (48) as follows:

y ¼

y1

y2

:

yp

26664

37775; z ¼

z1

z2

:

zp

26664

37775

Then from (46)–(48) and the using Tr from (54) and choosing Af = 1we have

z1 ¼ y1; z2 ¼ y2; z3 ¼ y3 � 1:1194y5; z4 ¼ y4; _z5 ¼ �z5 þ y5

Implementing the design method outlined above, and choosing sca-lar parameters q, c associated with the nonlinear switching term(53) as 1500 and 0.0001, respectively, the following observer gainsare found.

Gl ¼ Gn ¼

�0:0076 �0:0344 0:2477 0:0531 0:0149

�0:0148 �0:0318 �0:0401 �0:0247 0:0244

�0:0112 �0:0403 �0:0456 �0:0334 0:0191

�0:0083 0:0476 0:7441 0:6949 0:0067

�0:0054 0:0043 0:0040 0:0036 0:0038

�0:0069 0:0107 0:0089 0:0101 0:0055

0:0131 �0:0074 �0:0021 �0:0095 �0:0122

�0:0074 0:6588 0:9079 0:3008 0:0013

�0:0137 0:0365 2:1023 1:6759 0:0163

0:0132 0:0084 �0:0291 0:0476 �0:0154

0:2084 �0:0388 �1:0412 1:0056 �0:2118

0:0002 �0:0018 �0:0017 �0:0017 0

�0:0001 �0:0011 �0:0013 �0:0009 0:0005

�0:0004 0:0013 �0:0059 0:0050 0:0080

�0:0122 0:0013 �0:0128 0:0058 0:0371

2666666666666666666666666666666666664

3777777777777777777777777777777777775

8. Simulation results

In this section we investigate the performance of the SMO onthe sample power system described in (45). The following scenariois simulated: The power system initially runs unperturbed (steady-state) condition for 5 s. Then the excitation system reference com-mand, VREF, is increased by 5% at time t = 5 s. This is followed byapplying a solid three-phase line-to-ground fault half way alongline 2 at time t = 25 s.

The simulation results are shown in Figs. 9–14, where the fol-lowing initial conditions are used: The power system states statevector, x, was assigned the value calculated from the initialisationprocess, which isx = [0.9501,0.2311,0.6068,0.4860,0.8913,0.7621,0.4565,0.0185,-0.8214,0.4447,0.6154,0.7919,0.9218,0.7382]. The observer wasassumed to have zero initial conditions (this is normally the situa-tion in practice when the observer is first connected to the powersystem and switched on).

Fig. 9 shows the responses of the output variables, which are ex-actly the as those shown in Fig. 5 (please refer to Section 5 for com-mentary on the behaviour of these variables). Fig. 10 shows theoutputs of the augmented system as well as the outputs of the faultfilter (superimposed with dotted lines). Visually, there is no differ-ence between the two sets of responses despite the presence of thefault and the mismatch in the initial conditions (though uponzooming in, it can be seen that there is an initial mismatch be-tween the augmented system outputs and the observer outputs,but perfect convergence takes place after 0.05 s).

Fig. 10 also shows that the fault filter is also a state estimatorand is able to reconstruct exactly the output variables. Further-more the figure shows a very important characteristic of the faultfilter; that is it is only sensitive to faults and no any other internalor external inputs.

Fig. 11 shows the performance of the fault filter, which demon-strates quite clearly its ability to detect and reconstruct the fault innear real time despite the fact that it starts from a different set ofinitial conditions. In practice, however, the fault filter is continu-ously switched on and therefore is able to detect faults in real timewhenever faults occur. Note that from the period up to the applica-tion of fault, the fault filter does not respond despite the fact that achange in the external input is affected.

Figs. 12–14 show the case where the reference power, PREF, is in-creased. Here again the fault reconstruction takes place almost inreal time, despite the difference in the initial conditions.

9. Conclusions

This paper presents a new fault dependent model for a singlemachine connected to an infinite bus through double transmissionlines. The model is simulated for faults occurring in either of thetransmission lines and in both and the results are discussed in de-tail, where it is explained how the model behaves in the manor itdoes in real life.

Then a fault detection filter (locator) is designed using slidingmode observer theory. The filter is then tested, through extensivesimulations, on the power system where disturbances and faultsare applied. It is demonstrated quite clearly that the designed faultdetector is only sensitive to faults and not to any other externaldisturbance, such changes in the reference inputs, load demands,noise, etc. furthermore the filter is also capable of acting as a stateestimator where the states, and outputs can be reconstructed intheir entireties.

Extension of this research is now underway to the multi-machine case and also to the case where asymmetrical faults occurin the transmission network.


Appendix A. State equations parameters

Combining the state equations of the synchronous machine, ex-citer and governor results in the following state equation.

_x ¼ Axxþ AiI þ Buþ Ef

where the current I is defined as follows:

I ¼ ITx 0T

8

� ; Ix ¼ id iF iD iq iG iQ

� T

The current Ix can be expressed in terms of the states, through thefollowing flux linkage–current relationship.

Ix ¼ L�1xw; L ¼

�Ld Lad Lad 0 0 0

�Lad LF Lad 0 0 0

�Lad Lad LD 0 0 0

0 0 0 �Lq Laq Laq

0 0 0 �Laq LG Laq

0 0 0 �Laq Laq LQ

266666666664

377777777775

As a result, the state equation can now be written as

_x ¼ Axþ Buþ Exf ; A ¼ Ax þ AiLinverse:

The matrix A is 14 � 14 real matrix with the nonzero elementsbeing

Axð1;4Þ ¼ xbð1þ _d�Þ; Axð1;7Þ ¼ xbjVBj cos d�

Axð1;8Þ ¼ xbw�q

Axð2;11Þ ¼ xbrF

Lad

Axð4;1Þ ¼ �xbð1þ _d�Þ

Axð4;7Þ ¼ �xbjVBj sin d�; Axð1;8Þ ¼ �xbw�d

Axð7;8Þ ¼ 1

Axð8;1Þ ¼�xb

2Hi�q

Axð8;4Þ ¼xb

2Hi�d

Axð8;8Þ ¼�xb

2Hkd þ

T�m1þ _d�

� �

Axð9;7Þ ¼KR

TRacjVBj v�d cos d� � v�q sin d�

�

Axð9;9Þ ¼�1TR

; Axð10;9Þ ¼ �KAKF

TATF

Axð10;10Þ ¼ � 1TFþ KAKF

TATF

� �

Axð10;11Þ ¼ � KF

TATF

Axð11;9Þ ¼ �KA

TA; Axð11;10Þ ¼ �KA

TA; Axð11;11Þ ¼ �1

TA

Axð12;8Þ ¼ �1xbTSRRGov

; Axð12;12Þ ¼ �1TSR

Axð13;12Þ ¼ 1TSM

; Axð13;13Þ ¼ � 1TSM

Axð14;13Þ ¼ 1TST

; Axð14;14Þ ¼ � 1TST

The matrix Ai is also 14 � 14 real matrix with the following nonzeroentries

Aið1;1Þ ¼ xb r þ R2

� �; Aið1;4Þ ¼ �

xbX2

Aið2;2Þ ¼ �xbrF ; Aið3;3Þ ¼ �xbrD

Aið4;1Þ ¼xbX

2; Aið4;4Þ ¼ xb r þ R

2

� �Aið5;5Þ ¼ �xbrG; Aið6;6Þ ¼ �xbrQ

Aið8;1Þ ¼xb

2Hw�q; Aið8;4Þ ¼ �

xb

2Hw�d

Aið9;1Þ ¼KR

TRacv�d

R2þ KR

TRacv�q

X2

� �

Aið9;4Þ ¼KR

TRacv�q

R2� KR

TRacv�d

X2

� �

where

ac ¼1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

v�2d þ v�2q

qv0

d and v0q are initial values of vd and vq, respectively.

The matrix B is 14 � 2 real matrix with the following nonzeroentries

Bð10;1Þ ¼ KAKF

TATF;Bð10;1Þ ¼ KA

TA;

Bð12;2Þ ¼ 1TSR

Ex is a 14 � 2 matrix with the following nonzero elements.

Exð1;1Þ ¼ ðb1 � 1ÞxbjVBj sin d� þ b1 �12

� �xbRi�d � b1 �

12

� �xbXi�q

Exð4;1Þ ¼ ðb1 � 1ÞxbjVBj cos d� þ b1 �12

� �xbRi�q þ b1 �

12

� �xbXi�d

Exð9;1Þ ¼

ðb1 � 1ÞjVBj sin d� KRTR

acv�dþKRTR

acv�d b1 � 12

�Ri�d �

KRTR

acv0d b1 � 1

2

�Xi0

q

0@

1Aþ

ðb1 � 1ÞjVBj cos d� KRTR

acv�qþKRTR

acv�q b1 � 12

�Ri�q þ

KRTR

acv�q b1 � 12

�Xi�d

0@

1A

0BBBBBBB@

1CCCCCCCA

Exð1;2Þ ¼ ðb2 � 1ÞxbjVBj sin d� þ b2 �12

� �xbRi�d � b2 �

12

� �xbXi�q

Exð4;2Þ ¼ ðb2 � 1ÞxbjVBj cos d� þ b2 �12

� �xbRi�q þ b2 �

12

� �xbXi�d

Exð9;2Þ ¼

ðb2 � 1ÞjVBj sin d� KRTR

acv�dþKRTR

acv�d b2 � 12

�Ri�d �

KRTR

acv0d b2 � 1

2

�Xi0

q

0@

1Aþ

ðb1 � 1ÞjVBj cos d� KRTR

acv�qþKRTR

acv�q b2 � 12

�Ri�q þ

KRTR

acv�q b2 � 12

�Xi�d

0@

1A

0BBBBBBB@

1CCCCCCCA

Appendix B. State equations parameters

The output equation is defined in (44) as

y ¼ ½ d _d VN Ta PN �T

The first two are defined as state variables. The terminal voltage isderived in Eq. (38). The accelerating torque Ta is defined as


Ta = Tm � Te, where Te and Tm are given in (8) and (15), respectively.The terminal power of the generator, PN is written as

PN ¼ V�NIN þ I�NVN

where I0N and V0

N are initial values of terminal voltage and current,respectively, and

IN ¼idi�d þ iqi�q� ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffii�2d þ i�2q

q and VN ¼vdv�d þ vqv�qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

v�2d þ v�2q

qCombining these expressions, yields the following output equation

y ¼ Cxþ Eyf

where C = (Cx + CiLinverse), Cx is a 5 � 14 matrix with the followingnonzero elements:

Cxð1;7Þ ¼ 1; Cxð2;8Þ ¼ 1Cxð3;7Þ ¼ acv�djVBj cosðd�Þ � acv�qjVBj sinðd�ÞCxð4;1Þ ¼ �i�q; Cxð4;4Þ ¼ i�dCxð4;8Þ ¼ �T�m=ð1þ _d�Þ; Cxð4;14Þ ¼ 1=ð1þ _d�ÞCxð5;7Þ ¼ I�N acv�djVBjCosd� � a1v�c jVBjSind�

�Ci is a 5 � 14 matrix with the following nonzero elements

Cið3;1Þ ¼12

acv�dRþ 12

acv�qX; Cið3;4Þ ¼ �12

acv�dX þ 12

acv�qR

Cið4;1Þ ¼ w�q; Cið4;4Þ ¼ �w�d

Cið5;1Þ ¼ 12

acv�dRþ 12

acv�qX� �

I�N þ V�Nbi�d

Cið5;4Þ ¼ �12

acv�dX þ 12

acv�qR� �

I�N þ V�NbI�q

where

b ¼ 1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffii02

d þ i�2q

qEy is a 5 � 2 matrix with the following nonzero elements

Eyð3;1Þ ¼acv�dðb1 � 1ÞVB sin d� þ acv�d b1 � 1

2

�Ri�d

�acv�d b1 � 12

�Xi�q þ acv�qðb1 � 1ÞVB cos d�

þacv�q b1 � 12

�Ri�q þ acv�q b1 � 1

2

�Xi�d

0B@

1CA

Eyð5;1Þ ¼acv�dðb1 � 1ÞVB sin d0 þ acv�d b1 � 1

2

�Ri�d�

acv�d b1 � 12

�Xi�q þ acv�qðb1 � 1ÞVB cos d0þ

acv�q b1 � 12


2

�Xi�d

0BB@

1CCAI�N

Eyð3;2Þ ¼acv�dðb2 � 1ÞVB sin d� þ acv�d b2 � 1

2

�Ri�d

�acv�d b2 � 12

�Xi�q þ acv�qðb2 � 1ÞVB cos d�

þacv�q b2 � 12


2

�Xi�d

0B@

1CA

Eyð5;2Þ ¼acv�dðb2 � 1ÞVB sin d0 þ acv�d b2 � 1

2

�Ri�d�

acv�d b2 � 12

�Xi�q þ acv�qðb2 � 1ÞVB cos d0þ

acv�q b2 � 12


2

�Xi�d

0BB@

1CCAI�N

Appendix C. Data

Ld ¼ 0:2768; Lad ¼ 0:2497; Laq ¼ 0:2422; Lq ¼ 0:2692

Lf ¼ 0:2795; LG ¼ 0:3804; LQ ¼ 0:2647; R ¼ 0:0200

X ¼ 0:2000; kd ¼ 1;H ¼ 2:3700 KW s=KVA; xb ¼ 2pf rad=s

KR ¼ 1; TR ¼ 0:0001 s; KA ¼ 400; TA ¼ 0:0500 s

KF ¼ 0:0400; TF ¼ 1 s; Rgov ¼ 0:0500; TSR ¼ 0:1000 s

TSM ¼ 0:2500 s; TST ¼ 1 s

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