anÁlisis comparativo entre el pss y el statcom en el

ANLISIS COMPARATIVO ENTRE EL PSS Y EL ANLISIS COMPARATIVO ENTRE EL PSS Y EL ANLISIS COMPARATIVO ENTRE EL PSS Y EL ANLISIS COMPARATIVO ENTRE EL PSS Y EL STATCOM EN EL AMORTIGUAMIENTO DE STATCOM EN EL AMORTIGUAMIENTO DE STATCOM EN EL AMORTIGUAMIENTO DE STATCOM EN EL AMORTIGUAMIENTO DE

Universidad Tecnolgica de Pereira

STATCOM EN EL AMORTIGUAMIENTO DE STATCOM EN EL AMORTIGUAMIENTO DE STATCOM EN EL AMORTIGUAMIENTO DE STATCOM EN EL AMORTIGUAMIENTO DE OSCILACIONESOSCILACIONESOSCILACIONESOSCILACIONES

INTRODUCCIN

Los actuales sistemas de potencia son operados cerca delos lmites de estabilidad y estn continuamente sujetosa perturbaciones, provocando desbalances que causanoscilaciones de diferente magnitud en el ngulo del rotorde las mquinas sncronas.de las mquinas sncronas.

Tales oscilaciones deben ser amortiguadas por loselementos de control del sistema con el fin de evitarprdida de sincronismo de las mquinas. Estasoscilaciones pueden permanecer y crecer de manera quecausen la separacin del sistema si no se tiene unamortiguamiento adecuado [1].

Estos factores han sido muy importantes en el diseo yuso de dispositivos basados en la electrnica depotencia, cuyo objetivo principal es proporcionarrespaldo ante cambios en la estructura de la red yrespaldo ante cambios en la estructura de la red yperturbaciones severas.

ANTECEDENTES

Los sistemas de excitacin antiguos se controlabanmanualmente para mantener el voltaje en terminales y lasalida de potencia reactiva. Con la automatizacin, sereconoci en estos sistemas, el potencial para mejorar laestabilidad de pequea y gran seal a travs del uso de

reguladores continuos y de respuesta rpida [4].reguladores continuos y de respuesta rpida [4].

El papel de los sistemas de excitacin se expandi con lasuma de seales auxiliares de estabilizacin, paraamortiguar oscilaciones del sistema por medio del control

de voltaje de campo [4].

En los ltimos aos se ha venido estudiando lautilizacin de dispositivos estabilizadores FACTS(Flexible Alternating Current Transmission Systems);que han mostrado capacidad de amortiguamiento deque han mostrado capacidad de amortiguamiento deoscilaciones, aunque son utilizados principalmente

para la compensacin reactiva [3].

OBJETIVO GENERAL

Mediante el anlisis de la operacin de un sistemaconformado por una mquina sncrona conectada a unbarraje infinito a travs de una lnea de transmisin,barraje infinito a travs de una lnea de transmisin,comparar el efecto de el estabilizador del sistema depotencia (PSS) y de el compensador esttico sncrono(STATCOM) en el amortiguamiento de oscilaciones.

OBJETIVOS ESPECFICOS

Obtener el modelo no lineal del sistema conformado poruna mquina sncrona conectada a un barraje infinito atravs de una lnea de transmisin, con base en elmodelo Heffron-Phillips.

Modelar el estabilizador del sistema de potencia (PSS) yel STATCOM.

Realizar una simulacin lineal y no lineal del sistemabajo perturbaciones en la referencia de tensin paradiferentes puntos de operacin.

SISTEMA MQUINA SNCRONA BARRA INFINITA

La tendencia actual es trabajar con un sistema elctrico detransmisin y distribucin que interconecte las diferentescentrales generadoras y los puntos de consumo, de modoque una sola central y ms an, una sola mquinaque una sola central y ms an, una sola mquinarepresenta un pequeo porcentaje de la potencia total delsistema. Evidentemente esta mquina no estar capacitada

para alterar ni el voltaje ni la frecuencia del sistema elctrico.

MODELO DEL SISTEMA

El sistema puede modelarse mediante un conjunto deecuaciones diferenciales y algebraicas de la forma [2]

),,( uyxfx =

(1)

Siendo el vector de estados, el vector de variablesalgebraicas y representa las ecuaciones algebraicas delestator y de conexin a la red.

),,( uyxfx =

),(0 yxg=

x yg

(2)

))'('('

1' fddddq

do

q EIXXET

E +=

La mquina se modela por las siguientes ecuaciones [2]

(3)

sww =

)))()'('((2

sqddqqqm

s wwDIIXXIETH

ww +++=

)( trefAfdfdA VVKEET +=

(4)

(5)

(6)

Asumiendo , las ecuaciones del estator estn dadas por0SR

0= dqq VIX

0'' = ddqq IXVE

(7)

(8)0'' = ddqq IXVE

Figura 1. Sistema mquina sncrona barra - infinita

(8)

Asumiendo un ngulo de fase igual a cero en el barraje infinito

ee

j

qdj

qdjXR

VjVVjII

ee +

+=+

0)(

)(

)2

(

)2

(

Separando (9) en parte real e imaginaria

senVVIXIR =

(9)

(10)

22

qdt VVV +=

senVVIXIR dqede =

cos=+ VVIRIX qqede

Adems

(10)

(11)

(12)

Modelo linealizado

fd

dodo

q

do

q ETT

KE

TKE +=

'

1

''

'

1' 4

3

=

sw

(13)

(14) = sw

ms

q THH

Dw

H

KE

H

K +=

2

1

22'

2

12

))'(( 65 qrefAfdfdA EKKVKEET ++=

sww=

(15)

(16)

Donde

ref

A

q

s

s

dododoq

V

K

E

KKKKH

Dw

H

K

H

K

w

TT

K

TKE

+

=

0

0

0'

1

0222

000

'

10

''

1

'

12

4

3

(17)

A

A

fd

AA

A

A

Afd TETT

KK

T

KKHHH

E

1

0

222

56

Las constantes llamadas K1 K6 fueron desarrolladas por Heffron-Phillips [2], para el estudio de oscilaciones locales de baja frecuencia.

Figura 2. Diagrama de bloques del modelo linealizado del Sistema mquina sncrona barra - infinita

ESTABILIZADOR DEL SISTEMA DE POTENCIA (PSS)

Es el encargado de amortiguar los efectos de lasoscilaciones que se presentan cuando alguna perturbacinacta sobre el sistema elctrico.

Su funcin bsica es la de extender los lmites detransferencia de potencia aplicando una seal de controlsuplementario a travs del sistema de excitacin, paraque se produzca una componente del torque elctrico enfase con la velocidad, que se encargue de amortiguar las

oscilaciones del ngulo de par [1].

ESTRUCTURA GENERAL DEL PSS

Sensor

Filtro torsional

Red de adelanto de fase

Figura 3. Estructura del sistema PSS

Filtro washout

Lmitador

La seal estabilizadora derivada de la mquina (velocidad) esprocesada por el PSS con una funcin de transferencia G(s) y susalida es conectada a la entrada del excitador.

)1()1)(1(

)1)(1()(

42

31

W

WPSS

sT

sT

sTsT

sTsTKsG

+++++= (18)

)1()1)(1( 42 WsTsTsT +++

Figura 4. Seal auxiliar en el regulador

Tomando el PSS con solo una etapa de adelanto atraso yomitiendo la etapa washout, una ecuacin de estado extra debido alPSS es adicionada al sistema

(19)

++=

H

wD

H

KE

H

K

T

TK

T

Ky

Ty SqPSS

PSS

22'

2

1 12

2

1

22

COMPENSADOR ESTTICO SNCRONO (STATCOM)

El STATCOM (Static Synchronous Compensator) es unconvertidor VSI trifsico, el cual produce un conjunto de trestensiones de salida con frecuencia, relacin de fase y magnitudajustable; cuya conexin a un sistema se realiza a travs de unaajustable; cuya conexin a un sistema se realiza a travs de unareactancia, usualmente la de dispersin de un transformador deacople. Este dispositivo se instala en derivacin con un nodo detransmisin y se usa principalmente para regular el voltaje en lossistemas de transmisin.

ESTRUCTURA DEL STATCOM

2V 1VI1VSi > , se adelanta a ,entregndose potencia reactiva alnodo de conexin y el convertidorse comporta como un grancapacitor.

2V 1VI1VSi < , se atrasa de , seabsorbe potencia reactiva del nodoy el convertidor acta como unreactor.

Figura 5. Estructura del STATCOM

MODELO DEL STATCOM

_

(cos )o DC DCV cV jsen cV = + =

)cos( senIIC

c

C

I

dt

dVLoqLod

DCDC

DCDC +==

(20)

(21)

Figura 6. STATCOM conectado a un sistema mquina sncrona barra - infinita

MODELO NO LINEAL

))'('('

1' fdtLdddq

do

q EIXXET

E +=

sww =

(22)

(23)

)cos( senIIC

cV LoqLod

DC

DC +=

)))()'('((2

stLqtLddqqqms wwDIIXXIETH

ww +++=

)( trefAfdfdA VVKEET +=

(24)

(25)

(26)

Resolviendo el circuito se tiene que

cos

(1 )

LBB DC

SDTtLq

LB LBtL LB tL q

SDT SDT

XV sen cV

XI

X XX X X X

X X

+=

+ + + +

' (1 ) cosLB LBq B DCSDT SDT

X XE V cV sen

X XI

+ = (28)

(27)

(1 ) '

SDT SDTtLd

LB LBtL LB tL d

SDT SDT

X XI

X XX X X X

X X

=+ + + +

' ( ' )q d tL tLd DCLod

SDT

E X X I cV senI

X

+ =

cos ( )DC q tL tLqLoq

SDT

cV X X II

X

+= (30)

(29)

(28)

Reemplazando (27)-(30) en (22)-(26) y linealizando

sw

=

4 3

1' '

'q s s q Qc Qdc DC Q fd

do

E K K E K c K V K ET

= + + + + (31)

(32)

1 2

1'

2s s q Pc Pdc DC P sK K E K c K V K Dw

H

= + + + + +

( )5 61 'Afd fd ref s s q Vc Vdc DC VA A

KE E V K K E K c K V K

T T

= + + + + +

7 8 'DC s s q Dc Ddc DC DV K K E K c K V K

= + + + +

(33)

(34)

(35)

3 4

2 1

6 5

10 '' ' ' ' '

0 0 0 0

02 2 2 2

10

Q d cs s

qq

d o d o d o d o

s

s s s P d c

A s A s A V d cf df d

KK KEE T T T T

w

K K D w K

H H H H

K K K K K KE ET T T T

=

8 7 0 0

A A A A

D CD C

s s D d c

T T T TV VK K K

' '

0 0

2 2

Q c Q

d o d o

PP c

A VA V c

A A

D c D

K K

T T

cKK

H H

K KK K

T T

K K

+

(36)

SIMULACIONES Y RESULTADOS

La metodologa de la simulacin consiste en aplicar al sistemauna cambio en la referencia de tensin, para diferentespuntos de operacin.

Con el propsito obtener conclusiones acerca delCon el propsito obtener conclusiones acerca delamortiguamiento de oscilaciones en el sistema, se comparanlas grficas de la respuesta de voltaje en terminales delgenerador , del ngulo de par y de la velocidad del sistemano lineal, puesto que en este se reflejan mejor los resultadosobtenidos.

El software utilizado fue MATLAB 7.0, el cual dispone de dos

herramientas: Simulink (plataforma de simulacinmultidominio) y GUIDE (editor de interfaces de usuario GUI).

Los datos se digitan en la interfaz de usuario y por medio debotones se calculan las condiciones iniciales del sistema y sellaman los archivos del simulink con los respectivosdiagramas de bloques de los modelos desarrollados.

ESPECIFICACIONES DE RESPUESTA TRANSITORIA

Figura 7. Especificaciones de respuesta transitoria

SIMULACIN DEL MODELO NO LINEAL

Datos del sistema en p.u.[2]

377,1.0,39.0',5.2,1.2,0

,2.0,400,6.9',2.3,5.0,0

============

sddqs

AAdoee

wDXXXR

sTKsTsHXR

sddqs

05.0,83,1,1.0,15.0 ===== CDCSDTtL TkCXX

PUNTO DE OPERACIN DEL SISTEMA

pujS 2511.04828.0 =puVpuVt 005.1,151 ==

En el primer punto de operacin del sistema se realiza una

perturbacin en el voltaje de referencia a los 40 segundos.

1435.1,5435.0,0041.1

,6393.1,7923.0',3675.0,4014.0,6358.0,7718.0

0 ===

======

mreffdqqdqd

TV

EEIIVV

3708.7,3995.0,10,059.0,4152.1,108458.2 214 ====== nWPSS WTTTK

0 0 00.25, 12.41 , 1DCc V= = =

Se calculan las condiciones iniciales y los parmetros del PSS

Perturbacin en la referencia de tensin 1- 1.05 p.u.

1

1.01

1.02

1.03

1.04

1.05

1.06

1.07

1.08

Vol

taje

[p.

u.]

1

1.01

1.02

1.03

1.04

1.05

1.06

1.07

1.08

Vol

taje

[p.

u.]

39 40 41 42 43 44 450.99

Tiempo [s]39 40 41 42 43 44 45

0.99

Tiempo [s]

39 40 41 42 43 44 450.99

1

1.01

1.02

1.03

1.04

1.05

1.06

1.07

1.08

Tiempo [s]

Vol

taje

[p.

u.]

39 40 41 42 43 44 450.99

1

1.01

1.02

1.03

1.04

1.05

1.06

1.07

1.08

Tiempo [s]

Vol

taje

[p.

u.]

Figura 8. Voltaje en terminales del generador: (a) Sistema, (b) Sistema con PSS, (c) Sistema con

STATCOM, (d) Comparacin PSS y STATCOM

(a) (b)

(c) (d)

1.0455

1.046

1.0465

Vol

taje

[p.

u.]

45 50 55 601.0435

1.044

1.0445

1.045

Tiempo [s]

Vol

taje

[p.

u.]

Figura 9. Voltaje en terminales del generador en estado estacionario: Sistema (azul), Sistema con PSS (negro), Sistema con STATCOM (rojo)

0.9

1

1.1

1.2

1.3

Ang

ulo

delta

[ra

d]

377

377.2

377.4

Vel

ocid

ad [

rad/

s]

38 39 40 41 42 43 440.4

0.5

0.6

0.7

0.8

0.9

Tiempo [s]

Ang

ulo

delta

[ra

d]

38.5 39 39.5 40 40.5 41 41.5 42 42.5 43 43.5 44

376.4

376.6

376.8

Tiempo [s]

Vel

ocid

ad [

rad/

s]

Figura 10. Angulo de par: con PSS (negro),

con STATCOM (rojo)

Figura 11. Velocidad: con PSS (negro),

con STATCOM (rojo)

Perturbacin en la referencia de tensin 1- 0.95 p.u.

0.9

0.92

0.94

0.96

0.98

1

1.02

Vol

taje

[p.

u.]

0.9

0.92

0.94

0.96

0.98

1

1.02

Vol

taje

[p.

u.]

38 39 40 41 42 43 44 45 46 470.9

Tiempo [s]38 39 40 41 42 43 44 45 46 47

0.9

Tiempo [s]

38 39 40 41 42 43 44 45 46 470.9

0.92

0.94

0.96

0.98

1

1.02

Tiempo [s]

Vol

taje

[p.

u.]

38 39 40 41 42 43 44 45 46 470.9

0.92

0.94

0.96

0.98

1

1.02

Tiempo [s]

Vol

taje

[p.

u.]

(a) (b)

(c) (d)



1.1

1.2

1.3

1.4

1.5

Ang

ulo

delta

[ra

d]

377.4

377.6

377.8

378

Vel

ocid

ad [

rad/

s]

38 39 40 41 42 43 44 45 46 470.5

0.6

0.7

0.8

0.9

1

Tiempo [s]

Ang

ulo

delta

[ra

d]

38 39 40 41 42 43 44 45 46 47 48

376.4

376.6

376.8

377

377.2

Tiempo [s]

Vel

ocid

ad [

rad/

s]


con STATCOM (rojo)


con STATCOM (rojo)

PUNTO DE OPERACIN DEL SISTEMA

En el segundo punto de operacin del sistema se realiza una

perturbacin en el voltaje de referencia a los 40 segundos de1.05 a 0.97 p.u.

pujS 0799.08056.0 =puVpuVt 01,5.2105.1 ==

Se calculan las condiciones iniciales y los parmetros del PSS

pujS 0799.08056.0 =puVpuVt 01,5.2105.1 ==

1564.1,7696.0,0561.1

,4604.2,0130.1',3502.0,6859.0,7455.0,7393.0

0 ===

======

mreffdqqdqd

TV

EEIIVV

5109.7,3921.0,10,077.0,4496.1,102327.3 214 ====== nWPSS WTTTK

1,10.18,25.0 000 === DCVc

Perturbacin en la referencia de tensin 1.05- 0.97 p.u.

0.9

0.92

0.94

0.96

0.98

1

1.02

1.04

1.06

Vol

taje

[p.

u.]

0.9

0.92

0.94

0.96

0.98

1

1.02

1.04

1.06

Vol

taje

[p.

u.]

38 40 42 44 46 48 50 52 54 56 58 600.9

Tiempo [s]38 40 42 44 46 48 50 52 54 56 58 60

0.9

Tiempo [s]

38 40 42 44 46 48 50 52 54 56 58 600.9

0.92

0.94

0.96

0.98

1

1.02

1.04

1.06

Tiempo [s]

Vol

taje

[p.

u.]

38 40 42 44 46 48 50 52 54 56 58 600.9

0.92

0.94

0.96

0.98

1

1.02

1.04

1.06

Tiempo [s]

Vol

taje

[p.

u.]

(a) (b)

(c) (d)



0.97

0.975

0.98

0.985V

olta

je [p.

u.]

42 44 46 48 50 52 54 56 58 600.94

0.945

0.95

0.955

0.96

0.965

Tiempo [s]

Vol

taje

[p.

u.]

Figura 16. Voltaje en terminales del generador: Sistema (azul), Sistema con PSS (negro), Sistema con STATCOM (rojo)

1.2

1.3

1.4

1.5

Ang

ulo

delta

[ra

d]

377.5

378

378.5

Vel

ocid

ad [

rad/

s]

38 40 42 44 46 48 50 52 54 56 58 60

0.8

0.9

1

1.1

1.2

Tiempo [s]

Ang

ulo

delta

[ra

d]

38 40 42 44 46 48 50 52 54 56 58 60375.5

376

376.5

377

Tiempo [s]

Vel

ocid

ad [

rad/

s]


con STATCOM (rojo)


con STATCOM (rojo)

Para verificar los resultados obtenidos de la comparacin grfica,se calculan los valores propios de las matrices de estado de losmodelos linealizados en el punto de operacin para cada sistema.

S=0.4828-j0.2511 p.u.

S=0.8056-j0.0799 p.u.

Tabla 1. Valores propios de la matriz de estado en el punto de operacin 1

Tabla 2. Valores propios de la matriz de estado en el punto de operacin 2

CONCLUSIONES

Cuando el sistema trabaja cerca del lmite de operacin,el amortiguamiento propio del sistema no garantiza laestabilidad ante una perturbacin. La situacin empeorasi la ganancia del regulador es alta.

Es necesario incluir en el sistema un elemento auxiliarque proporcione el amortiguamiento adecuado paraextender los lmites de transferencia de potencia.

El estabilizador del sistema de potencia (PSS) y elSTATCOM amplan los lmites de operacin y mejoran laestabilidad de pequea seal del sistema de potencia.

Cuando aumenta la carga del sistema, el STATCOMofrece mejor amortiguamiento que el PSS ante unaperturbacin en la referencia de tensin; conservandouna rpida respuesta, con un tiempo de establecimiento

menor y eliminando las oscilaciones por completo.

Con el clculo de los valores propios de la matriz deestado de los sistemas linealizados, en condiciones decarga media y cuando el sistema esta cerca del lmite deoperacin, se confirm la influencia del PSS y delSTATCOM en la estabilidad del sistema.STATCOM en la estabilidad del sistema.

RECOMENDACIONES TRABAJOS FUTUROS

Considerar un modelo mas complejo del PSS con masetapas en la red de adelanto y otras seales auxiliares ouna combinacin de estas.

Combinar el sistema PSS con el dispositivo STATCOM u

otro dispositivo FACT.

BIBLIOGRAFA

[1] ALZATE, Alfonso. Dinmica de sistemas elctricos:Estabilidad y control. Maestra en Ingeniera Elctrica,Universidad Tecnolgica de Pereira, 2000.

[2] Sauer, Peter W. y Pai, M. A. Power systems dynamics [2] Sauer, Peter W. y Pai, M. A. Power systems dynamicsand stability. Upper Saddle River, New Jersey: Prentice-Hall, 1998. 357 p. ISBN 0-13-678830-0.

[3] WANG, H. F.. Phillips Heffron model of powersystems installed with STATCOM an applications. IEEProc. Gener. Distrib., septiembre 1999. Vol 146, No 5:521-527.

[4] VANFRETTI, Luigi. Sistemas de control de excitaciny estabilizadores de sistemas de potencia. Department ofelectrical, computer and systems engineering,Ressenlaer Polytechnic Institute, NY, Abril 2007.Disponible en:http://www.rpi.edu/~vanfrl/pdfs/AVRPSS_LV_2007.pdfhttp://www.rpi.edu/~vanfrl/pdfs/AVRPSS_LV_2007.pdf

Phillips-Heff ron model of power systems installed with STATCOM and applications

H . F. Wa ng

Abstract: The paper establishes the linearised Phdlips-Heffron model of a power system installed with a static synchronous compensator (STATCOM) and demonstrates the application of the model in analysing the damping effect of the STATCOM and designing a STATCOM stabiliser to improve power system oscillation stability. Both cases of single-machme infinite-bus and multimachme power systems are studied and example power systems are presented. These show the negative damping influence of a STATCOM DC voltage regulator on power system oscillations and the effectiveness of the STATCOM stabiliser superimposed on a STATCOM AC voltage regulator to counterattack the negative damping effect.

1 Introduction

The static synchronous compensator (STATCOM) is based on the principle that a voltage-source inverter generates a controllable AC voltage source behind a transformer-leakage reactance so that the voltage dlfference across the reactance produces active and reactive power exchange between the STATCOM and the transmission network. The STAT- COM is one of the important 'FACTS' devices and can be used for dynamic compensation of power systems to pro- vide voltage support and stability improvement [1-91.

In [lo, 111, a unified Phillips-Heffron model [I21 of a power system is established for three major types of FACTS devices installed in the system: static VAR compensator (SVC), thyristor-controlled series compensator (TCSC) and thyristor-controlled phase shifter (TCPS). Sub- sequently, the model has been used for the analysis and design of damping functions of these three FACTS devices [13-151. Hence, in this paper, the Phillips-Heffron model of power systems installed with a STATCOM is derived, whch turns out to have the same configuration as the unified model for SVC, TCSC and TCPS. Thus the STAT- COM is added to the category of FACTS devices to be described by the unified Phillips-Heffron model. Note, however, that there are fundamental differences between these FACTS devices. The first is between SVC and TCSC as controlled-impedance sources and the STATCOM and TCPS as controlled-voltage sources. The second difference is that STATCOM has developed from a switch-mode voltage-source converter configuration with an energy-storage device (DC capacitor) whilst SVC, TCSC and TCPS are based on phase-controlled thyristors/diodes which do not have any energy-storage devices.

In the paper, the Phillips-Heffron model is established for both single-machine infinite-bus and multimachine

OIEE, 1999 IEE Proceedings online no. 19990333 DOL 10.1O49hpgtd 19990333 Paper fmt received 30th October 1998 and in revised form 9th February 1999 The author is with the Department of Electrical and Electronic Engineering, University of Bath, Bath BA2 7AY, UK

power systems installed with a STATCOM. Applications of the model established are demonstrated by an example single-machine infinite-bus power system and an example three-machine power system to investigate the effect of the STATCOM on power system oscillation stability. A simple analysis indicates that the STATCOM DC-voltage regulator contributes negative damping to power-system oscillations, which is confirmed by both eigenvalue computation and nonlinear simulation. To counterattack the negative- damping effect brought about by the STATCOM DC voltage regulator, a STATCOM stabiliser superimposed on the STATCOM AC-voltage regulator is designed and its effectiveness is demonstrated by both eigenvalue computation and nonlinear simulation.

- - "L - 'b t -

ILB - ItL -

rn Y

cDC STATCOM

Fig. 1 A STATCOM installed in a singlemachine mfmite-h power system

2 Single-machine infinite-bus power systems installed with a STATCOM

Fig. 1 is a single-machine infinite-bus power system installed with a STATCOM which consists of a step-down transformer (SDT) with a leakage reactance X , , , a three- phase GTO-based voltage source converter (VSC) and a DC capacitor. The VSC generates a controllable AC-volt-

521 IEE Proc-Gener. Transm. Distrib., Vol. 146, No. 5, September 1999

age source vo(t) = Vo sin(wt - q) behind the leakage reactance. The voltage difference between the STATCOM-bus AC voltage vL(t) and vo(t) produces active and reactive power exchange between the STATCOM and the power system, whxh can be controlled by adjusting the magnitude Vo and the phase q. In Fig. 1, we have [16]

- ILO = ILOd + j ILOq - Vo = CVDC(COS 11, + j sin $) = CVDCL$

where, for the PWM inverter, c = mk and k is the ratio between AC and DC voltage, depending on the inverter structure; m is the modulation ratio defined by the PWM; and the phase qj is defined by the PWM. The STATCOM model of eqn. 1 is the enhanced dynamic model recom- mended for stability study of power systems. Although this model may not be valid for transient (internal short-circuits, for example) phenomena and unsymmetrical conditions, it is good enough for the study of power system osciuation stability [16].

From Fig. 1, - -

Substituting eqn. 2 into eqn. 3 gives

X L B - vo -vB X S D T X S D T

i.e.

(CVDC COS 11, + ~ C V D C sin 11,) X L B X S D T

_ -

- VB sin b - j V B COS 6 from whch it is possible to obtain

VB sin S + ~ C V D C COS 11, I tLq =

L C t L + X L B + x t L z + (1+%)2,

E: - VB cos6 - z c V ~ c sin'$

X t L + x L B + X t L E + ( 1 + ~ ) " & I tLd =

(4) The nonlinear dynamic model of the power system of Fig. 1 is [lo]

S = wbw W = (P, - Pe - D w ) / M

2 vt = \/(Eh - X & I t L d ) + ( 2 q I t L q I 2 By linearising eqns. 4 and 5 it is possible to obtain

A8 = wbaw AW = ( -Ape - D A w ) / M AEi = ( -AEq + A E j d ) / T ; ,

By denoting

K A K TA

K v = [ K A K f ] K-+ the linearised model of eqn. 10 can be shown by Fig. 2. From 1, = v, -jxtLItL - v&xSDT one can obtain

522 IEE Proc-Gener. Transm. Distrib., Vol. 146, No. 5, September 1999

I I , ' \ I

I I T I . I '4 I

Fig.2 instulled with U STATCOM

Phill@s-Hefion model of a single-mhke mfmite-bu power system

Hence by linearising the thrd equation in eqns. 1 ,4 and 11, one obtains

AVD, = K7Ad + KgAEi + K~AVDC + KdcAC + Kd+A$ (12)

From eqns. 10 and 12, the full-state system model can be obtained as follows:

wb 0 0 0 -.E -5 K,DC 0 -- M M

0

0 0

3 STATCOM

Multi-machine power systems installed with a

Without loss of generality, it is assumed that, in an it- machine power system, a STATCOM is to be installed on the transmission line between nodes 1 and 2 as shown by Fig. 3. It can also be assumed that, before the STATCOM is installed into the power system, a network adrmttance matrix P t is formed, where only it generator nodes plus node 1 and 2 are kept. The circuit equation of the network is

(14) where Ig = [I,, Tg2 ... Ig,JT, Vg = [Vgl Vg2 ... VgJT. With the installation of the STATCOM between nodes 1 and 2,

Tg21 G2 Tg1 G' p v g 2 ITVg,

+ %

cDc STATCOM

ifi

Fig. 3 STATCOM instdled in an n-muchke power system

the network equation of eqn. 14 becomes

where Fll' and Y22' are obtained from Tl1 anc excluding x12 = xlL + xL2. From Fig. 3,

Substituting eqn. 16 into eqn. 15 and then deleting [VI VdT, one can obtain

\ L - - J /

= i - y v g t r o v o

where

IEE Proc-Gener. Transm. Distrib., Vol. 146, No. 5, September 1999 523

For the n-mache power system, the terminal voltage of the generators can also be expressed in the common co- ordinates as [17]

V , = - jx.&Tg - j ( z q - z & ) I , (18) - -

From eqns. 17 and 18 one can obtain

Ig = cg {z, - j ( zq - z&)7, + cove} ( 1 9 ) where Cg = (Fyl + j x i ) - l , CO = F y l F O . In dj - qj co- ordinates, - IG. - 7 .ej&

z - gz

= 2 E g i k [ E ; k exp(j(90" + 6 k - 6,)) k = l

+ ( x q k - 5 & k ) e x P { j ( d k - & ) } r q k + c ( ) k t f O e 3 6 ' ]

(20)

By denoting Cgik = Cgk exp(jDgjk), C& = Cok exp(j&) and using eqn. 1, eqn. 20 becomes

n

Idi = c g i k { - E ; k sin d i k g + ( x q k - x & k ) r q k cos 6 z k g k=l

1 r4i = 2 C g i k { E h k cos d i k g + (xqk - X & k ) r q k sin & k g

}

+ c O k c v D C cos 601,

k=l

+ C O k C v D C sin 601,

(21) where Skg = S, - Si + Pgk, 4, = qj + Si + /3g(jgik + Dot Linearising eqn. 21, one can obtain

AI, = Y, A6 + F, AE, + G,AVDC + Hq AC + R, A$ = YdAd+FdAE; +GdAvDC + H d A C + R d $

(22 ) Substituting eqn. 22 into the following linearised equations of the n-machine power system [l 11,

ad' = w O ~ w a; = M - ~ ( - A T ~ - D A ~ ) A&': = Tgi [-AE, + AEFD] AE, = AE; - ( X D - X ~ ) A I D AEFD = (-AEFD - KAAVT)Til

ATE = AIQE~,+ I Q ~ A E ; + AIQ(XQ - X ~ ) I D O

+ IQO(XQ - X ~ ) A I D = XQAIQ AvTQ = - X',AID

(23) one can obtain the linearised model of the iz-machine power system installed with the STATCOM:

- -

+ 1 - M - l K p c -T ; ;~K,~ -T;;~ M-lKp* K , ~ I 1 - T , - ~ K ~ K , , -T,-~ K A K ~ + 1

(24) where

AT, = KlA6 + KzAE; + K,DCAVDC + Kp,Ac + Kp+A$

AE, = K4A6 + + K,ocAVDc + K,& + Kq+A$

A& = KsA6 + KcAE; + K,DCAVDC + K,,Ac + Ku+A$ (25)

which can be shown by Fig. 4, where

Af,=[AvDC Ac A$]

From Figs. 2 and 4 it can be seen that the configuration of the linearised model obtained for the STATCOM is exactly same as that of the unified model for the SVC, TCSC and TCPS [lo, 111. Thus the STATCOM is added to the category of FACTS devices in the unified model.

Fig. 4 a STATCOM

PhiNips-Heeffron mode[ ofa dtinachine power system installed with

From eqns. 15 and 16, - rLO = I I L + T z L

= [1 11 T,, 1 524 IEE Proc.-Gener. Transm. Distrib.. Vol. 146, No. 5 , September 1999

- 7 0 W " I 0 0 0

- M - ~ K ~ - M - ~ D - M - ~ K ~ o - M - ~ K , * ~

- T ; ; ~ K ~ o - T ; ; ~ K ~ T ; ; ~ - T ; ; ~ K , ~ ~

- T , - I K * K ~ o - T , - ~ K . , , K ~ -T;' - T ; ' K ~ K , , , , ~

- K r 0 K8 0 h'9 -I

L

'DCref KDCl KDCP -+ 7

4 Application

The DC-voltage regulator controls the DC voltage across the DC capacitor of the STATCOM. Fig. 5 shows the dynamic model of the DC-voltage regulator, which adopts PI control [16]. The DC-voltage regulator functions by exchanging active power between the STATCOM and the power system. Hence its influence on power system oscillation damping should be expected and can be investigated based on the Phillips-Heffron model. For example, if it is assumed that the active power input to the STATCOM installed in the single-machme infinite-bus power system of Fig. 1 is PcoM = VDcZD, The power balance equation of the power system should be Pm - P, = PucL + PcoM, where P,, (constant) is the mechanical power input to the generator, P, is the electric power output from the generator and Pa,, is the accelerating power to the rotor movement of the generator. In the steady-state operation mode, Pmo - Pd = 0 since Pacto = 0 and PcoMo = V D d D a = 0 (IDa = 0).

1 Y b

1 +sT,

'DC

PI DC voltage regulator converter dynamics Fig. 5 STATCOM d$nmnic nwakl with the P WM converter

However, during the dynamic process, power balance is achieved to ensure that Ape + AP,,, + APcoM = 0. A P C ~ M varies inversely with AP, as does Mac, so that the active power is kept in balance. Therefore, APcoM is opposite to AP, in phase and thus in 90" of phase lead with respect to generator speed Am. From eqn. 1 one can obtain

APCOM = VDCOAIDC + I D C O ~ V D C = VDCOAIDC = C V D C ~ A V D C = S C V D C ~ A V D C

Thus AVDC lags APcoM by 90" in phase and hence is in phase with Am, which can be expressed as AVDc = KDcJ,.

Therefore, from Figs. 1 and 5 one can obtain the 'direct- electric-torque' [lo, 1 11 contribution from the DC-voltage regulator to the electromechanical oscillation loop of the generator to be (converter dynamics is neglected, Tc = 0)

ATEDC = Kp+A$

Since, with increasing +, more active power is sent into the power system from the STATCOM, Kpw = dPJd+ > 0 whch indicates that the proportional control of the DC- voltage regulator of the STATCOM provides the power system with negative damping torque, while the integral control of the DC-voltage regulator contributes no damping to system oscillations.

In fact, the conclusion obtained above from the simple analysis can be confirmed by eigenvalue computation from the full-state system model of power systems installed with the STATCOM derived in the Sections above. Parameters of an example single-machne infinite-bus power system are given in the Appendix (Section 7). Table 1 presents the results of the eigenvalue computation from system linearised model of eqn. 13, where the STATCOM controls its bus voltage by the AC-voltage regulator. From Table 1 it can be seen that: (a) the STATCOM DC-voltage regulator contributes negative damping to the power system so that the damping of the oscillation mode is reduced as the result of the installation of STATCOM DC-voltage regulator; (b) the STATCOM DC-voltage regulator's integral control has no effect on system damping; and (c) the STATCOM AC-voltage control has little influence on system damping.

Table 1: Result of eigenvalue computation

Without the STATCOM

With STATCOM DC-voltage regulator

With STATCOM DC-voltage regulator

With STATCOM AC- and DC-voltage regulator

With STATCOM AC- and DC-voltage regulator and stabiliser

h, = -0.3546 f j4.2654 & = -0.2024 2 j4.2332

A, = -0.2024 f j4.2331

A, = -02001 f j4.3019

A, = -1.3205 * j4.6517

(Kocp = 0.2, = 0.0)

(KDCp = 0.2, KDC, = 0.2)

These confirm the conclusions drawn from the simple analysis. To combat the negative damping effect imposed by the STATCOM, an auxdiary stabilising signal can be introduced to and superimposed on the AC voltage control

IEE Pro,.-Gener. Transm. Distrib., Vol. 146, No. 5. September 1999 525

loop of the STATCOM as shown by Fig. 6, where the feedback signal for the STATCOM stabiliser is the locally available active power delivered along the transmission line. On the basis of the linearised model of the power system of eqn. 13, a phase-compensation method can be used to design this STATCOM stabiliser in the exactly same way as for the design of SVC, TCSC and TCPS-based stabilisers on the basis of their udied model [13-151. The result of the design is

Ks = 10.0, Tw = 10.0 s Ti = 0.9 S, Tz = 0.1 S, T3 = 0.7 S, T4 = 0.2 s

which moves the oscdlation mode to A,, = -1.3205 2j4.6517 (the feedback signal of the STACOM stabiliser is the active power delivered along the transmission line).

VL PI AC voltage regulator converter dynamics I r

auxiliary signal

feedback STATCOM stabiliser signal

Fig.6 (with P WM converter)

STATCOM a'ynamic model of AC-voltage regulator und stabilk

W 0 I 0 2.0 4.0 6.0 8.0 10.0

t,s Fig.7 ness of STATCOM-stabiliser &sign (i) with STATCOM DC-voltage regulator (KDcp = 0.2, K D , = 0.2) (ii) with STATCOM DC-voltage regulator (KDcp = 0.2, KDcr = 0) (iii) STATCOM stabiliser added (iv) with STATCOM DC- and AC-voltage regulator (v) without STATCOM

Nonl~i~ear simulation: confmtion of results in Table I andefftive-

' . O T I

-1.0 I V I I I I 0 2 4 6 8 10

t,s Fig. 8 site phase with dp, whik AVDc b in phase with the generator speed Aw

Nonlinear simulation: hvwnstrutwn that MCOM varies in the o p p

Figs. 7 and 8 present the result of nonlinear simulation. Fig. 7 conf i i s the results given in Table 1 and the effectiveness of the STATCOM stabiliser design. Fig. 8 demonstrates that U,,, vanes in the opposite phase with Me whilst AV,, is in the same phase as the generator speed Am as indicated above in the simple analysis.

526

The second application example is a three-machme power system shown by Fig. 9, parameters of which are given in the Appendix (Section 7). Table 2 presents the results of eigenvalue computation from system linearised model of eqn. 28 and Fig. 10 is the confirmation of nonlinear simulation. Thls example demonstrates the negative damping effect from the STATCOM DC-voltage regulator installed in the multimachine power system. With the introduction of an a d a r y stabilising signal from the STAT- COM stabiliser, this negative damping effect can be overcome and system-oscillation stability can be restored.

I m3

+ 1 L3 F$ 5b

Fig.9 Exmple three-machine power system

Table 2: Result of eigenvalue computation

Without the STATCOM A, = -1.0495 a2 = -0.4610 15.9123 j4.4999

STATCOM installed with A, =-1.1201 f h2 = 0.0191 f DC-voltage regulator 15.9879 j4.7274 STATCOM installed with AC-voltage regulator and p.9904 j4.5978 stabiliser added

A, = -1.2098 f k2 = -0.4891 *

80 T (iii) 70

$ 60 5 50 40 30

IJ)

C O

0 2 4 li i - 10 t.S

Fig. 10 Nonlinear shulatwn (i) without STATCOM (U) with STATCOM DC-voltage regulator (iii) with STATCOM AC-voltage regulator and stabiliser added

5 Conclusions

Major contributions of ths paper are: (i) establishment of the Phillips-Heffron model for both single-machine infinite-bus and multimachme power systems installed with STATCOM; and (ii) demonstration of applications of the model established by an example single-machme infinite-bus power system and a three-machine power system to reveal the negative- damping effect of STATCOM DC-voltage regulator and to design a STATCOM stabhser which, however, may not always exist in any multimachme power system installed with a STATCOM.

6 References

1 GYUGYI, L., HINGORANI, N.G., NANNERY, P.R., and TAI, T.: 'Advanced static Var compensator using gate turn-off thynstors for utility applications'. CIGRE, Pans, 1990, pp. 23-203 GYUGYI, L.: 'Reactive power generation and control by thyristor circuits', ZEEE Trans., 1979, IA-15, (5), pp, 521-532

2

IEE Proc-Gener. Transm. Distrib., Vol. 146, No. 5, September 1999

3 SCHAUDER, C., and MEHTA, H.: Vector analysis and control of advanced static VAR compensator, ZEE Proc. C, 1993, 140, (4), pp.

4 SCHAUDER,C., GERNHARDT,M., STACEY,E., CEASE, T.W., and EDRIS, A.: Development of a 2 100 MVAR static condenser for voltage control of transmission systems, IEEE Truns. Pwr. Deliv., 1995, 10, (3), pp. 14861496 EKANAYAKE, J.B., JENKINS, N., and COOPER, C.B.: Expen- mental investigation of an advanced static Var compensator, IEE Proc. Gener. Trunsm. Distrib., 1995, 142, (2), pp. 202-210

6 SAAD-SAOUD, Z., LISBOA, M.L., EKANAYAKE, J.B., JENKINS, N., and STRBAC, G.: Application of STATCOMs to wmd farms, IEE Proc. Gener. Trunsm. Distrib., 1998, 145, (5), pp. 51 1-516

7 TRAINER, D.R., TENNAKOON, S.B., and MORRISON, R.E.: Analysis of GTO-based static VAR compensators, IEE Proc. Gener. Trunsm. Distrib., 1994, 141, (6), pp. 293-302

8 AINSWORTH, J.D., DAVIES, M., FITZ, P.J., OWEN, K.E., and TRAINER, D.R.: Static Var compensator (STATCOM) based on single-phase chain circuit converters, IEE Proc. Gener. Trunsm. Dis- trib., 1998, 145, (4), pp. 381-386

9 MORI, S., MATSUNO, K., TAKEDA, M., and SETO, M.: Devel- opment of a large static Var generator using selfcommutated inverters for improving power system stability, IEEE Truns. Pwr. Syst., 1993,

10 WANG, H.F., and SWIFT, F.J.: An unified model for the analysis of FACTS devices in damping power system oscillations. Part I: single- machine infinite-bus power systems, IEEE Truns. Pwr. Deliv., 1997,

11 WANG, H.F., SWIFT, F.J., and LI, M.: An unified model for the analysis of FACTS devices in damping power system oscillations. Part 11: multi-machne power systems, IEEE Truns. Pwr. Deliv., 1998, 13, (4), pp. 1355-1362

12 HEFFRON, W.G., and PHILLIPS, R.A.: Effect of a modem ampli- dyne voltage regulator on under excited operation of large turbine generator, AIEE Trans., 1952,71

13 WANG, H.F.: Selection of operating conditions for the co-ordinated setting of robust fm-parameter stabilisers, IEE Proc. Gener. Trunsm. Distrib., 1998, 145, (2), pp. 11 1-1 18

14 WANG, H.F.: The phase compensation method to design FACTS- based stabilizer. Part I: single-machine infinite-bus power systems, Adv. Model. Anal., C, 1998, 1, (52)

299-306

5

8, (I), pp. 371-378

12, (2), pp. 941-926

15 WANG, H.F.: The phase compensation method to design FACTS- based stabilizer. Part I1 multi-machine power systems, Adv. Model. Anal., C, 1998, 1, (52)

16 Modelling of power electronics equipment (FACTS) in load flow and stability programs. CIGRE

17 YU, Y.N.: Electric power system dynamics (Academic Press, 1983)

7 Appendix

Parameters of example single-machine infinite-bus power system (in p.u. except indicated): H = 3.0~, D 4.0, Td = 5.044~, xd = 1.0, Xq = 0.6, Xi = 0.3, xtL = 0.3, xLB = 0.3, X ~ D T = 0.15, KA = 10.0, T A = 0.01~, Tc = 0.05~, CDc = 1.0, VD, = 1.0, C, Pd 0.8, Va = 1.0, Vm = 1.0, VLo = 1.0, CO = 0.25, WO = 52 Parameters of example three-machme power system (in p.u. except indicated): Generator: Hl = 20.09s, H2 = 20.09~, H3 = 11.8~, D1 = 10.0, 0 2 = 0.0, D3 = 60.0, Tdl = 7.5~, Ta2 = 7.5~, Td3 = 4 .7S , x d l = 0.19, x& = 0.19, x& = 0.41, xql = 0.163, xq2 = 0.163, Xq3 = 0.33, Xdl = 0.0765, X& = 0.0765, Xd = 0.173. Exciter: KA1 = KA3 = K A ~ = 100, TA1 = T A ~ = T A ~ = 0.01s. Network: z4L =jO.l, zL3 =jO.2, 2 3 5 xjo.3, zn = 2~ = + jl.0. jo.03, TIt = 1.OL14, T7zt 1.0L5, P3, = l.OLOo, L3 = 1.07

STATCOM: V D , = 1.0, CDc = 1.0, CO = 0.25, WO = 13, Tc = 0.05~, K D c p = K D C - = 10.0, K A c p = KAcI = 10.0, Kpss = 12.0, Tw = 10.0~, Ti = 2.5~, T2 = O.~S, T3 = 2.2~, T4 = 0.9s.

IEE Proc-Gener. Transm. Distrib.. Vol. 146, No. 5. September 1999 521

STATCOM Modeling for Voltage and Angle

Stability Studies

Claudio A. Canizares University of Waterloo, Dept. Electrical & Computer Eng., Waterloo, ON,

N2L-3G1, Canada

Massimo Pozzi, Sandro Corsi

CESI, Via R. Rubattino 54 , 20134 Milan, Italy

Edvina Uzunovic

New York Power Authority, 123 Main Street, 6th floor, White Plains, NY, 10601,USA

Abstract

This paper proposes and validates models to accurately represent STATic Syn-chronous Shunt COMpensators (STATCOM) in voltage and angle stability studiesof powers systems. The proposed STATCOM stability models are justified basedon the basic operational characteristics of this Flexible AC Transmission System(FACTS) controller for both phase and PWM control strategies. These models arefirst validated by means of EMTP simulations on a test system, and then are imple-mented into two different programs used to study voltage and angle stability issuesin the system. All details of the model implementation, the controls used, and thedata for the test system are provided in the paper.

Key words: STATCOM, FACTS, modeling, voltage stability, angle stability, smallsignal stability, transient stability

This work was supported by the Italian Ministry of Industry under the grantRicerca di Sistema DM 17/04/2001, and by the National Science and EngineeringResearch Council (NSERC) of Canada. Corresponding author.

Email addresses: [email protected] (Claudio A. Canizares),[email protected] (Massimo Pozzi), [email protected] (EdvinaUzunovic).

Article to be published in Electrical Power & Energy Systems 25 (2003) 120

1 Introduction

The development and use of FACTS controllers in power transmission systemshas led to many applications of these controllers to improve the stability ofpower networks [1,2]. Thus, many studies have been carried out and reportedin the literature on the use of these controllers in a variety of voltage andangle stability applications, proposing diverse control schemes and locationtechniques for voltage and angle oscillation control [2].

Several distinct models have been proposed to represent FACTS in static anddynamic analyses [3]. The current paper concentrates on describing in detailadequate STATCOM models for these types of studies, based on an energybalance criterion previously used in the modeling and simulation of this Volt-age Sourced Converter (VSC)-based controller [47]. It is demonstrated herethat the proposed models allow to accurately and reliably represent a STAT-COM, operating under either phase or PWM control schemes, for voltage andangle stability studies using power flow, steady state and transient stabilityprograms, as the models allow for an appropriate representation of the typicalcontrol limits for this controller [5,8,9].

Details of the implementation and use of the STATCOM models proposed hereare discussed in regards to two programs used for steady state and transientstability analyses of power systems. These programs are UWPFLOW [10], aprogram designed for voltage stability analysis of power systems, and EASY5[11], a program designed for modeling general linear and nonlinear controlsystems, and used at ENEL and CESI for model validation studies as wellas steady state and transient stability analyses of sample power systems. Theresults of using these programs for the study of the stability of a test system arepresented and thoroughly analyzed here, together with all the data requiredto reproduce these results in any simulation program.

Section 2 describes the proposed STATCOM models, based on the fundamen-tal operation of this controller under both phase and PWM control strate-gies; improvements of these models with respect to models previously usedto represent this controller in power flow and transient stability studies arealso discussed in this section. In Section 3, the results of implementing theproposed STATCOM models into two programs used for voltage and anglestability analyses of a sample system are discussed in detail. Finally, Section4 summarizes the implementation issues associated with the proposed modeland discusses its limitations.

2

PWM

(PWM)m

Magnitude

Vref

Magnitude

Vi

Vdc

C

Filters I

V

Vrefdc

a:1

SwitchingLogicPLL

ZeroCrossing

Controller

Fig. 1. Block diagram of a STATCOM with PWM voltage control.

2 STATCOM Models

The basic structure of a STATCOM with PWM-based voltage controls isdepicted in Fig. 1 [5,8]. Eliminating the dc voltage control loop on this figurewould yield the basic block diagram of a controller with a typical phase anglecontrol strategy.

The STATCOM models proposed here is based on the power balance equation

P =Pdc + Ploss (1)

which basically represents the balance between the controllers ac power Pand dc power Pdc under balanced operation at fundamental frequency (theseare the basic assumptions on which steady state and transient stability studiesof power systems are based). For the models to be accurate, it is importantto represent the losses of the controllers (Ploss), as discussed below; previouslyproposed models in [3] do not consider this issue.

PWM controls are becoming a more practical option for transmission systemapplications of VSC-based controllers, due to some recent developments onpower electronic switches that do not present the high switching losses ofGTOs [12], which have typically restricted the use of this type of controltechnique to relatively low voltage applications. In PWM controls, switchinglosses associated with the relatively fast switching of the electronic devices andtheir snubbers play an important role in the simulation, as these have a directeffect on the charging and discharging of the capacitor, and hence should be

3

RCCk (PWM)

PWM

P+jQ

Filters

a:1

dck V

V

I

Vrefdc

R+jX

Controller

Vref

Magnitude

dcV

Magnitude

Fig. 2. Transient stability model of a STATCOM with PWM voltage control.

considered in the modeling. The models discussed in this paper assume theuse of PWM control techniques, as these allow for developing more generalmodels that can readily be adapted to represent other control techniques (e.g.phase angle control).

2.1 Transient Stability Model

Assuming balanced, fundamental frequency voltages, the controller can be ac-curately represented in transient stability studies using the basic model shownin Fig. 2 [5,7]. The p.u. differential-algebraic equations (DAE) correspondingto this model can be readily written as follows:

xc

m

= fc(xc, , m, V, Vdc, Vref , Vdcref ) (2)

Vdc =V I

C Vdccos( ) GC

CVdc R

C

I2

Vdc

4

0=

P V I cos( )

Q V I sin( )

P V 2 G + k Vdc V G cos( )+k Vdc V B sin( )

Q + V 2 B k Vdc V B cos( )+k Vdc V G sin( )

where most of the variables are explained on Fig. 2. The admittance G+jB =(R+ jX)1 is used to represent the transformer impedance and any ac seriesfilters (e.g. smoothing reactors), whereas GC is used to model the switchinginertia of the converter due to the electronic switches and their associatedsnubber circuits, which have a direct effect on the capacitor voltage dynamics.

The constant k =3/8 m is directly proportional to the modulation index

m.

The variables xc and functions fc() in (2) stand for the internal control systemvariables and equations, respectively, and hence vary depending on whether aPWM or phase control technique is used in the controller. For example, in thesimple PWM voltage controller shown in Fig. 3 [13], the variables and differ-ential equations associated with the various control blocks directly define xcand fc(). Observe that in this PWM controller, the ac bus voltage magnitudeis controlled through the modulation index m, as this has a direct effect onthe VSC voltage magnitude, whereas the phase angle , which basically deter-mines the active power P flowing into the controller and hence the chargingand discharging on the capacitor, is used to directly control the dc voltagemagnitude. Note also that the controllers have a bias, which corresponds tothe steady state value of the modulation index mo for the voltage magnitudecontroller, and to the phase angle of the output voltage of the STATCOMfor the dc voltage controller (this value changes as the system variables changeduring the simulation). Although the latter complicates the simulation, it isneeded to guarantee a direct control of the charging and discharging of thecapacitor, which basically depends on the power flow between the VSC andthe ac bus, i.e. it depends on ( ). (This can be simplified by setting thebias of the dc voltage control to the constant value o = o, where o standsfor the bus voltage phase-shift when the STATCOM is not connected [14].)Typically, the modulation index control would be faster than the phase an-gle control, as there is a significant charging and discharging inertia of the

5

o

K + S T 2D

MK

MK

maxI

minI

1

M

dc

dc

M

ac

ac + S T

1

+ S T

K ( + S T )

1

1

mindc

maxdc

m

+

+

KI

SKP

+

-ref

Vdc

+

+

+

m

-+dc

V

V

V

V

ref

V

Fig. 3. Basic STATCOM PWM voltage control.

capacitor due to its relative large value, whereas the modulation index has animmediate effect on the output voltage of the controller.

The second equation in (2) is the direct result of applying the power balanceequation (1), and allows to represent fairly accurately the dynamics of the dcvoltage in the controller model, as demonstrated in Section 3 for a realistic testsystem. The adequate modeling of the Vdc dynamics is important, given thefact that the time constants associated with the dc voltage on the capacitorare in the order of the time constants of interest in stability studies. These dcvoltage dynamics are basically defined by the GC parameter in the proposedmodel, as its value directly affects the capacitors charging and dischargingtime constant. The losses and dc voltage dynamics are considered in the modelsproposed in [5,7], whereas in the STATCOM model proposed in [3], these arenot fully considered, since GC is not represented in the model, thus introducingerrors in the controller representation as demonstrated here.

The control limits of the controller are directly defined in terms of both thecurrent limits in the electronic switches, which is the main limiting factor inVSC-based controllers, and the dc voltage, which is a secondary operationallimit in this controller. This direct implementation of limits allows to closelyduplicate the steady state V-I characteristics of the controller shown in Fig. 4,as well as allowing for an adequate representation of the basic control limits onan actual STATCOM [2]. In time domain simulations, the integrator blocks,

6

such as those shown in Fig. 3, are stopped whenever the converter currentI or dc voltage Vdc reach a limit. An alternative way of handling these limitsfor both PWM and phase control techniques to allow temporary controlleroverload is discussed in Section 3. (Another way to simulate these limits isto determine the values of the modulation index m and phase angle cor-responding to the current and dc voltage limits, respectively, by solving thesteady state equations of the converter, as discussed in [14].)

2.2 Steady State Model

The steady state or power flow model can be readily obtained from (2) byreplacing the corresponding differential equations with the steady state equa-tions of the dc voltage and the voltage control characteristics of the STATCOM(see Fig. 4 [2]). Thus, the steady state equations for the PWM controller are

0=

V Vref XSLI

Vdc Vdcref

P GC V 2dc R I2

P V I cos( )

Q V I sin( )

P V 2 G + k Vdc V G cos( )+k Vdc V B sin( )

Q + V 2 B k Vdc V B cos( )+k Vdc V G sin( )

(3)

where on the first equation, the positive sign is used when the device is op-erating on the capacitive mode (Q < 0) and negative for the inductive mode(Q > 0), since I 0.

7

I (Q>0)

V

Vref

II min max

(mo , o )

I (Q Imin as the electronic switchesself commutate on the inductive region. Furthermore, Vdcmax and Vdcmin aretypically not an issue on steady state models, given their corresponding rela-tively high and low values with respect to the typical range of application ofthese models.

A phase control technique can be readily modeled by simply replacing thedc voltage control equation in (3) with an equation for k, i.e. for a 12-pulseVSC, replace 0 = Vdc Vdcref with 0 = k 0.9. In this case, the dc voltagechanges as changes, thus charging and discharging the capacitor to controlthe converter voltage magnitude.

These equations can be directly used to compute the control steady statevalues and biases as well as its limits. For example, the modulation index biasand steady state is determined by setting I = 0, yielding

mo =

8

3

VrefVdcref

ko

Modulation index and phase-shift control limits corresponding to the con-troller ac current and dc voltage limits can be readily determined by solvingequations (3) [14].

The limits on the current I, as well as any other limits on the steady statemodel variables, such as the modulation ratio represented by k or the voltage

8

I=

Vref > ref Vo

Vref ref Vo

I =

ref VoVref

Vref ref Vo

& Q > 0

& Q < 0

I< maxI

< IminI

Vref ref Vo= 0ImaxI >

I

Imin

I

Fig. 5. Handling of limits in the STATCOM steady state model.

phase angle , can be directly introduced in this model. It is important toproperly represent the control mode switching when these limits are reached,as this is needed to properly model FACTS controllers in voltage stability stud-ies [15]. The switching logic depicted in Fig. 5 is proposed here to representthe steady state control mode switching for the STATCOM, which is mainlyassociate with the controller ac current, as previously discussed. When eithermaximum or minimum current limits are reached, depending on whether thecontroller is operating in the inductive or capacitive region, respectively, volt-age control is lost; the controller is allowed to recover from its limits when thevoltage is again within the control voltage range as defined by the controllervoltage droop.

The model presented here allows for a more adequate representation in steadystate analysis of the STATCOM than the typical power flow models basedon reactive power source representations of the controller (e.g. [7]). In thesetypes of models, limits are usually represented through limits in reactive power,i.e. the STATCOM is basically modeled as a synchronous condenser using astandard PV bus model. This would somewhat represent the controller currentlimits if its terminal voltage is known; however, this is not always the case, asthis voltage depends on control droops, the system conditions, the STATCOMcontrolled bus, and other controller limits. Hence, the PV bus model presentsthe following limitations:

The controller droop cannot be readily modeled. Certain controller limits, such as limits on the dc voltage, phase angle and/ormodulation ratio cannot be properly represented,

If the STATCOM controls the voltage at a remote bus, the limits in thereactive power will not adequately model the controller current limits.

The proper representation of control droops and controller limits is of partic-ular importance in voltage stability studies [16], as controller limits may leada power system to voltage collapse problems. These limitations are clearlyillustrated through simulations on a realistic test system in the next section.

9

3 Implementation and Results

The STATCOM model described here was implemented into two softwarepackages that may be used for the stability analysis of power systems, namely,UWPFLOW [10] and EASY5 [11]. The results obtained with these programswere compared with results extracted from [5], where the ElectromagneticTransient Program (EMTP) is used to validate the proposed model. The de-tails of the STATCOM model implementation in these programs and the re-sults obtained from the stability analysis of a realistic test system are discussedin this section.

3.1 Test System

The test system depicted in Fig. 6 is used here to validate the implementationof the STATCOM model into the programs UWPFLOW and EASY5 basedon detailed EMTP simulation results.

The EMTP is used in [5] to simulate a detailed switching model of the STAT-COM operating under phase control, and to compare the results obtainedfrom this model against those obtained for the controller model described inSection 2. The results of this comparison are depicted on Fig. 7. A load re-jection fault is simulated by connecting a large load at Bus 6 at 4.5s, andthen opening breaker 3-5 at 4.65 s. Observe how close the results are for theexternal variables, i.e. voltage magnitudes and angles, for the detailed andthe reduced models. The internal STATCOM variables Vdc and do notmatch exactly, although the general trends are similar, as the switching cannotbe represented in the stability model. Observe that a value of GC = 0, whichbasically corresponds to a typical STATCOM stability model [3], yields fairlydifferent results from those obtained for the detailed switching studies.

All the data and controls required for typical stability studies of the given testsystem were extracted from the detailed 3-phase EMTP data of the system,and are depicted in Figs. 8 and 9, and Tables 1, 2 and 3. It is importantto highlight the fact that the value of GC was carefully chosen to match thesystem losses and dc voltage dynamics obtained from the EMTP detailedswitching studies.

3.2 Voltage Stability Results

The program UWPFLOW, as described in [10], is a tool that can be usedto determine the steady state operating conditions of power systems as cer-

10

STATCOM

Infinite bus Z Thevenin

Bus 1 Bus 2

Bus 3

Bus 8 Bus 9

Bus 6

15 mi

15 mi

90 mi

230 kV

Bus 5

Transf.

125 mi

13.8 kV

Bus 4

245.5 MW

Generator

Bus 10

321 MW43 MVar

952 MW267 Mvar

Filter

Fig. 6. Test system.

4 4.2 4.4 4.6 4.8 5 5.2 5.4 5.6 5.8 650

0

50

Gen

. Ang

le [d

eg.]

Detailed ModelReduced ModelG

C=0 Model

4 4.2 4.4 4.6 4.8 5 5.2 5.4 5.6 5.8 60

0.5

1

VB

us 4

[p.u

.]

4 4.2 4.4 4.6 4.8 5 5.2 5.4 5.6 5.8 60

0.5

1

1.5

VB

us 1

0 [p

.u.]

4 4.2 4.4 4.6 4.8 5 5.2 5.4 5.6 5.8 60

5

10

15

Vdc

[kV

]

4 4.2 4.4 4.6 4.8 5 5.2 5.4 5.6 5.8 610

0

10

[d

eg.]

t [s]

Fig. 7. EMTP results for the test system with different models for a phase-controlledSTATCOM.

11

750

1 + S 0.001

4

S 0.04

1 + S

1

(0.11+S 0.01)(0.17+S 0.95)f

+

-

-

1 +

+

1

V

V

Fig. 8. EMTP generator AVR model for the test system.

min=-.1745 rad

(1+S 0.001)(11+S 0.03)

3.9652 (1.5+S 0.0115)

8

8

max= .7145 rad (10 )

(-10 )

o

o

V

-

+

-

+

1

Fig. 9. EMTP phase control for the STATCOM in the test system.

Table 1Generator data in p.u. with respect to a 200 MVA and 13.8 kV base.

Variable Value [p.u./s] Variable Value [p.u./s]

Poles 2 H 2.7113

Ra 0.001096 D 0

Xl 0.15 T do 6.19488

Xd 1.7 T qo 0

Xq 1.64 T do 0.028716

X d 0.238324 Tqo 0.07496

X d 0.18469 Xq 0.185151

tain system parameters change. It can be used to partially study the steadystate stability of these systems, especially the issue of voltage stability withrespect to a variety of system changes (e.g. load changes). UWPFLOW isbasically a continuation power flow program with fairly detailed steady statemodels of generators and HVDC links, including some of their control systemsand corresponding limits. It also contains SVC and TCSC controller modelsto represent these popular Thyristor Control Reactor (TCR)-based FACTScontrollers in power flow and voltage stability studies [15].

12

Table 2Transmission system data in p.u. with respect to a 240 MVA base.

Element R X B/2

1-2 0.0003 0.0684 0

2-3 0.0159 0.2275 0.0754

3-8 0.022 0.316 0.1047

4-3 0 0.08 0

5-6 0.0026 0.0379 0.0126

6-7 0 0.12 0

8-9 0.0026 0.0379 0.0126

9-10 0 0.12 0

Filter 0.0087 -4.3 0

Table 3STATCOM data in p.u. with respect to a 150 MVA and 12 kV base.

Variable Value

R 0

X 0.145

GC 2.16

C 0.0432

k (Phase) 0.9

Vdcref (PWM) 1

Imax 1

Imin 1

The model corresponding to equations (3) was programmed into UWPFLOW,which was used to obtain the PV curves and maximum loading conditions forthis system as shown in Fig. 10. As expected, the loading margin and voltageprofiles of the system are significantly improved by the introduction of theSTATCOM, especially for the phase control mode, as the dc voltage is free tochange while the current is within its limits, whereas in PWM control mode,the dc voltage is kept constant at a value that could be considered low forthe test system. In all cases, current limits are reached before the maximumloading point.

For comparison purposes, the STATCOM was also modeled as a PV bus.Note that modeling the STATCOM as a PV bus with fixed reactive powerlimits (Qmax = Qmin = 1 p.u. for Imax = Imin = 1 p.u.) does not properly

13

0 20 40 60 80 100 120 140 160 180 20060

80

100

120

140

160

180

200

220

240

kV B

US

8

L.F. [MVA]

No STATCOM With PWM STATCOMWith phase STATCOMPV STATCOM model

Fig. 10. PV curves obtained with UWPFLOW for the test system with and withoutSTATCOM.

max1 + S 0.5

8

ref

S 0.5

0.25

0.2

M

dc

dc

A

0.25

0.2X

MIN

dcmin

dc

8o

max

min

0.25V

V

Q/V

V+

+=1.2

=-1

V

Q/V

V+

+=0.8

=-1

-

-

+

-I

I

V

-

-

-

-

+= 1

Fig. 11. STATCOM phase controller in EASY5.

represent the controller for these types of studies, as previously discussed.

14

min

0.2N

0.25

0.2M

dcmin

dc

AX

o

1 + S 0.01

S 0.01

M

8o

1 + S 0.5

S 0.50.25

dcref

dc

0.25

0.25dcmax

dc

I

8

max

ref

V

Q/V

+=-1

Q/V

V+

=0.8

=-1

V

+

+

+

m

m

-

+

V

= 1

-

+

V

=1.2V

+

I

-I

V

-

-

-

-

-

+= 1V

Fig. 12. STATCOM PWM controller in EASY5.

3.3 Transient Stability Results

EASY5 is a program by BOEING used for the simulation of linear and non-linear control systems [11]. This program allows to graphically represent anylinear and nonlinear control system through the definition of its equations andassociated graphical icons. Linear matrix analysis tools and nonlinear integra-tion tools allow for the analysis of the steady state as well as the transientstability of any control system defined by the user. Libraries can be read-ily developed, so that new systems and elements can be easily defined andintegrated into other simulations. The different element models and the con-nections between these elements in a given system must be defined togetherwith the numerical analysis techniques required for its analysis. Thus, as withSIMULINK-MATLAB, the program can be used to graphically model powersystems. This program has been successfully used at ENEL and CESI to modeland test a variety of power system controllers [17].

The model corresponding to equations (2) for a phase and PWM controllerswere implemented in EASY5. The STATCOM phase and PWM controllers aredepicted in Figs. 11 and 12, respectively. The generator AVR and STATCOMphase controllers implemented in this program for the test system are notexactly the same as the ones used in the EMTP, particularly for the STAT-

15

COM phase and PWM controls, which are significantly different from the onesdiscussed in Section 2 in the way the limits are implemented. The reason forthese modeling differences, particularly in the implementation of the STAT-COM limits, is to improve the dynamic voltage control characteristics of thesystem controllers, as transient overload of the STATCOM is allowed to im-prove its dynamic response; this does not affect the steady state behavior ofthe controller.

The results of using this program and the corresponding voltage controls tomodel the test system fault depicted in Fig. 6 are illustrated in Figs. 13 and14. As in the EMTP example, a load rejection problem is simulated by sud-denly applying a large load on Bus 6 at 4.5s, and then disconnecting it at 4.65sby opening the breaker 3-5. Observe that the results are somewhat similar tothose obtained with the EMTP; however, one cannot expect an exact match,as the STATCOM controls are basically different and the generator loadingconditions are not exactly the same (there is an approximate 2 differencebetween the EMTP and ESAY5 simulations in the internal generator angleat the initial steady state conditions). It is interesting to notice that underPWM control, the STACOM internal variables, i.e. Vdc and , show less varia-tions than under phase control; however, the controlled load voltage basicallypresents the same transient response in both cases.

4 Conclusions

The STATCOM transient stability and power flow models proposed in thispaper are basically improved versions of models previously proposed in theliterature. Thus, the current paper concentrates in discussing and justifyingthe improvements to these models so that proper voltage and angle stabilitystudies can be performed on networks that contain this kind of FACTS con-troller. The implementation of the STATCOM model into a couple of stabilityanalysis tools, and the results presented and discussed for a simple test systemshow how these models can be readily and reliably used for stability studiesof power systems.

The models discussed here are all based on the assumption that voltages andcurrents are sinusoidal, balanced, and operate near fundamental frequency,which are the typical assumptions in transient stability and power flow studies.Hence, these models have several limitations, especially when studying largesystem changes occurring close to FACTS controllers:

(1) These models cannot be reliably used to represent unbalanced systemconditions, as they are all based on balanced voltage and current condi-tions.

16

4 4.5 5 5.5 60.7

0.75

0.8

0.85

0.9

0.95

1

4 4.5 5 5.5 60.92

0.94

0.96

0.98

1

1.02

4 4.5 5 5.5 60.275

0.3

0.325

0.35

0.375

0.4

0.425

s

radi

ans

Generator Phase Angle

s

p.u.

Generator Terminal Voltage

s

p.u.

Generator Active Power

4 4.5 5 5.5 6-0.02

-0.01

0

0.01

0.02

4 4.5 5 5.5 67.5

7.8

8.1

8.4

8.7

9

9.3

4 4.5 5 5.5 60.9

0.93

0.96

0.99

1.02

1.05

1.08

1.11

s

p.u.

Load Voltage

s

kV

Statcom DC Voltage

s

radi

ans

Alpha

Fig. 13. Fault simulation results obtained with EASY5 for the STATCOM operatingunder phase control.

17

4 4.5 5 5.5 60.7

0.75

0.8

0.85

0.9

0.95

1

4 4.5 5 5.5 60.92

0.94

0.96

0.98

1

1.02

4 4.5 5 5.5 60.275

0.3

0.325

0.35

0.375

0.4

0.425

s

radi

ans

Generator Phase Angle

s

p.u.

Generator Terminal Voltage

s

p.u.

Generator Active Power

4 4.5 5 5.5 6-0.02

-0.01

0

0.01

0.02

4 4.5 5 5.5 67.4

7.6

7.8

8

8.2

8.4

8.6

4 4.5 5 5.5 60.9

0.93

0.96

0.99

1.02

1.05

1.08

1.11

s

p.u.

Load Voltage

s

kV

Statcom DC Voltage

s

radi

ans

Alpha

Fig. 14. Fault simulation results obtained with EASY5 for the STATCOM operatingunder PWM control.

18

(2) Large disturbances that yield voltage and/or currents with high har-monic content, which is usually the case when large faults occur nearpower electronics-based controllers, cannot be accurately studied withthese models, as they are all based on the assumptions of having sinu-soidal signals.

(3) The above also applies for cases where voltage and current signals undergolarge frequency deviations.

(4) Internal faults as well as some of the internal variables of the controllercannot be reliably represented with these models.

For all of these cases, detailed EMTP types of studies are required to obtainreliable results. It is important to highlight the fact that these limitationsalso apply to most models typically used to represent a variety of devices andcontrollers in transient stability and power flow studies.

References

[1] N. G. Hingorani, Flexible AC Transmission Systems, IEEE Spectrum (1993)4045.

[2] FACTS Applications, Technical report 96TP116-0, IEEE PES (1996).

[3] Modeling of Power Electronics Equipment (FACTS) in Load Flow and StabilityPrograms: A Representation Guide for Power System Planning and Analysis,Technical brochure no. 145, TF 38-01-08, CIGRE (Aug. 1999).

[4] C. Schauder, H. Mehta, Vector Analysis and Control of Advanced Static VarCompensators, IEE ProceedingsC 140 (4) (1993) 299306.

[5] E. Uzunovic, C. A. Canizares, J. Reeve, Fundamental Frequency Model ofStatic Synchronous Compensator, in: Proc. North American Power Symposium(NAPS), 1997, pp. 4954, Laramie, Wyoming.

[6] E. Uzunovic, C. A. Canizares, J. Reeve, Fundamental Frequency Model ofUnified Power Flow Controller, in: Proc. North American Power Symposium(NAPS), 1998, pp. 294299, Cleveland, Ohio.

[7] D. N. Koseterev, Modeling Synchronous Voltage Source Converters inTransmission System Planning Studies, IEEE Trans. Power Delivery 12 (2)(1997) 947952.

[8] C. Schauder, M. Gernhardt, E. Stacey, T. Lemak, L. Gyugyi, T. W. Cease,A. Edris, Development of a 100 MVAr Static Condenser for Voltage Controlof Transmission Systems, IEEE Trans. Power Delivery 10 (3) (1995) 14861493.

[9] P. Rao, M. L. Crow, STATCOM Control for Power Applications, in: Proc. NorthAmerican Power Symposium (NAPS), 1997, pp. 172178, Laramie, Wyoming.

19

[10] C. A. Canizares, UWPFLOW: Continuation and Direct Methods to LocateFold Bifurcations in AC/DC/FACTS Power Systems, University of Waterloo,available at www.power.uwaterloo.ca (Sep. 2000).

[11] EASY5 Users Guide, Programs manual, BOEIGN (Jul. 1989).

[12] P. K. Steimer, H. E. Gruning, J. Werninger, E. Carrol, S. Klaka, S. Linder,IGCT-A New Engineering Technology for High Power, Low Cost Inverters,IEEE Industry Applications Magazine (1999) 1218.

[13] E. Uzunovic, C. A. Canizares, J. Reeve, EMTP Studies of UPFC PowerOscillation Damping, in: Proc. North American Power Symposium (NAPS),1999, pp. 405410, San Luis Obispo, California.

[14] C. A. Canizares, Modeling and Implementation of TCR and VSI Based FACTSControllers, Internal report, ENEL and Politecnico di Milano, Milan, Italy (Oct.1999).

[15] C. A. Canizares, Z. T. Faur, Analysis of SVC and TCSC Controllers in VoltageCollapse, IEEE Trans. Power Systems 14 (1) (199) 158165.

[16] Voltage Stability Assessment: Concepts, Practices and Tools, Power SystemDynamic Performace Committee special publication, IEEE/PES, Available atwww.power.uwaterloo.ca (Aug. 2002).

[17] M. Pozzi, Implementation of Dynamic Models and Controls for HVDC andFACTS Systems: The Macro Components of the Power System Library in theSimulation Tools Easy5x from Boeing, Report no. 99/594, ENEL Ricerca, AreaTrasmissione e Dispacciamento (Dec. 1999).

20

1

1. Amortiguamiento de oscilaciones con el compensador de VARs esttico

Introduccin

Se utiliza un compensador de VARs esttico para ampliar los lmites de estabilidad del sistemaelctrico, en este caso una mquina sincrnica conectada a un barraje infinito a travs de unalnea de transmisin. Se parte del supuesto de que se conoce la metodologa para encontrar elmodelo linealizado del sistema elctrico, tal como de planteo en captulos anteriores. Se tienecomo objetivo en este caso obtener el modelo linealizado del sistema elctrico considerando losefectos del compensador de VARs esttico.

Aunque el propsito fundamental del compensador de VARs esttico es el de suministrar lapotencia reactiva que requiere el sistema para mantener a un nivel adecuado los perfiles detensin, se aprovecha el hecho de que su instalacin tiene efectos benficos al ampliar los lmitesde estabilidad del sistema elctrico como tambin ayudar a amortiguar los efectos de lasoscilaciones que se producen cuando se presenta perturbaciones.

El esquema utilizado en este caso contiene una parte para el control de los reactivos y uncompensador adicional del tipo PI para amortiguar las oscilaciones que aparecen cuando sepresenta una perturbacin. Este esquema provoca una variacin en los coeficientes del modelolinealizado, tal que el torque sincronizador es positivo y esto asegura la estabilidad.

En la figura 1 se muestra la forma de conexin del SVC en el sistema conformado por unamquina sincrnica conectada a un barraje infinito a travs de una lnea de transmisin. Seobserva que el SVC se conecta en el barraje del generador.

Figura 1 Esquema general a estudiar

2

2. Modelo del compensador de VARs esttico

Desde hace un tiempo se han planteado diferentes esquemas para los compensadores de VARsesttico, encontrndose entre ellos el SVC basado en un condensador fijo y una reactancia cuyacorriente es regulada en ambos semiciclos por medio de dos tiristores conectado en antiparalelo.La estructura de este tipo de SVC es mostrada en la figura 2.

La funcin primaria de este SVC es controlar la potencia reactiva y estabilizar el voltaje delsistema. La seal auxiliar U es aplicada a la entrada del controlador del SVC y al sistema deexcitacin de la mquina para amortiguar las oscilaciones del sistema cuando aparecenperturbaciones. El SVC est equipado con un regulador de tensin el cual incrementa el torquesincronizador. En general, la contribucin al torque amortiguador con slo el regulador detensin es pequea. Si se quiere amortiguamiento adicional, se requiere una accin de controlsuplementario, que en este caso es realizada por el controlador PI y el trmino washout paraevitar que opere en rgimen permanente. La seal usada como entrada es la velocidad, peropueden usarse otras tales como la potencia elctrica, la frecuencia o una combinacin de variasseales.

3. METODO PROPUESTO

Se parte del modelo linealizado del sistema plateado inicialmente por HeffronPhillips y luegopor DeMello-Concordia, en le cual se calculan un conjunto de constantes K1 hasta K6 las cualesdependen de los parmetros del sistema y del punto de operacin.

Figura 2 Modelo del SVC planteado

3

Los criterios para el calculo de K1 hasta K6 son dados en el captulo 2 y algunas aplicacionesse muestran en los captulos 2 y 8.

3.1 MODELO DEL SVC

El modelo usado para el SVC es el mostrado en la figura 2, y el punto de operacin de estadoestable es dado por

S tI BV= (1)

La ecuacin (1) se linealiza alrededor de un punto de operacin quedando:

)2(00 BVVBI ttS +=

La seal de entrada al sistema de control del SVC est expresada como:

)3(sIti IGVUV =

Partiendo de las ecuaciones (2) y (3) y planteando el resto con el modelo de la figura 2, se tiene:

Figura 3 Modelo linealizado del sistema sin SVC

4

)7()(

)6(

)5(

)4(

43'

'21

'65

=

++=

+=

+=

KEKE

VKUKVKE

EKKT

EKKV

fdq

refAAtAfd

qe

qt

Reemplazando (7) en (4)

)8(63)5( 643 fdEKKKKKKtV +=

Haciendo Vref=0 en (8):

)9(11

UK

KK

KV

c

c

c

dt +

++

=

con

63

6435

KKKKKKKKK

Ac

d

==

donde

c

c

c

dQ K

Ky

KK

H++

=11

t Q DV H H U = + (10)

Reemplazando (10) en (3):

[ ] )11()1(1)1( UBoGHBVtGBoGHV IDoIIQi +++=

Ecuacin del SVC

{ })13(

)1()]1(1[

)1()1(

)]1(1[)1()12(

UVtGK

BoGHKVtGK

BoGHKB

UBoGHBVtGBoGHKBVKB

oIr

IDr

oIr

IQr

IDoIIQr

ir

++

++

+=

+++==

5

Simplificando:

8

9 9

1 KU BK K

= + (14)

reemplazando la ecuacin (6) en la (7){ }3' A A AE q K K Vt K U K = +

Ahora de la ecuacin (14)

83

9 9

1' (A A AKE q K K Vt K B K

K K

= + +

de (10)por tanto

BK

HKKK

KKK

KKK

KHKHKKE DAAADAQAq ++= )()](['

9934

9

8

9

83

entonces

BKKE q += 1110'

donde

)(][][

]([

111021

1121021

49

8

9

8310

BKKKKTBKKKKKT

KK

KKK

KHKHKKK

e

e

ADAQA

++=

++=

+=

donde ahora

12 13

8

9 9

(16)

( ) (17)

e

D DQ

Q D

T K K BK H HVt H B

K KVt H H U

= +

= + +

= +

Despus de manipular el conjunto de ecuaciones se llega a:

6

)49()48(1312

BwCVBKKT

t

e

+=+=

La ecuacin (48) representa una nueva formula para expresar el torque electromagntico. Elprimer trmino representa el coeficiente del torque sincronizador, el cual debe ser positivo paraasegurar la estabilidad del sistema. El segundo trmino representa el grado al cual un cambioen la susceptancia del SVC puede causar una relativa aceleracin de la mquina. Entre ms altosea este factor mejor la estabilidad del sistema. En la figura 4 se modelan las ecuaciones (49)y (49) mediante un conjunto de ecuaciones diferenciales de segundo orden. Este tambin es unaversin modificado del modelo mostrado en la figura 1 pero teniendo en cuenta los efectos delSVC. En el anexo 1 se muestra el desarrollo realizado para la obtencin de las ecuaciones (48)y (49).

Los criterios de diseo para obtener los parmetros del controlador PI, Kp y Ki son dados enel captulo 5.

4. Simulaciones y resultados

Se muestran los resultados obtenidos de realizar la simulacin del sistema propuesto bajodiferentes condiciones de operacin. Para todos los casos analizados se tiene que a los 20segundos se aplica una perturbacin en la referencia de tensin al incrementarla de 1 a 1.05.

Posteriormente a los 30 segundos se produce un variacin fuerte de carga del 50%. Asimismo

Figura 4 Modelo linealizado resultante obtenido al iniciar el SVC

7

Figura 5 Respuesta de la tensin sin y con SVC

se muestran los resultados para la simulacin lineal del sistema y luego del sistema no lineal,puesto que en este ltimo se reflejan mejor con respecto a la realidad los resultados obtenidos.

En la figura 5 se muestran los resultados obtenidos para la tensin en el barraje del generadorsin y con SVC. Cuando no se tiene SVC el sistema trabaja slo con el regulador automtico detensin, por lo que cuando se presenta una perturbacin el AVR acta, pero se siguenpresentando oscilaciones. Cuando se utiliza el SVC las oscilaciones desaparecen, pero apareceun sobrepaso de de elevada amplitud y corta duracin que refleja en cierta forma el hecho deque estos elementos introducen armnicos en el sistema y cuyos efectos deben tenerse en cuentapara posteriores estudios. Para esta misma situacin se muestra tambin, en la figura 6, larespuesta del ngulo de par para ambos casos

8

Figura 6 Respuesta del ngulo del par sin y conSVC

Figura 7 Respuesta de la tensin en la simulacin no lineal

9

Figura 8 Respeusta del ngulo del par en la simulacin no lineal

La diferencia que se observa en los valores de rgimen permanente muestran que la inclusindel SVC ampla los lmites del estabilidad del sistema puesto que la excursin que realiza esteparmetro ante una perturbacin es menor.

Se realiz luego la simulacin no lineal del sistema puesto que en esta se refleja de mejormanera los efectos de las perturbaciones y de las acciones de control que se apliquen.

Para este caso se aplica a los 20 segundos una variacin en la referencia y a los 30 una variacinde la carga.

Inicialmente se muestra el comportamiento de la tensin sin y con SVC en la figura 7 donde seobservan de nuevo los efectos benficos del uso del SVC al hacer el sistema menos oscilatorioy con tensin ms regulada.

En la figura 8 se muestra el comportamiento del ngulo del par para la simulacin no lineal antelas mismas perturbaciones.

De nuevo se observa de ella los efectos benficos del uso del SVC.

10

5. Conclusiones

Se ha mostrado el uso de un compensador de VARs esttico para el control de reactivos yampliacin de los lmites de estabilidad del sistema. El SVC usado consta de un condensadorfijo y de una reactancia cuya corriente es regulada por dos tiristores , lo que garantiza que elSVC se comporte como una fuente o un sumidero de reactivos de acuerdo a las exigencias delsistema en el cual esta conectado. Se observaron mejoras en cuanto a la regulacin de tensiny tambin se observ disminucin en las oscilaciones cuando se presentaron perturbaciones. Deacuerdo los resultados obtenidos se puede tambin concluir que una de las desventajas al usarestos dispositivos es que el contenido armnico en el sistema se incrementa, al utilizar laconmutacin de dispositivos de estado slido.

7. Datos

Se utilizaron los siguientes datos

Punto de operacin: P+jQ=0.6+j0.2 Vt=1.025

Mquina sincrnica:

Xd=1.930, Xq=1.77, Xq=0.5, Xd=0.230, Rs=0 H=3.74 D=1.0 w=377 Td0=5.2 s, Tq0=0.81

AVR: IEEE tipo 1 rpido

Ka=100 Ta=0.05

Lnea de transmisin:

Re=0.0 Xe=0.81

SVC:

Gi=0.5 Kr=10 Tr=0.15 B0=0.6 Bc=1.326

Kp=14 Ki=7

Constantes del modelo linealizado de la mquina para el punto de operacin definido

K1=0.64 K2= 0.75 K3= 0.38K4=1.28 K5=0.121 K6=0.63

CINVESTAV Centro de Investigacin y de Estudios Avanzados del I.P.N.

Unidad Guadalajara

Metodologas TCAD para disear diodos epitaxiales de recuperacin rpida de silicio

usando una estructura con contacto tipo mosaico P+/N+

Tesis que presenta: Hector Eduardo Aldrete Vidrio

para obtener el grado de:

Maestro en Ciencias

en la especialidad de: Ingeniera Elctrica

Director de Tesis

Dr. Juan Martn Santana Corte Dr. Juan Luis del Valle Padilla

Guadalajara, Jal., Junio del 2002.

Unidad Guadalajara

Anlisis del STATCOM trifsico en estado estacionario y dinmico para la estabilidad

de voltaje

Jos Luis Murillo Prez

Maestro en Ciencias

Ingeniera Elctrica

Dr. Juan Manuel Ramrez Arredondo

Guadalajara, Jalisco, Noviembre de 2005.

Metodologas TCAD para disear diodos epitaxiales de recuperacin rpida de silicio

usando una estructura con contacto tipo mosaico P+/N+

Tesis de Maestra en Ciencias Ingeniera Elctrica

Por: Hector Eduardo Aldrete Vidrio

Ingeniero en Comunicaciones y Electrnica Universidad de Guadalajara 1992-1996

Becario del CONACyT, expediente no. 143876

Director de Tesis Dr. Juan Martn Santana Corte Dr. Juan Luis del Valle Padilla

CINVESTAV del IPN Unidad Guadalajara, Junio del 2002.

Anlisis del STATCOM trifsico en estado estacionario y dinmico para la estabilidad

de voltaje

Tesis de Maestra en CienciasIngeniera Elctrica

Jos Luis Murillo PrezIngeniero Electricista

Instituto Tecnolgico de Morelia 1998-2003

Becario de CONACYT, expediente no. 182449

Dr. Juan Manuel Ramrez Arredondo

CINVESTAV del IPN Unidad Guadalajara, Noviembre de 2005.

Dedicatoria

Con todo mi cario, admiracin y respeto: A mis Padres

Mara de Jess Prez Quiroz Guillermo Murillo Ponce

A mis Hermanos

Gabriel Murillo Prez Guillermo Murillo Prez

Ya que por ellos siempre he tenido grand

anÁlisis comparativo entre el pss y el statcom en el

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