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Dynamic Braking Resistor for Control of Sub-synchronousResonant Modes

A. H. M.A.Rahim,Senior Member,IEEE

H.M.Al-Maghraby,Member,IEEE

Department of Electrical Engineering,

K,F, University of Petroleum and Minerals

Dhahrrm,Saudi Arabia.

Abstract: A dynamic braking resistor control strategy has been

proposed to damp the slowly growing sub-synchronous resonant

frequency oscillations. It employs generator speed variation, rotor

angle and power variation signals to switch in braking resistors at

the generator terminal. The proposed control has been tested on the

IEEE second benchmark model for sub-synchronous resonance

studies. The dynamically switched braking resistors have been

found to control the unstable modes very effectively. The control

algorithm is simple and its realization will require very little

hardware.

Keywords: Damping Control, SSR, Series Capacitor Compensation,

Dynamic Brake.

I. INTRODUCTION

Investigations

phenomena

be seen from the large number

bibliographycompiled by the IEEE Working Group on SSR

[1-3].Depending on theextent

capacitorcompensatedlines may have

oscillations, which are weakly or even negatively damped. A

system with such natural modes of oscillation

more actively to inputs that contain frequencies near to these

naturalmodes.Althougha generator

damped so as to prevent steady state oscillation, the damping

factorsmaybe smallenough

torques can be created by a particular disturbance.

proposedintheliterature

oscillationarisingatsmaller

control, static var compensators,

bypass filter and shunt reactors [4-8].

The use of resistive brake to damp the oscillations seen in

the mechanicalmass-spring system of a turbine

reported in [9,10]. The resistor switching was primarily done

depending on machine speed changes. Since the nature of the

low frequencyoscillations

instabilitysituation,a search

appropriate stabilizing signals should be carried out. Also the

power ratingsofthebrakes

oscillationswould bedifferent

situations.

in sub-synchronousresonance(SSR)

continue to be a subject of intense interest as can

of publicationsincluded in

of compensation,

natural

series

modesof

will respond

shaft is sufficiently

such thatdamagingshaft

Measures

type

excitation

for countering

frequencies

HVDC, static phase shifter,

this

are

of

shaft was

isdifferent

for other possible

fromtransient

or more

forsuchslowly

transient

growing

stabilityfrom

This paper proposes a dynamic braking resistor switching

strategy for control of unstable

derived as an optimal combination

acceleratingpower of the generator.

tested on the IEEE second benchmark model and found to be

very effective in controlling the SSR oscillations.

requirement of the brake is significantly

SSR modes. The control is

of speed deviation

The strategy has been

and

The power

small.

II. SECOND BENCHMARK MODEL WITH DYNAMIC

BRAKE

The IEEE second benchmark

shownin Fig. 1 is considered

consists of a synchronous

over two parallel transmission

model for SSR studies [5,11]

for this study.

generatorfeeding an infinite bus

lines, one of which is

The system

I

1

xl

Bus

If

c

BRAKE

Fig.1.Configuration of the IEEE second bench -mark model with

SMES.

capacitor

generating

masses shown in Fig.2. The high-pressure

low-pressureturbine

exciter (EX) are all coupled on the same shaft.

compensated.

unitis a mass-spring

Themechanical

system

partofthe

fourcontaining

turbine (HP), the

(GEN),(LP), the generatorand the

HP

LP

GENEX

Fig.2. The generator turbine system,

The dynamic resistor brake is connected

the generator terminals. The IEEE Type 1 exciter is included

to represent the excitationsystem of the second benchmark

model.

Thedynamic modelof

comprisingof the electromechanical

to the system at

thesystem

swing equation

givenin Fig. 1

of the

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generator, the voltage-current-flux

circuits, theexcitation

containingthe series capacitor,

on the generator shaft can be written in the form

relationships

the

and the mass-spring

of its various

transmission system,line

system

x = f[x,u]

where, the state vector X is

X=[xl

X2 X3 X4]T

xl=[o)H, OH,QL,eL, ~>~>‘BY eE]

X2=[id, if,i~,iq,i~, is]

X3=[ ecd, ecq, icd >icq 1

X4=[ VA>Efd , Vs 1

Control u is the brake output power P~

1

REAL PART OF EIGENVALUES(l/SEC.)

0.2

/\

li

I

I

1

I

‘\

0.1

‘. mode 1

\

(1)

(2)

1

:-

0

2040 60

80 100

$1X,

(%)

Fig. 3 The real part of the eigenvalues of the sub-synchronous

modes for various degrees of compensation.

For small perturbations

(1) can be linearized

point and expressed as,

of the states, the system of equation

aroundthe steady state equilibrium

AX= AAX+BU

(3)

where, AX is the change of the state vector. The matrix A in

the above equation depends on the degree of series capacitor

compensation.The compensation

normally expressed in percent.

eigenvalues of the sub-synchronous

of compensationfactors are plotted in Fig. 3. The eigenvalue

correspondingto theelectromechanical

mode O,while the other sub-synchronous

factor, given as Xc/XL, is

The realparts

modes for various values

of

the

swing

modes in ascending

is termed

order are termed modes 1, 2, and 3. Fig. 3 shows that at lower

values of compensationthe electromechanical

O)is unstable. This is expected for a weakly connected power

system. With the increase in compensation,

the electromechanical mode starts to decrease, while that of

mode 1 starts to increase. At 56°A capacitor

mode 1 has the largestreal part.

imaginary parts of the modes are not significant.

mode (mode

the real part of

compensation,

The variationsof the

III. THE PROPOSED BRAKE CONTROLSTRATEGY

Braking

system under transient

energy of the system following

severe disturbances.

systemsthatare prone

improved with dynamic braking resistor. The brakes can be

switched in to the networkwhen the generator

triggered by inputs having supplement of the SSR mode. The

following analysispresentsa switching

brakingresistorsthat will restore

operating condition in minimum possible time.

Considertheelectromechanical

generatorincluded in the general dynamic

the single machine system. This is rewritten as

resistorsare known

conditions

to help stabilize

by absorbing

three-phase

characteristics

to SSR conditions

a power

the excess

faults or other

of power

canalso be

The damping

accelerates,

algorithm

system

for the

to normalthe

swingequation

equation

of the

(1) for

M pzh = P. – P. –P~

(4)

where,

mechanical

generator, respectively.

p is the derivative

power input and electrical power output of the

The equation can be rewritten as

operatordldt. Pm, P, are the

p28=bAP+bu(5)

Here, b is the reciprocal of inertia constant. Contro u which is

the powerabsorbedby the brake,

maximum, say 1; hence the constraint on the control variable

can be written as

can vary from O to a

O<u<l

(6)

Since one of the main objectives

system is to bring the states back to their equilibrium

quickly, this can be formulated

minimizing the cost index

in stabilizingthe power

values

problemas an optimization

J = j:: dt

(7)

A quasi-optimum

above can be written as

controlstrategyfor the problemstated

{

1

(braking resistor on) if E >0

(braking resistor ofl if Z <0

u=

0

(8)

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where,

z=z$-

paz

2[b.AP - b.sgn{p~}]

Normally,

operating point; r~ther it may be the s~able manifold of ‘Oto

7c/2.Equation (9) in that case will be modified accordingly.

The relations (8) and (9) show that if the rotor angle has to

be restored to the original value following a disturbance,

brake switching depends on the weighted value 6 and pti, the

weightdepending onthepower

generator. However, if there is no constraint on & the braking

strategy dependson the speed variation

alone.

the target valuefor 6 may

(9)

not be the original

the

unbalance AP of the

of the generator

IV. SIMULATIONRESULTS

The IEEE second benchmark

one generating unit and two parallel transmission

dynamic resistor brake unit connected to the generator bus, as

given in Fig. 1, was simulated.

was solved withoutand with the proposed

scheme.56% series capacitor

transmission lines which gives the largest real part of SSR

mode 1 was considered.A torque pulse of 20°/0 for duration

of 3 cycles was applied to the generator shaft. Figure 4 shows

the response in the absence of any control. The response is

slowlygrowing.The torque

sections of the turbine as well the generator is oscillatory and

growing. Amongstall the three torsional

pressure and low-pressuresection,

generator, and the generator and exciter, the LP-GEN torque

is the largest. The top plot in Fig. 4 shows the variation of

this torsional torque. The variations of angular frequency and

the terminal voltage are also shown in the same figure. Fig. 5

shows the variationof the respective

proposed brake control is applied.

brake power applied for the response shown in Fig. 5 is 10%.

The variation of P~ as determined by the optimum strategy is

shown in Fig.6. Examination of Figs. 5 and 6 indicate that the

small frequency oscillations have been completely

in about 1 second.

The optimum strategy(8) gives the output power

bang-bangfunction. Realization

control is, generally, difficult because of instrumentation

lags. In this simulationa control u=JSZ was used instead. K

was selected by trial and error so that the control reaches the

ceiling value when the oscillations

a dead-zonewas providedto minimize

brake power whenthe oscillations

down.

The effectiveness ofthe

examined with various amounts of maximum brake power. It

was observed that the unstable

model for SSR studies with

lines and a

The system of equation (1)

braking

compensation

control

of one of the

on the shaft of the various

torques, the high

the low pressure and

quantities when the

The maximum amount of

captured

as a

of this type of discrete

time

are relatively large. Also,

unnecessary

havesufficiently

use of

died

brake control strategywas

modes could be controlled

with a maximum

disturbance

little longer while to stabilize as can be seen in Fig. 7.

brakepower

Expectedly,

of as low as 20/0 for this

the oscillationscondition. take a

LP-GEN Torque Dev. (p.u.)

0.8,

I

0.6

0.4

0.2

0

-0.2

-0.4

-0.6

-0.81

I

2

I

3

I

I

o

1

4

Time(sec.)

Ang. Freq. Dev (%)

0“4~

-o.~

o

1

2

3

4

Time(sec.)

_ _Term. Voltage Dev. (%)

-1I

I

6

I

7

I

8

I

9

5

1

Time(sec.)

Fig. 4 Variation of LP-GEN torsional torque, generator angular speed and

terminal voltage variation following a 20’Ko torque pulse on the generator

shaft for 3 cycles without any stabilizing control.

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0.2

0.1

0

-0.1

-o.~

o

1

2

34

Time(sec.)

Ang. Freq. Dev (%)

0.1

0

1

2

34

Time(sec.)

Term. Voltage

O.f

Dev. (%)

o.

0.

0.’ -

0r

-o.’

-o.: !

/

-0.3

o

I1

2

I

3

I

1

4

Time(sec.)

Fig. 5 Variation of LP-GEN torque, generator angular speed and terminal

voltage corresponding to Fig. 4 with the proposed dynamic braking control.

The SSR behavior

severe disturbance

generator

voltage variations

power of 50% is shown in Figs. 8 and 9. Fig. 10 shows the

variation of the brake power output as a function of time. Fig.

11 shows that with as little as 5’%0 maximum brake powers the

SSRoscillationsfor this severe

controlled. The number of switching in that case is more than

the condition shown in Fig. 10.

I

0.02 -

of the system was examined

of a symmetrical

bus for 3 cycles. The angular speed and terminal

of the generator

with a

three-phase fault on the

with a maximum brake

fault conditionscan be

The Brake Control (p.u.)

0.1

0.1

0.0

0.0

0.04-

I

o

I

o

123

4

Time(sec.)

Fig. 6 The variation of brake power P~corresponding to Fig.5.

Ang. Freq. Dev (%)

o.

0

1

23

4

Time(sec.)

Fig. 7 Generator angular speed variation for a 20% torque pulse disturbance

with a maximum brake power of only 0.02 pu.

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. .Ang. Freq. Dev (%)

0.I

o

o.

-0.2-

-o.~

o

1

234

Time(sec.)

Fig. 8 Generator angular speed variation following a three-phase fault for 3

cycles at the generator terminal with a maximum brake power of 0.5pu.

1 -

0.5-

0

-o.F

-1.51

o

-1

I

1

I

I

3

I

24

Time(sec.)

Fig. 9 Terminal voltage variation following a three-phase fault for 3 cycles

with a maximum brake power of 0.5pu.

V. CONCLUSIONS

A method

oscillations through the control of dynamic braking resistors

connectedto the generator bus is presented.

strategy is formulatedsuch that the generator speed and rotor

angle are broughtback to the desired steady conditions

shortest possible time.

Transient response

recorded

synchronous frequencyoscillations

with judicious use of brake power. It has been observed that

evenfora severe three-phase

oscillationscan be controlled with a maximum brake output

power of as low as 0.05 pu. This is quite different from the

transient stability conditions where brake power requirement

of controlling the sub-synchronous resonant

The control

in

show

are eliminated

thatthe sub-

quickly

fault condition,the SSR

of 1.0 pu or more is not unusual.

installed to cater for both transient

type oscillations,

minimize brake control power.

If dynamic

stability as well as SSR

be designed

brakes are

circuits should properlyto

The Brake Control (p.u.)

0.6,

0.5 T

0.4

0.3 -

0.2 -

0.1 -

00

1

2

3

4

Time(sec.)

Fig. 10 The variation of brake power P~corresponding to Fig.7.

Ang. Freq. Dev (%)

1,,

I

I

., .’~

o

1

2

3

4

Time(sec.)

Fig. 11 Generator angular speed variation following a three-phase fault for 3

cycles at the generator terminal with a maximum brake power of only

o.05pu.

In actual

proposed requires primarily the measurement

in generatorspeed, rotor angle and its power

proposed strategy is simple

hardware to implement.

implementation,the brakingcontrol

of the changes

output. The

requirevery little

strategy

and would

VI. ACKNOWLWEDGEMENTS

The authors wish to acknowledge

King Fahd University of Petroleum and Minerals towards this

research.

the facilities provided by

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