# Simulation of Wound Rotor Synchronous Machine under voltage sags

**ABSTRACT** This paper presents a study of the Wound Rotor Synchronous Machine (SM) stability under voltage sags (dips). Machine behavior has been analyzed by considering different magnitudes (or depths) and duration of the sag. Voltage sags cause speed variations on SM (with possible lost of synchronism), current and torque peaks and hence may cause tripping and equipment damage, which leads to financial losses. Three machines of different nominal power have been modeled and simulated with MATLAB in order to study its behavior under voltage sags. These SM's have been analyzed operating as generator, as motor, and in both cases, working under-excited and overexcited.

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**ABSTRACT:**A 1000 kW/19 200 rpm wound-rotor synchronous machine has been designed as a draft based on 2-D transient finite-element method (FEM). The magnetic fields distribution, the iron losses, the electromagnetic torque and the terminal characteristics of the prototype machine are investigated in this paper by using 2-D finite element analysis. The d-axis inductance Ld and the q-axis inductance Lq of the sample machine are predicted in analytical models. All studies indicate that the wound-rotor synchronous machine is potential to apply in high speed generator or motor application.IEEE Transactions on Magnetics 11/2012; 48(11):4650-4653. · 1.21 Impact Factor

Page 1

Simulation of Wound Rotor Synchronous Machine

under Voltage Sags

D. Aguilar1, G. Vazquez, Student Member, IEEE, A. Rolan, Student Member, IEEE,

J. Rocabert, Student Member, IEEE, F. Córcoles, Member, IEEE, and P. Rodríguez, Member, IEEE.

Electrical Department of Engineering

Technical University of Catalonia

Colom 1- 08222-Terrassa, Spain

1Contact Author: aguilar.seer@gmail.com

Abstract- This paper presents a study of the Wound Rotor

Synchronous Machine (SM) stability under voltage sags (dips).

Machine behavior has been analyzed by considering different

magnitudes (or depths) and duration of the sag. Voltage sags

cause speed variations on SM (with possible lost of synchronism),

current and torque peaks and hence may cause tripping and

equipment damage, which leads to financial losses. Three

machines of different nominal power have been modeled and

simulated with MATLAB in order to study its behavior under

voltage sags. These SM’s have been analyzed operating as

generator, as motor, and in both cases, working under-excited and

overexcited.

I.

INTRODUCTION

Nowadays, electrical power systems are constituted by many

generation sources. Nevertheless, the vast majority of the

electrical power generation systems consist of synchronous

generators coupled to the electrical grid through a transformer.

This is due to its capability of generating electricity in large

scale power plants [1]. Large (in the MVA range) and small (in

the kVA range) synchronous generators are part of electrical

system and hence its behaviour is a very important issue.

Moreover, during the last years, the power quality has

gained a great importance in both industry and research, due to

technical and situational factors, such as the equipment

sensibility (which are also the cause of perturbations),

distributed generation, new grid code requirements for specific

power supply, by mentioned just a few.

The grid may present many types of disturbances, being the

voltage sags the most common ones. The principal causes of

voltage sags are faults, large transformer energizing and the

starting of large motors. The interest in studying this kind of

disturbances is mainly due to the problems that they cause in

electrical equipments [2]. Voltage sags may cause large torque

peaks on SM, and it can damage the shaft or the equipment

connected to the shaft [3]. In addition, voltage sags may cause

tripping, which leads to financial losses.

On the other hand, the simulation of the SM’s behaviour

under voltage sags permits defining different criteria for

protecting and preventing potential damages in such machines,

as well as the effects on the power supply interruptions [4].

Hence, a valid model for SM is essential in order to obtain a

reliable analysis of stability and dynamical performance.

In [5] it has been presented the transient behaviour of SM

under sags according to their duration, depth, and initial point-

on-wave. The aim of this paper is to simulate the wound rotor

SM and study its stability under a wide variety of symmetrical

and unsymmetrical voltage sags, considering its performance

as generator, as motor and in both cases, operating

underexcited and overexcited. Section II presents the voltage

sags classification and characterization. Section III shows the

wound rotor SM model. Finally, based on the simulations

using MATLAB, voltage sags effects on SM stability are

presented in section IV.

Type A

Type BType C

Type D

Type E

Type FType G

Fig. 1. Voltage sag types: Symmetrical (type A) and unsymmetrical (type B -

G). All sags have a depth of 50%. [2]

TABLE I

POSITIVE, NEGATIVE AND ZERO SEQUENCE EXPRESSIONS FOR

VOLTAGE SAGS [2]

PositiveTypeNegative Zero

A

VhVA

⋅=

+

0

=

−

A

V

0

0=

A

V

B

V

h

VB

⋅

+

3

=

+

2

V

h

VB

⋅

−

3

−=

−

1

V

h

VB

⋅

−

3

−=

1

0

C

V

h

VC

⋅

+

2

=

+

1

V

h

VC

⋅

−

2

=

−

1

0

0=

C

V

D

V

h

VD

⋅

+

2

=

+

1

V

h

VD

⋅

−

2

−=

−

1

0

0=

D

V

E

V

h

VE

3

21

⋅+

=

+

V

h

VE

⋅

−

3

=

−

1

V

h

VE

⋅

−

3

=

1

0

F

V

h

VF

3

21

⋅+

=

+

V

h

VF

⋅

−

3

−=

−

1

0

0=

F

V

G

V

h

VG

3

21

⋅+

=

+

V

h

VG

⋅

−

3

=

−

1

0

0=

G

V

??????????????????????????????????????????

Page 2

II. VOLTAGE SAGS (DIPS)

A voltage sag is a short-duration (from half cycle to 1

minute) drop between 10% and 90% in the magnitude of the

rms voltage. When the voltage sag is caused by short-circuit,

the tripping time must limit the fault between 3 and 30 cycles.

Nevertheless, voltage sags duration due to large motors can be

of some seconds [2] [6].

Voltage sags can be either symmetrical or unsymmetrical,

depending on the causes. If the individual phase voltages are

equal and the phase relationship is 120o, the sag is

symmetrical. Otherwise, the sag is unsymmetrical.

According to [2], voltage sags can be grouped into seven

types denoted as A, B, C, D, E, F and G (of which C and D

cover the majority of sag). This classification is based on the

fault type and connection: if the fault is three-phase the sag

type is A (both star and delta connection). When the fault is

single-phase the sag types are B (star connection) and C (delta

connection). The sag types are also C (star connection) and D

(delta connection) if the fault is phase-to-phase. Finally, in

two-phase-to-ground faults the sag types are E (star

connection) and F (delta connection). Sag types E and G only

differ in the zero-sequence voltage.

Table I and II show their expressions (where h is the sag

depth), both in positive-negative-zero variables and in per-

phase variables, respectively. The first table make evident the

difference between sag type E and G mentioned above and also

that symmetrical ones are the most severe voltage sag types.

Fig. 1 shows their phasor diagrams.

In order to simulate the behaviour of SM, in this paper it has

been considered that the sag shape is rectangular, no phase

jump occurs and the sag type does not change when the fault is

cleared. The depth (or magnitude) and duration are the main

characteristics of the voltage sags, but do not completely

characterize the sag. Hence, the voltage sags will be defined by

their depth h (0.1 ≤ h ≤ 0.9), their duration Δt, and their initial

point-on-wave ψi [7].

III. WOUND ROTOR SYNCHRONOUS MACHINE MODEL

In this paper a SM eight-order model is used in order to

evaluate its behaviour under voltage sags (damper winding

exist in both d and q axis). The dynamic equation in the dq

reference frame of SM is given by [8]. The dynamic expression

in form of state equations can be written as:

( )

pppp

i

dt

( )( ){}

tt

Jdt

d

ω

{}

pp

d

iMRvMi

ω−−=

−1

(1)

d

res

m

Γ−Γ=1

ω

(2)

m

dt

θ=

(3)

Where ip is the vector which contains the Park’s stator and

rotor currents, vp is the vector that includes the transformed

stator and rotor voltages, R is the winding resistance matrix

and Mp is the transformed inductance matrix. The mechanical

equations are given in (2) and (3).

TABLE II

SAG TYPES IN EQUATION FORM (OBTAINED FROM [2])

Type A Type B

hV j hV

c

V

hVj hV

b

V

hV

a

V

2

3

2

1

2

3

2

1

+−=

−−=

=

VjV

c

V

VjV

b

V

hV

a

V

2

3

2

1

2

3

2

1

+−=

−−=

=

Type C Type D

hVjV

c

V

hVjV

b

V

hV

a

V

2

3

2

1

2

3

2

1

+−=

−−=

=

VjhV

c

V

Vj hV

b

V

hV

a

V

2

3

2

1

2

3

2

1

+−=

−−=

=

Type E Type F

hVjhV

c

V

hVjhV

b

V

V

a

V

2

3

2

Type G

)

h

+

1

2

3

2

1

+−=

−−=

=

()

()Vhj hV

c

V

VhjhV

b

V

hV

a

V

++−=

+−−=

=

2

12

1

2

1

2

12

1

2

1

(

()

()

hVjVh

c

V

hVjVh

b

V

V

a

V

2

3

2

6

1

2

3

2

6

1

2

3

1

++−=

−+−=

=

Where:

h = 0.1 …0.9 sag depth

V = UL /

3 phase-to-ground

ac voltage

Saturation is not considered. An equivalent circuit is shown in

Fig. 2. The instantaneous electromagnetic torque Г(t) and the

load angle δ(t) (measured in steady-state from phasor Es to

phasor Us) are expressed as:

(

)(

+=Γ

dFd sq

MiMit

(

qd

LL −+

...

(

)()(

θϕωδ−+=

ttt

ss

When the SM has round-rotor (Ld = Lq), the instantaneous

electromagnetic torque is reduced to:

(

dFdsq

MiMit

+=Γ )(

Moreover, the matrix of magnetic flux ϕp is expressed as:

iMφ =

)

i

() ...

+−

Qq

sdD

iMii

)

sqsd

i

(4)

) ()

2

π

+

(5)

)

()

QdsdD

iMii

−

(6)

ppp

(7)

????

Page 3

Three SM have been simulated: a high-speed steam turbine

generator (835MVA), a low-speed hydro turbine generator

(325MVA), both obtained from [8], and a small generator.

Rated parameters of these machines are included in Appendix.

ML −

FD

M

()

ω

mQq sqq

iMiL

+−

dv

sr

dFs

dFF

ML −

dF

M

−

dFD

ML −

dF

M

Fr

Dr

D

v

F v

=−

m sqωφ

()

ω

mD dFF dFsdd

iMiMiL

++

q v

sr

dFs

ML −

dFQ

ML −

dF

M

Qr

Q

v

=

msdωφ

Fig. 2. Synchronous machine equivalent circuits.

IV. VOLTAGE SAGS EFFECTS ON WOUND ROTOR

SYNCHRONOUS MACHINE

In [5] it has been demonstrated the effects on SM using

recursive simulation for each sag characteristic: depth, duration

and initial point-on-wave. The depth influence on the torque

peaks is linear and the duration influence on torque peaks is

periodical, for all sag types. In addition, different initial point-

on-wave can produce peaks with different amplitude for

unsymmetrical sags.

The most relevant voltage sag effects on the SM behaviour

are speed variations, with possible loss of synchronism, current

and torque peaks, as well as tripping protection. The machine

behaviour is different when symmetrical and unsymmetrical

sags are produced in the grid.

Fig. 3 shows the transient effects caused by symmetrical sag

on the 5kVA SM, whose duration and depth are 100

milliseconds and 20 per cent of rms voltage, respectively.

When symmetrical sag is applied on this machine, the most

severe peaks occur at the beginning and at the end of the sag,

because symmetrical sags do not have negative sequence

voltage, thus the waveform is not oscillatory.

Fig. 4 shows the SM behaviour against an unsymmetrical

voltage sag type C on the 5kVA SM with the same

characteristics. In this case, the instantaneous active and

reactive power has a higher distortion, but their peak values are

smaller when the sag starts as well as when the sag ends. It is

due to the fact that the positive sequence voltage applied to the

machine is higher for the unsymmetrical sag than for the

symmetrical sag [7]. As can be deduced from Table I, the

minimum positive sequence voltages available (when h = 0) in

00.050.10.150.20.25

-2

0

2

vsa vsb vsc

00.050.10.150.20.25

-20

0

20

isa isb isc

00.050.10.150.20.25

-20

-10

0

10

p q

00.05 0.1

Time [sec]

0.150.2 0.25

-20

-10

0

10

Tm

Fig. 3. 5kVA Synchronous Machine behaviour for a symmetrical voltage sag:

type A, h = 0.2, Δt = 100ms, ψi = 30º

0 0.050.10.15 0.20.25

-2

0

2

vsa vsb vsc

0 0.050.1 0.150.2 0.25

-20

0

20

isa isb isc

0 0.050.1 0.150.2 0.25

-10

-5

0

5

p q

0 0.050.1

Time [sec]

0.150.2 0.25

-20

-10

0

10

Tm

Fig. 4. 5kVA Synchronous Machine behaviour for a unsymmetrical voltage

sag: type C, h = 0.2, Δt = 100ms, ψi = 30º

????

Page 4

the all unsymmetrical sags are 0.66V (type B), 0.5V (types C

and D) and 0.33V (types E, F and G). Moreover, due to fact

that the machine losses are small and the machine speed is

nearly constant, torque and active power have the same

waveform. The reactive power (shown in dotted line in Fig. 4)

has an oscillatory behaviour for all the unsymmetrical sags

because of their negative sequence voltage.

According to the simulation that has been carried out on

three machines, an increase in the sag duration leads to an

increase in the current peaks when the sag ends. On the whole,

the transient wave shape caused by voltage sag depends on

different factors such as its depth, duration, initial point-on-

wave and machine parameters.

On the other hand, when voltage sags occurs on the SM

operating as generator, for example due to a three-phase short

circuit at some point common coupling (PCC) near to the

machine, the rotor speed increases steadily. This fact occurs

because the terminal voltage (which decreases due to sag) is

proportional to the electrical torque, thus it also decreases; in

consequence, the rotor speed increases in order to compensate

the electrical torque reduction. Since a determined point of

view, the rotor speed could increase unlimitedly and cause an

over-speed on the machine, which could lead to collapse the

system. At this point, of course, the generator should be

disconnected by the protection system and returning to a new

steady state.

Fig. 5 confirms the above mentioned. During the voltage

sags, the SM accelerates if it is working as a generator (or

reduces its speed if it is operating as a motor), so that it may

become unstable due to loss of synchronism. In this case, rotor

speed behaviour of both hydro and steam machine have been

simulated when a symmetrical sag occurs at terminal voltage.

Voltage sag, which depth is 20% of rms voltage, beginning at

50 milliseconds and it finish at 250 milliseconds.

0 0.1 0.2 0.30.4 0.50.6 0.70.8 0.91

0.99

0.995

1

1.005

1.01

1.015

time [sec]

ωmec [rad/sec]

ωhydro - Gen

ωsteam - Gen

ωhydro - Motor

ωsteam - Motor

Fig. 5. Rotor speed (ωm) transient shape of both hydro turbine and steam

turbine synchronous machines. (Sag type A, h = 20%, Δt = 200 ms, ψi = 15º).

In both case, the machines are operating as a generator as a motor.

Due to the fact that synchronous machine may lose

synchronism for the most severe voltage sags (large duration

and depth), one important issue is to determinate the transient

stability limit and the critical clearing time for each specific

case. Consequently, in order to analyze the stability machine is

necessary to consider various methods to permit to define these

stability limits of the machine. This is treated below.

depth [%]

duration [sec]

1.01

1.02

1.03

1.04

1.05

1.05

1.05

1.06

1

←← 1

2 →→

0.10.20.30.40.50.60.70.80.9

0

5

10

15

20

25

30

35

40

(a)

depth [%]

duration [sec]

1.01

1.01

1.02

1.03

1.04

1.05

1.05

1.06

1.07

1.07

← 1

2 →

0.10.2 0.30.40.50.60.70.80.9

0

5

10

15

20

25

30

35

40

(b)

Fig. 6. Symmetrical voltage sag’s durations and depth influence on the

rotor speed (ωm) of a hydro turbine (a) and a steam turbine (b) SM. (Sag

type A, h = 0% - 100%, Δt = 0 - 1 second, ψi = 15º)

Fig. 6(a) shows simulation results of several symmetrical

sags applied on a hydro turbine synchronous generator. This

figure is the result of recursive simulations with depth h = 0%

to 100% and duration Δt = 0 to 1 second. In this case, the

initial point-on-wave is kept at 15º for all simulations. Thus,

this figure indicates the rotor speed peaks in absolute values

and per unit (p.u.).

In addition, real and approximated stability machine limits

have been represented. The thick dotted line (identified with

number one) in Fig. 6(a) indicates the real stability machine

limits (when the synchronous generator is unable to return to

its steady state value), obtained by using the dynamic equations

of the SM. Regarding to the thin dotted line (identified with

number two) in the same figure, it indicates the approximated

limits of the machine’s stability, obtained by using the equal-

area criterion described in [8], not only to short circuits, but

also extended to voltage sags in order to determine the critical

clearing time.

This method considers the approximated transient torque-

angle curve, which along with the equal-area criterion is often

used to predict the large excursion dynamic behaviour of a

synchronous machine during a system fault (Fig. 7). Assume

that the input torque Tin is constant and the machine is

operating steadily, delivering power to the system with a rotor

angle δ0 (point O). When the voltage sag occurs at the

terminals, the power out (and the approximate torque) drops

lineally proportional to sag’s depth, and then the machine

????

Page 5

accelerates. The fault is cleared at δ1, and in this case the

torque immediately becomes the value of the approximate

transient torque (point D). In Fig. 7, the area OABCO is the

energy stored in the rotor during the acceleration. After the

clearing of the fault the rotor decelerates back to synchronous

speed at δ2. The energy given up by the rotor during this time is

represented by area CDEFC.

Therefore, in order to maintain synchronism in the machine

after voltage sag, the follow mathematic expression must be

accomplished.

( ) ( )

δδδ

δδ

∫∫

10

[]

d

( )

δ

( )

δ

[] δ

d

δδ

load aprox

Γ

sagload

Γ−≤Γ−Γ

21

(8)

20 406080 100120 140160 180

0

1

2

3

4

5

6

Taprox

Tss

Tsag

Tload

Fig. 7. Equal-area criterion extended to voltage sags: Torque vs. load angle

curve (Symmetrical sags on hydro turbine synchronous generator)

Fig. 6(b) show the rotor speed peaks on the steam turbine

synchronous generator, which has a similar behaviour with

respect to the above one, but the synchronism is lost for shorter

sag duration. The voltage sags characteristics in this case are

the same ones. In both machines, all simulations were made

considering symmetrical voltage sags because they are the

cause of worst peaks.

Another method for determining the loss of synchronism in

the machine is to take into account the load angle. The ability

to keep the synchronism may be defined as the case when the

load angle is below 180º for all the time during and after the

voltage sag [4].

This can be seen clearly in Fig. 8, where several symmetrical

sags have been simulated on both hydro turbine and steam

turbine SM. This figure shows the load angle peaks and has

been obtained through recursive simulations with depth h = 0%

to 100% and duration Δt = 0 to 1 second and the initial point-

on-wave is kept at 15º for all simulations. In this figure the

dotted line indicates the stability machine limits.

The SM behaviour when it is overexcited is different from

the behaviour when it is underexcited (considering either motor

or generator mode of operation).

100

100

100

200

200

200

300

400

500

300

400

500

500

500

600

600

600

700

800

900

700

700

800

900

depth [% ]

duration [sec]

(a)

0.30.40.50.60.70.80.91

0

5

10

15

20

25

30

35

depth [%]

duration [sec]

(b)

200

400

600

600

600

800

800

1e+003

1.2e+

0.30.40.50.60.70.80.9

0

5

10

15

20

25

30

35

40

Fig. 8. Symmetrical voltage sag’s durations and depth influence on the load

angle (δ) of a hydro turbine (a) and a steam turbine (b) SM. (Sag type A, h

= 0% - 100%, Δt = 0 - 1 second, ψi = 15º)

1.01

1.01

1.02

1.03

1.04

1.05

1.06

1.07

depth [%]

duration [sec]

(a) Overexcited

0.10.20.30.40.50.60.70.80.9

0

20

40

60

80

100

1.01

1.01

1.02

1.02

1.03

1.04

1.05

1.06

1.07

1.08

depth [%]

duration [sec]

(b) Underexcited

0.10.20.30.40.50.60.70.80.9

0

20

40

60

80

100

Fig. 9. Symmetrical voltage sag’s durations and depth influence on the rotor

speed (ωN) of a 5kVA SM (Sag type A, h = 0% - 100%, Δt = 0 - 1 second, ψi =

15º)

????