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Modeling and Simulation of Internal Faults in Salient-pole SynchronousGenerators with Wave WindingsAmrita Sinhaa; D. N. Vishwakarmaa; R. K. Srivastavaa

a Department of Electrical Engineering, Institute of Technology, Banaras Hindu University, Varanasi,India

Online publication date: 28 December 2009

To cite this Article Sinha, Amrita , Vishwakarma, D. N. and Srivastava, R. K.(2010) 'Modeling and Simulation of InternalFaults in Salient-pole Synchronous Generators with Wave Windings', Electric Power Components and Systems, 38: 1,100 — 114To link to this Article: DOI: 10.1080/15325000903273338URL: http://dx.doi.org/10.1080/15325000903273338

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Electric Power Components and Systems, 38:100–114, 2010

Copyright © Taylor & Francis Group, LLC

ISSN: 1532-5008 print/1532-5016 online

DOI: 10.1080/15325000903273338

Modeling and Simulation of Internal Faults

in Salient-pole Synchronous Generators

with Wave Windings

AMRITA SINHA,1 D. N. VISHWAKARMA,1 and

R. K. SRIVASTAVA1

1Department of Electrical Engineering, Institute of Technology,

Banaras Hindu University, Varanasi, India

Abstract When an internal fault occurs in the wave-connected windings of salient-

pole synchronous generators, the symmetry between the parallel windings is broken,and different currents flow in them since unsymmetrical magnetic linkage exists

between the stator windings. The aim of this article is to present a model to investigatethe internal fault currents in large hydrogenerators with wave windings. This model

is based on a modified winding function approach, where the machine inductancesare calculated directly from the machine winding distribution, and the space harmon-

ics produced by them are also taken into account. The calculation of the machineinductances are made easier by the use of machine electrical parameters instead of

geometrical parameters. The fast Fourier transform analysis of the simulated resultshas been tabulated for different cases of internal faults. By using the simulated

internal fault data, suitable numerical protection schemes for hydrogenerators canbe developed.

Keywords internal faults, salient-pole synchronous generators, hydrogenerators,modified winding function, parallel windings

1. Introduction

Hydrogenerators used in hydel power stations are large salient-pole synchronous gener-

ators that form an important part of the modern power system and usually have wave-

connected stator windings. Knowledge about the internal faults in the stator windings

of large synchronous machines is essential, as the large fault currents may cause severe

damage to the windings and possibly to the shaft and coupling of the machine. In order to

design an appropriate numerical protection scheme for synchronous generators to protect

against internal faults, it is important to get the fault data. However, it is impractical and

very difficult to get actual synchronous generator fault data. Therefore, a detailed analysis

of a variety of situations is done by simulating a synchronous generator under internal

fault conditions, i.e., turn-to-turn, inter-turn, turn-to-frame, insulation failure, etc., which

is more versatile and cost effective. An internal fault simulation requires access to the

stator winding using the modified winding function approach.

Received 25 March 2009; accepted 8 June 2009.Address correspondence to Dr. Amrita Sinha, Department of Electrical Engineering, Institute

of Technology, Banaras Hindu University, Varanasi, 221 005, India. E-mail: amrita_bhu@yahoo.co.in

100

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Simulation of Faults in Synchronous Generators 101

Several articles have been published analyzing the internal faults of synchronous

machines. The fundamental and only third harmonic components of time and space are

taken into consideration in the symmetrical component method [1]. However, under an

internal fault condition, the symmetry between the machine windings no longer exists,

and the faulted windings would produce stronger space harmonics in the air gap magnetic

field.

Due to the inconvenience of the dq0 model, internal faults of synchronous machines

using direct phase quantities were simulated in [2]. The voltage and flux equations are

directly developed in the fixed phase reference without coordinate transformation. Under

internal fault conditions, the ratio between the winding inductances has been assumed to

be proportional to the ratio between the effective number of turns of the corresponding

windings [3]. However, this is correct only when the windings are concentrated or a

sinusoidal distribution of magneto motive force (MMF) exists in the air gap.

The physical arrangement of the conductors and only the fundamental component

of the MMF of the faulted section were considered in [4]. A straight-forward method

for partitioning the sinusoidally distributed stator windings was used in [5–7]. However,

the actual stator windings are never perfectly sinusoidally distributed in space. A model

based on the multi-loop circuit method of AC machines was presented in [8] to calculate

the inductance parameters of the windings. However, the difficulties and inaccuracies

involved in the calculation of loop inductances make the method impractical.

The accuracy of the model for the simulation of internal faults in synchronous

machines depends on the accuracy of the calculation of the inductance of the faulty

windings, i.e., the asymmetry developed due to the fault must be considered in the

modeling of the machines. The winding function theory has been developed as a veritable

tool for the calculation of AC machine inductances [9]. The turns and modified winding

function theory presented in [10] has the capability to take any winding distribution of

stator or air gap distribution around a rotor into account and can be used for modeling and

analyzing the fault conditions of synchronous machines. The space harmonics produced

by a machine winding can be determined by its winding function, which depicts MMF

distribution in the air gap in terms of space distribution. In [11], this approach was used

in the transient analysis of asynchronous machines for broken rotor bars, broken end

ring segments, or fault conditions in stator windings. The same approach was used in

the analysis of air gap eccentricity in the synchronous machines [12–14]. The combined

winding function theory and direct phase quantities for simulating the internal faults

in the synchronous machines wave used in [15]. The distribution factor and the pitch

factor are taken into account for the calculation of inductances. A synchronous machine

model for internal fault analysis based on the modified winding function theory, using

machine electrical parameters, was presented in [16–19]. The calculation of inductances

is directly derived from the original waveforms of the winding function of the actual

machine winding distribution. Hence, all the space harmonics generated by the windings

are accounted for in the modeling of the synchronous machines.

To choose an adequate type and trip level of relay for the protection of synchronous

machines, not only is the current in the faulted winding important, but the change in other

parallel paths should be taken into consideration as well. This article presents a simple and

accurate model to simulate internal faults in large hydrogenerators with parallel windings

using machine electrical parameters. The model is based on the modified winding function

theory. The calculation of inductance is derived directly from the original waveforms

using winding functions of the actual machine winding distribution; hence, the space

harmonics are taken into account. The machine electrical parameters are used to simplify

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102 A. Sinha et al.

the calculation of the machine inductances. The advantage of the simulation method

described in this article takes into account the mutual inductances between all possible

pairs of windings and, hence, provides the unbalanced currents in all the individual

parallel paths of the three phases under internal fault conditions.

2. Modified Winding Function Approach

A modified winding function theory, as expressed by Eqs. (1) and (2) and discussed

in [20], is used in this article to take into account the effect of the non-uniform air

gap and non-sinusoidally distributed windings in the case of internal faults in large

synchronous generators:

Lxy D 2��0rlhg�1.'; �/nx .'; �/ny .'; �/i

� 2��0rlhg�1.'; �/nx .'; �/ihg

�1.'; �/ny .'; �/i

hg�1.'; �/i; (1)

where

�0 is the permeability of the free space;

r is the average radius of the air gap;

l is the axial stack length of the machine;

g�1.'; �/ is the inverse air gap length;

nx.'; �/ and ny.'; �/ are the turns function of the windings x and y, respectively;

' is the angle along the inner surface of the stator; and

� is the angular position of the rotor with respect to the stator reference axis.

Operator hf iis defined as the mean value of function f over Œ0; 2�� as follows:

hf i D1

2�

Z 2�

0

f .'/d': (2)

For a 2p pole synchronous machine analysis, the inverse air gap length is usually

approximated as [21]

g�1.'; �/ D ˛0 � ˛2 cos.2p � �/; (3)

where the minimum air-gap length is .˛0 C ˛2/�1, and the maximum air-gap length is

.˛0 �˛2/�1. Substituting Eq. (3) into Eq. (1) and neglecting the higher order harmonics,

the mutual inductance Lxy of the stator windings is given by

Lxy D K0Œhnxnyi � hnxihnyi�

�K2Œhnxny cos.2p.' � �//i � hnxihny cos.2p.' � �//i

� hnyihnx cos.2p.' � �//i�; (4)

where coefficients K0 and K2 are given by

K0 D 2��0rl˛0 and K2 D 2��0rl˛2:

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Simulation of Faults in Synchronous Generators 103

The coefficients K0 and K2 are the machine geometrical constants and can be represented

by the machine electrical parameters Lmd and Lmq , as expressed in Eqs. (5) and (6) [18]:

K0 DLmd C Lmq

2Œ.hn2xi � hnxi2/ � .hnxnyi � hnxihnyi/�

; (5)

K2 DLmd �Lmq

2

�

1

2

q

A2xx CB2

xx Cq

A2xy C B2

xy

� ; (6)

where

Axy D hnxny cos.2p'/i � hnxihny cos.2p'/i � hnyihnx cos.2p'/i; (7)

Bxy D hnxny sin.2p'/i � hnxihny sin.2p'/i � hnyihnx sin.2p'/i; (8)

for Axx and Bxx, substitute y D x.

By substituting Eqs. (5) to (8) into Eq. (4), the mutual inductances between two

arbitrary windings x and y can be calculated. To calculate the self-inductance, y D x

in Eq. (4) can be used. The winding distribution arrangements of x and ycan be easily

found out with the help of the turns function nx and ny .

Expressions to Calculate Mutual Inductances Between Stator

and Rotor

The internal faults in the stator winding do not affect the winding distribution and the

electrical parameters of the rotor windings. However, since the winding distribution of

the faulted windings on the stator has changed, the mutual inductances between the rotor

and the stator should be calculated. The mutual inductance between an arbitrary stator

winding x and the damper windings kd , kq are given as [18]

Lfdx

Lkdx

)

D Lmd

hnx sin.p.' � �//ip

hna sin.p'/i2 C hna cos.p'/i2; (9)

Lkqx D Lmq

hnx cos.p.' � �//ip

hna sin.p'/i2 C hna cos.p'/i2: (10)

3. Mathematical Model of Synchronous Machines

The synchronous machines are usually described as a system having three stator phases

and three rotor circuits (one field winding and two damper windings). If the number of

parallel paths in the stator is n, then the machine can be represented by 3n stator circuits.

Under internal fault conditions, the number of stator circuits is determined by the number

of independent currents flowing in the stator. Therefore, it is convenient to assume m

stator circuits for the arbitrary stator configuration. Therefore, in terms of flux linkages,

, voltage relationships for the stator and rotor circuits, e, are linked with the winding

resistances, R, and instantaneous current, I , as follows:

e Dd

dt�Ri; (11)

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104 A. Sinha et al.

where

e D Œe1 e2 � � � em efd e1d e1q�t ;

D Œ 1 2 � � � m fd 1d 1q�t ;

R D diagŒR1 R2 � � � Rm Rfd R1d R1q�;

and

i D Œi1 i2 � � � im ifd i1d i1q�t :

The flux linkage is expressed as

‰ D Li; (12)

where L is the inductance matrix, given as

L D

2

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

4

L11 M12 M13 � � � � � � � � � M1m M1f M1kd M1kq

M12 L22 M23 � � � � � � � � � M2m M2f M2kd M2kq

M13 M23 L33 � � � � � � � � � M3m M3f M3kd M3kq

::::::

::: � � � � � � � � �:::

::::::

:::

::::::

::: � � � � � � � � �:::

::::::

:::

M1m M2m M3m � � � � � � � � � Lmm Mmf Mmkd Mmkq

M1f M2f M3f � � � � � � � � � Mmf Lff MfD MfQ

M1kd M2kd M3kd � � � � � � � � � Mmkd MfD LDD MDQ

M1kq M1kq M3kq � � � � � � � � � Mmkq MfQ MDQ LQQ

3

7

7

7

7

7

7

7

7

7

7

7

7

7

7

7

7

7

7

5

: (13)

Here, m D 3n C 1 for a fault in any one path, m D 3n C 2 for a fault between two

paths, and n D the number of parallel paths. Therefore, the inductance matrix is of the

order of m C 3, i.e., the m stator paths and three rotor circuits. When the machine is

supplying through a short transmission line (Figure 1), the machine terminal voltages can

be expressed as

e D V � iRE � idLE

dt� iRG � i

dLG

dt; (14)

where RE and LE are bothm-by-m diagonal matrices, and the elements are the resistance

RE and inductance LE of the short transmission line, respectively; RG and LG are

both m-by-m diagonal matrices, and the elements are the grounding resistance RG and

inductance LG , respectively, of the generator to limit the fault current. V represents

the m-by-1 infinite bus phase voltage vector. The equation of motion of a synchronous

generator can be expressed by

d 2ı

dt2D

!

2H.Tmech � Telec/; (15)

where

! is the synchronous speed,

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Simulation of Faults in Synchronous Generators 105

Figure 1. Winding distribution diagram of path a1 of phase a in the stator slots.

H is the inertia constant,

Tmech is the mechanical input torque, and

Telec is the electrical torque.

The electrical torque is given by

Te D1

2iTs@Lss

@�is C iTs

@Lrs

@�ir ; (16)

where

is D Œi1 i2 � � � im�,

ir D Œif i1d i1q�,

Lss is the stator inductance matrix, and

Lrs is the rotor to stator mutual inductance matrix.

The synchronous machine modeling during an internal fault can be done with the

help of Eqs. (11)–(16). The elements of the inductance matrix can be calculated from

Eqs. (4)–(10).

4. Internal Fault Simulation

The modified winding function approach has been used to model a three-phase hydro-

generator with two parallel paths per phase, and its electrical parameters are given in

the appendix. The winding arrangement of path a1 are in Figure 1. This salient-pole

synchronous generator has 18 poles, 216 slots, and double-layer wave winding. A single-

turn coil, in which the coil sides are made in the form of roebel bars, has been used in the

stator winding. These coils are constructed after considering the skin effect losses. In

the straight slot portion, the conductors or strips are transposed by 360ı. The transposition

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106 A. Sinha et al.

Figure 2. Turns function of paths a1 and a2 of phase a.

is done to ensure that all the strips occupy equal length under similar conditions of the

flux. The transposition provides for a mutual neutralization of the voltages induced in the

individual strips due to the slot cross-field and ensures that no or only small circulating

currents exists in the bar interior. This also reduces the eddy current losses and helps in

obtaining uniform e.m.f.

The turns functions of the phase distribution of the two parallel paths of phase a (a1

and a2) along the entire periphery of the stator are shown in Figure 2, and are apparently

non-sinusoidal.

Figure 3 shows the system representation, where a11 and a12 are the two sub-

windings under an internal fault condition in a1. Under a normal condition without

internal fault, the machine can be seen as a system of nine magnetically coupled circuits

in which six are stator circuits. The stator-to-stator mutual inductances are evaluated using

Eqs. (4)–(8). The stator-to-rotor mutual inductances are evaluated using Eqs. (9)–(11).

The parameters of the rotor winding are directly calculated from the machine standard

electrical parameters.

Figure 3. System representation of hydrogenerator with two parallel paths during a turn-to-ground

fault.

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Simulation of Faults in Synchronous Generators 107

The machine model is simulated in the MATLAB/SIMULINK environment and has

been solved using a TR-BDF2 solver (an implicit Runga-Kutta formula with a first stage

that is a trapezoidal rule step and a second stage that is a backward differentiation formula

of order two). Since the elements of the inductance matrix depend on the position of

the rotor � , the matrix L and its inverse have to be evaluated at each step in order to

determine the current.

4.1. Turn-to-ground Fault

The turn-to ground fault in path a1 is shown in Figure 1. Under an internal fault condition,

winding a1 is divided into two sub-windings, a11 and a12. The part adjacent to the

machine terminal that is connected to the infinite bus through the transmission line is

referred to as a11, while the part adjacent to the neutral that is short circuited through the

ground is referred to as a12. Therefore, the machine can be viewed as ten magnetically

coupled circuits, as given in the inductance matrix in Eq. (14), instead of nine under a

normal condition. Hence, the current, voltage, resistance, and inductance matrices are all

augmented by one to take the faulted path into account.

Thus, in the matrices Re and Le , the element corresponding to sub-winding a12 is

zero, taking into account that it is not connected to the transmission line. For the same

reason, the element corresponding to a12 is also zero in the voltage matrix. Similarly, in

the matrices Rg and Lg, the element corresponding to a11 is zero, as it is not connected

to the ground terminal. In the matrix Ra, the element corresponding a11 will be (Ra � %

of unfaulted portion), and a12 will be (Ra � % of faulty portion).

The turns function of a11 and a12 is shown in Figure 4 for a fault at 36% in

path a1 from the neutral, referring to their placements in the stator. From the figure,

it can be seen that due to the different winding distribution and position of the two

sub-windings a11 and a12, the neutral inductances between the sub-windings and the other

windings are different; hence, there exists different magnetic linkages between them. The

self-inductances of paths a11, a12, and a2 of phase a and the mutual inductances between

them for a fault at 36% from neutral after the inception of the fault are shown in Figure 5.

This shows that different magnetic linkages exist between them due to the different

position and winding distribution of the two sub-windings a11 and a12. The current in

each parallel path has been simulated and is shown in Figure 6. Before the inception

of the fault, the currents in each parallel path are balanced, but as the fault occurs at

0.04 sec, the currents become unbalanced in both phase and amplitude. From the result, it

Figure 4. Turns function of sub-windings of a1 winding for a fault in slot number 111 at 36%

from neutral.

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108 A. Sinha et al.

Figure 5. Self-inductance of paths a11, a12, and a2 of phase a and mutual inductances between

them for a fault in slot number 111 at 36% from neutral.

is clear that the mutual inductances between the faulted path and all other paths change.

Since parallel paths have their individual proximity to the various faulted paths, they are

affected in a different way.

4.2. Inter-turn Fault

An inter-turn fault has been simulated for path a1 in slot number 40 as per the winding

distribution shown in Figure 2. Under this condition, winding a1 is also divided into two

sub-windings, a11 and a12. The unfaulted portion is taken as a11, and the faulty portion

of the winding is a12. The turns function of a11 and a12 is shown in Figure 7 for a fault

in path a1 at 28% and 68% from the neutral, referring to their placements in the stator.

The current in all parallel paths has been simulated and is shown in Figure 8.

Figure 6. Plot of currents in all parallel paths for fault in a1 winding in slot number 111 at 36%

from neutral.

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Simulation of Faults in Synchronous Generators 109

Figure 7. Turns function for inter-turn fault in a1 winding in slot number 40.

4.3. Turn-to-turn Fault

If a fault between two parallel paths is simulated, then the machine can be viewed as

11 magnetically coupled circuits, as per Eq. (14). Therefore, all of the matrices have to

be augmented by two to take both the faulted paths into account. The Re , Le , Rg, Lg ,

Ra, and the voltage matrix elements corresponding to the faulted windings have to be

re-evaluated as explained in the case of turn-to-ground fault.

A turn-turn-ground fault between paths a1 and a2 has been simulated near slot

number 185, i.e., at 81% of a1 and 61% of a2 from neutral. The turns function of the

healthy and faulty portions of the two affected windings and the simulated currents are

as shown in Figures 9 and 10, respectively.

Similarly, a fault between two paths of different phases has been simulated, i.e., in

slot number 97, i.e., at 67% of a1 and 70% of b1 from neutral. In Figure 11, the turns

function of a1 and b1 for the healthy and faulty portions is shown. The simulated results

of all the parallel path currents are shown in Figure 12.

Table 1 gives, in detail, the values of the DC component, fundamental frequency

component, and total harmonic distortion in all parallel paths of all the phases of the stator

winding at different values of the fault inception angle under different types of internal

fault. These values have been obtained using fast Fourier transform (FFT) analysis, taking

200 samples per cycle for a period of ten cycles from the instant of inception of a fault

for the faults discussed above.

Figure 8. Plot of currents in all parallel paths for inter-turn fault in a1 winding in slot number 40.

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110 A. Sinha et al.

Figure 9. Turns function of sub-windings a1 and a2 for turn-to-turn fault near slot number 185 at

81% of a1 and 61% of a2 from neutral.

Figure 10. Plot of currents in all parallel paths for turn-to-turn fault between sub-windings of a1

and a2 winding near slot number 185 at 81% of a1 and 61% of a2 from neutral.

Figure 11. Turns function of sub-windings of a1 and b2 for turn-to-turn fault in slot number 97.

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Simulation of Faults in Synchronous Generators 111

Figure 12. Plot of currents of all parallel paths for turn-to-turn fault between a1 and b2 in slot

number 97.

Therefore, the machine model has the capability of simulating a variety of internal

faults, i.e., the turn-to-ground fault in any one parallel path, the inter-turn fault or turn-

to-turn fault between parallel paths of the same phase or between two phases, etc. The

results show that the transients occurring at the inception of the fault is significant and

can be useful in developing the design of the suitable intelligent protection scheme for

salient-pole synchronous generators.

5. Conclusion

In this article, a simulation model of internal faults in large salient-pole synchronous

generators with wave windings has been presented. The currents of all the parallel

paths can be investigated as the machine winding voltage equations are established

according to the machine winding connections. The model is based on a modified

winding function approach, which takes into account the effect of all space harmonics,

as the faulted winding inductance is integrated automatically. In order to avoid the

difficulty of calculating the machine inductances by using the geometrical parameters,

the same have been calculated by the use of machine electrical parameters. The simu-

lation results illustrate the unbalanced currents flowing in the parallel windings under

internal fault conditions. The values of the DC component, the fundamental frequency

component, and total harmonic distortion have been obtained at different fault incep-

tion angles and under different fault conditions using FFT analysis. Accuracy and the

simplicity make this model more reliable and convenient for the study of internal faults

in large hydrogenerators with wave windings. Since the simulation method tabulates

an inductance matrix based upon the winding distribution of a particular machine, this

is applicable to a specific machine, and hence, the results cannot be generalized. For

internal faults very near to the terminal and the neutral, the inductance matrix becomes

almost singular where inversion is not possible. Hence, simulation under these cases is

not possible.

Using the simulated fault data under different internal fault conditions, a suitable

intelligent protection scheme for salient-pole synchronous generators can be developed.

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112

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Simulation of Faults in Synchronous Generators 113

Using simulated numerical fault data, various architectures of artificial neural networks

(ANNs) will be trained and tested to identify the most suitable architecture of the ANN

that will give the correct information regarding the occurrence of the internal fault in the

hydrogenerator. For on-line application, the relaying of data in numerical form will be

acquired through the data acquisition system, and the same will be processed using the

trained ANN to give the signal regarding the occurrence of fault. After receiving the signal

of the occurrence of the fault, the proper tripping command will be issued to the circuit

breaker to isolate the faulty hydrogenerator.

References

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machine,” IEEE Trans. Power Apparatus Syst., Vol. 87, No. 8, pp. 1675–1679, August 1968.

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Appendix

Hydrogenerator electrical parameters used in the simulation of the model

Rated power of generator: 76 MVA

Rated voltage: 11 KV

Rated frequency: 50 Hz

Number of poles: 18

Armature resistance: 0.0043 p.u.

Armature leakage reactance: 0.15 p.u.

Direct axis synchronous reactance: 1.03 p.u.

Direct axis transient reactance: 0.26 p.u.

Direct axis sub-transient reactance: 0.17 p.u.

Quadrature axis synchronous reactance: 0.68 p.u.

Quadrature axis transient reactance: 0.66 p.u.

Quadrature axis sub-transient reactance: 0.2 p.u.

Direct axis transient open circuit (OC) time constant: 6.45 sec

Direct axis sub-transient OC time constant: 2.4 sec

Quadrature axis transient OC time constant: 0.95 sec

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