static var converters

8/10/2019 Static VAR Converters

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267

where 0 is the angle hetween the voltage and the current vectors.

Clearly, only that component

of

the current vector which is in phase

w i t h t h e i n s t a n t a n e o u s v o l t a g e v e c t o r c o n t r i b u t e s t o t h e

instantaneous power. The remaining current component could be

r e m o v e d w i t h o u t c h a n g i n g t h e p o w e r a n d t h i s c o m p o n e n t i s

therefore the instantaneous reactive current. The se observations can

be extended to the following definit ion of instantaneous reactive

power:

Q = f I ~ I

ilsin(0)

( 4 )

where the constant 312 is chosen

so

that the definition coincides with

t h e c l a s s i c a l p h a s o r d e f i n i t i o n u n d e r b a l a n c e d s t e a d y - s t a t e

conditions.

- _

(Vdsiqs vqsids)

Figure 4shows how further manipulation of the vector coordinate

frame leads to a

useful

separation of variables for power control

purposes. A new coordinate system is defined where the d-axis is

always coincident with the instantaneous voltage vector and the q-

axis is in quadrature with i t . The d-axis current component, id .

accounts for the instantaneous power and the q-axis current, i is the

instantaneous reactive current.

The

d,q axes

are

n o t s t a t i o n s i n h e

plane. They follow the trajectory

of

the voltage vector, and the d,q

coordinates within this synchronously-rotating reference frame are

given by the following tim e- vq ing transformation:

Isz

1

I C 1 1

= f

IC11,

and substituting in ( I ) we obtain

Under balanced steady state conditions the coordinates

of

the voltage

and current vectors in the synchronous reference frame are constant

quanti t ies . This feature is useful for analysis and for deco upled

control of the two current components.

Equivalent Circuit and Fqations

Figure

5

show s a simplified repre sentation of the ASVC, including a

dc-side capacitor, an inverter, and series inductance in the three lines

connecting to the transmission line. This inductance accounts for the

leakage of the actual power transformers. The circuit also includes

resis tance in shunt with the capacitor to represent the switching

losses in the inverter, and resistance in series with the ac-lines to

rep resen t the inver te r and t rans fo rmer conduct ion lo s ses. The

inverter block in the circuit is treated as an ideal , lossless powe r

transformer.

In terms of the instantaneous variables shown in Fig.

5,

the ac-side

circuit equations can be written as follows:

7 )

where p.

=

dldt, and a per-unit system has been adopted according to

the following definitions:

O L

L = b B

.

C ' = L

. R r = A

..=A

base b bas e base ase

w c z '

e

i =

;

v =

; e '= ; = base

base

ase

base i

base ase

i

( 8 )

Using the transformation of variables defined in

3,

quations (7)

can be transformed to the synchronously-rotating reference frame as

follows:

where

w =

dQ1dt. Figure

6

illustrates the ac-side circuit vectors in the

synch ronous f rame. When i i s pos i t ive , the ASVC i s d rawing

inductive vars from the line, an% for negative i' it is capacitive.

Types Of Voltage-Sourced Inverter

Neglecting the voltage harmonics produced by the inverter, we can

write a pair of equations for e i and

e '

q

e

= kvLCcos(m) ( 1 0 )

e

= k v; ic s in ( o) ( 1 1 )

where k is a factor for the inverter which relates the dc-side voltage

to the amplitude (peak) of the phase-to-neutral voltage at

the

inverter

ac-side terminals, and 01 is the angle by which the inverter voltage

vector leads the line voltage vector. It is important to distinguish

hetween two basic types of voltage-sourced inverter that can

be used

in ASVC systems.

Inverter Type I allows the instantaneous values of both 01 and k to be

vane d for control purposes. Provided that vi c is kept sufficiently

high,

e i

and

e

can be independently controlled. T his capability can

he achieved b4y various pulse-width-modulation (PW M) techniques

that invariably have a negative impact on the efficiency, harmonic

content,

or

utilization of the inverter. Type I inverters are presently

considered uneconomical for transmission-line applications and their

co nm l will only he briefly considered here.

Inverter T ype I1 is of primary interest for transmission line A SVCs.

In this case, k is a constant factor, and the only available control

input is the angle,

01, of

the inverter voltage vector. This case will be

discussed in greater detail.

Inverter Type I Control System

Inspection of

(9)

leads directly to a rule that will provide decoup led

control of ih and i i . The inverter voltage vector is controlled

as

follows:


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268

Substitution of (12) and (13) in (9) yields

Equation (14) shows that nd . espond to x1 and x2 respectively

through a simple first order trans & function. with no cross-coupling.

The control rule of

(12)

and

(13)

s thus completed by defining the

feedback loops an d proportional plus integral compensa tion as

follows:

x

- (k +

5 )

( i f * -

i

(15)

1 a p . d d )

The control is thus actually performed using feedback variables in

the synchronous reference frame. The reactive current reference,

v,

s supplied from the ASVC outer-loop voltage control system,

and the real power is regulated by varying iJ

in

response to error in

the dc-link voltage via a proportional plus integral compensation. A

block diagram of the control scheme is presented in Figure 7.

Further Model Develooment For Inverter T y ~ e Control

For Type II nverter control it is necessary to include the inverter and

dc-side circuit equation into the model. The instantaneous power at

the ac- nd dc-terminals of the inverter is equal, giving the following

power balance equation:

v&ci&c=

2

(e?

+

e ' i ' )q

(17)

and the &-side circuit equation is

Combining (9), (10). (11). (17) and (18). we obtain the following

state equations for the

ASVC:

p . [ l = [ A ]

li]-l i'i

- R ' W

ko

dc

o b c o s ( = )

b

L

L'

[ A I

=

Steady state solutions for (19) using typical system parameters

are

plotted in Fig. 8 as a function of a (subscript 0 denotes steady state

values.) Note that ';lovaries almost linearly with respect to

a@

and

t h e r a n g e o f

mo

fo r one per un i t swing in i ' o i s very smal l .

Neglecting losses (i.e.

R;

= 0, R; = m) the ste&y state solutions

would

be

as ollows:

= o b ; i = i

u o

o

; ih0 = o ; v -

4

qo

; I V ' I

=

b

ihOL 20)

dcO - E [

vb

Linearization

of

ASVC Fauations f or Small Perturbations

The AS VC state equations (19)

are

non-linear if a s regarded as an

input variable. We can, however, find useful solutions for small

deviations about a chosen steady state equil ibrium point . The

linearization process yields the following perturbation equations:

Standard frequency domain analysis can

be

used to obtain transfer

functions from (21). Numerical methods have been used to obtain

specific results, but it is useful to first consider some general results,

neglecting the system pow er losses (i.e. R;

=

0,

R; =

-.) For this

case, the block diagram of Fig.

9

shows how the control input,

Aa

influences the system states. The corresponding transfer function

relating

A';l

and Aa is

as

follows:

AI' 5 )

~ - * [ 3 ' L ~ - l v & ~ ~ +

~ c w

i

s I s 2 + o2 + L C 1

( 2 2 )

- b q o

_

Au(s)

ko 3kw

C'

L

=

.

C'.

=

L '

The undam ped poles of the system are hus at

t .

s

= 0

and

s = - ]ob 11 t 3klC

L '

(23)

The transfer function, (22), also has a pair of complex

zeroes

on the

imaginary axis. These move along the imaginary axis as a function

of ';lo.occuring at lower frequency than the poles only when

2 V '

(24)

cO i,

3kC

q O X

ho

A numerical computation of AI$s)/Aa(s) from (21), including the

losses, has been done for two operating points to i l lustrate the

movement of the complex zeroes. Figure 1Oapresents the result

for

each case in a plot

of

log gain and phase vs. frequency.

Case 1: Full Capacitive Load

;lo= -1.01 pa., a0 = -0.011 rad (0.63 )

9 -

')

2893 (3t8.7tj1330) (st8.7-jl330)

- -

A-

5 )

(st23.8) ~t15.4 +j1476) st15.4- 1476)


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269

Case 2: Full Inductive Load

;lo= 1.07

p a . , 0

=

0.01 rad (0.57 )

' () 2111(~+ 11.4+j 1557) s+11.4-j1557)

- - -

Ao(s) (s+23.8) (s+15.4+j1476) (s+15.4-j1476)

While Case 1 is amenable to feedback control , Case 2 clearly has

little phase margin near the system resonant frequency. The latter

si tuation is typical for the co ndit ions O>

;lox

(

= 0.44

p.u. in this

example.) A controller has been designed to overc ome this problem

by using non-linear state-variable feedback to improve the phase

margin when

' '0

> ox The non-linear feedback function, Aq', has

the following?ormb

qo - ibex 1.Av

25)

q - g.[ i

dc

q, = Ai

where

g

is a gain factor to be set by design. Figure 10b shows the

transfer function, AQ(s)/A a(s), for the same operating points a s Fig.

loa, with g

= 2.0.

The improved phase margin in Case 2 is clearly

seen. The control scheme block diagram is shown in Fig. 11,with

the additional integral compensation required to obtain zero steady

state error in This scheme has been implemented in the ASVC

s c a l e d m o d e w i t h a c l o s e d - l o o p c o n t r o l b a n d w i d t h s e t t o

approximately 20 0 rads. This makes i t possible to swing between

full inductive mode and full capacitive mode in slightly more than a

quarter of a cycle.

y

L I N E V O L T A G E U N B A L A N C E A N D H A R M O N I C

DISTORTION

With balanced sinusoidal line voltage and an inverter pulse-number

of

24 or

greater, the ASVC draws no low-order harmonic currents

from the l ine. However, harmonic currents of low order d o occur

when th e l ine vo l t age i s unba lanced

or

d is tor t ed . As migh t be

e x p e c t e d f r o m a n o n - li n e a r l o a d , t h e A S V C c u r r e n t s i n c l u d e

harmon ics no t p resen t in the l ine vo l t age . I t

is

impor tan t to

understand ASV C behaviour under these condit ions s ince i t can

influence equipment rating and component selection.

The ASV C harmonic currents can be calculated by postulating a set

of harmonic voltage sources in series with the ASVC tie l ines as

shown in Fig. 12. If we further neglect losses (Le. RI =

0,

R;1

=

m)

and as sume the s t eady s t a te cond i t ion , 01 =

0

and w = wb. then

equations (19) are modified as follows:

( 2 6 1

wher e vid . v i are the d ,q components of the harmonic voltage

vector. E qua tsns (26) are inear and can be solved using Laplace

transforms. Consider the effect of a single balanced harmonic set of

order , n, whe re negative values of n denote negative sequence. The

associated harmonic voltage vector has magnitude, v; and rotates

with angular velocity nob. In the synchronous reference frame i t

rotates with angular velocity (n-l)% as shown in

Fig. 13

and

vhq=

v;

sin((n-1)o

t)

2 7 )

These sinusoidal inputs on the d- and q-axes give rise to sine wave

responses ihd. i

,

and vidc of frequency (n-l)wb. Generally i i d

and ii do not form a balanced two -phase sinusoidal set. They can

be re A ve d into a positive sequence set and a negative seque nce set

using normal two-phase phasor symmetrical components. We thus

find

two dist inct current component vectors in response to the n-

order harmonic voltage vector. Within the synchronous reference

9

frame, these rotate with frequency (n-l)wb and (l-n)wb respectively.

The corresponding ASVC line currents have frequency nub and

(2-n)wb respectively. Note also that the inverter develops an

a l t e rna t ing vo l t age componen t o f f requency (n - l )o b a t i t s dc

terminals.

Equat ions (26 ) and (27 ) have been so lved to ob ta in a lgeb ra ic

expressions for the magnitudes of these harmonic currents in the

particular case where n = -1 (Le., fundamental negative sequence

voltage.) In this case the ordinals of the harmonic currents are 1 and

3 and the magnitudes

are

calculated from the following:

V'

lijl

=

4L

l

-

2LI

kzC

( 2 8 1

( 2 9 1

These expressions have be en evaluated using typical parameters with

v i

= 1

PA., and are plotted ag ainst per-unit capacitive reactance in

Fig. 14. Notice that for

C' =

2L'/k2 both

iL1

and

i j

become infinite.

This condition occurs if the second harmonic of the line frequency is

equal to the ASVC-resonant-pole frequency defined in equation

(23).

Also when C'= 8L'/k2, ill is

zero

and the ASVC draws no negative

sequence fundamental current from the line.

EXPERIMENTAL RESULTS FROM ASVC SCALED MODEL

It is beyond the scope

of

this paper to discuss the

EPN

ASVC scaled

model in detail. However, two Sets of measured waveforms from the

model are presented in Figs. 1 5 and 1 6 to i l lustrate th e system

behaviour under transient conditions. Figure

15

shows the dynamic

response of the instantaneous reactive current controller. In this case

a square-wave reference, ;1*, is injected, and the oscillogram shows

i i , a nd the ASVC l ine cu rren t s . F igu re 16 shows the ASVC

response to a simulated transient unbalanc ed fault. In this case the

full ASV C control system is functional and i '* come s from the

system voltage controller. Initially, the ASVCqis supplying 1 p.u.

capac i t ive vars to the line . A phase- to -neu t ra l fau l t , l as t ing

approximately 5 cycles, is simulated and the oscillogram shows the

associated ASV C currents. Note that the reactive current reference,

i y s l imited in magnitude

to

2 p.u.

CONCLUSION

There

is

every indication that ASVCs will be an important part of

powe r transmission system s in the future . A sound analytical basis

has now been established for studying their dynamic behaviour. The

mathemat ica l m odel der ived here can read i ly be ex tended to

represent the ASVC in broader system studies. The ASVC analysis

has also led to control system designs for both Typ e I and Type

I1

v o l t a g e - s o u r c e d i n v e r t e r s . T h e T y p e I 1 i n v e r t e r c o n t r o l i s

par t i cu la r ly s ign i f i can t because it makes i t poss ib le to ob ta in

excellent dynamic performance from the lowest cost inverter and

transformer combination.

ACKNOWLEDGEMENT

The ASVC scaled model was designed and built at the Westinghouse

Science and Technology Center through the combined efforts of

severa l ind iv idua l s . In par t i cu la r , the au tho rs wou ld l ike to

acknowledge the important contributions made by

Mr.

M.

Gernhardt

and Mr. M. Brennen.


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270

REFERENCES

1.

Gyugyi, L., et al., 1990, Advanced Static VAR Com pensator

Using Gate Turn-O ff Thyristors For Uti l i ty Applications,

CIGRE Paper No. 23-203.

2. Edwards, C.W., et al., 1988, Advanced Static VAR Generator

E m p l o y i n g G T O T h y r i s t o r s , I E E E P E S W i n t er P o w e r

m aper No. 38WM109-1

3.

Hingorani, N.G., 1988, High Power Electronics and Flexible

AC Transmission System, IEEE Power Engineering Review,

J

4.

Electric Power Research Institute, 1990, Development of an

Advanced Static VAR Compensator, Conuact

No.

RP3023-1.

Park , R.H., 192 8, Definition of an

Ideal

Synchronous Machine

and Formula for the Armature

Flux

Linkages, General Elecmc

Review, 31.

Lyon, W.V., 1954, Transient Analysis of Alternating Current

Machinery, John Wiley & Sons, New

Yo ,

USA.

5.

6.

\\ -ESE

+qs-AXIS

i

B- xis)

bt

''''\\\\\\\,,,,,&V *

'\

iqs

'\ v4s

_ _ _ _ _ _ _ _ _ _ _ _

ids Vds +ds- mS

Figure 3.

Definition of Orthogonal Coordinates. (A-axis)

+C-PHASE

AXIS

Vector Representation of Instantaneous

Three-phase Variables.

Figure

1.

Figu re 6.

ASVC Vectors in Synchronous Frame.

25

FIFTH HARMONIC

Example of Vector Trajectory.

J '

Figure

2.

+ds-axis

*-axis)

Figu re

4.

Definition of Rotating Reference Fra me

VOLTAGE

INVERTER

iC

Figure

5.

Equivalent Circuit Diagram for ASVC

+d- AXIS


6/7

27

1

3.0

2.5-

2.0-

1.5-

p

E

R

0.5-

Figure

7 .

Block Diagram of Inverter Type I Control

I

I

I

I

I

I

I

I

I

v i c o

L

I

I

I

-3.0

I

I

R+

=

0.0

I R, = 10C

I = 1.P

un =

377

4

= 377 I

A

vdc

Figure

9.

Small Signal Block Diagram Showing

Dynamic Behaviour of

ASVC

System with

Type I1 Inverter.

40

10

B

-10

-20

' (1)

apacit ive

Rp

= lOO.O/k

D -90

E

G

E (2) nduct ive

s

-270 IQO

=

1.07

0

100 2

300 4

FREQUENCY

(HZ)

Figure loa. Transfe r Function AI' (s)/Aa(s).

I

Figure lob. Transfer Function AQ'(s)/An(s).

R e a c t i v e

CONTROL

c u r r e n t

I

l e

r e f e r e n c e

Iia

Iic

Figure 11.

Block Diagram for Inverter Type I1 Control


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272

Figure 12. ASVC Equivalent Circuit with Harmonic

Voltage Sources.

Figure

13.

Harmonic Vectors in the Synchronous

Reference Frame.

k

=

4/x

=

-

L =

0.15

p.u.

R,=

0

PER

UNIT DC-LINK

CAPACITANCE (C')

Figure 14.

ASVC Current Components with

Fundamental Negative Sequence Voltage

on

the Line.

I I I I I I I I ~ ~ ~

0 3 6.4 96

128

160 192

T i m e ( m s )

Figure 15.

Measured Transient Response of Reactive

Current Control System.

I , , , I , I

0

32 61 96

130

160 193

ASVC System Response to Line to Neutral

Fault.

Time ( ins )

Figure 16.

static var converters

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