[B] Advanced SVC models for NewtonRaphson.pdf

8/17/2019 [B] Advanced SVC models for NewtonRaphson.pdf

1/8

IEEE TRANSACTIONS

ON POWER SYSTEMS. VOL.

15. NO.

1,

FEBRUARY

2000

129

Advanced SVC Models for Newton-Raphson Load

Flow and Newton Optimal Power Flow Studies

H. Amhriz-PBrez,

E. Acha, and C.

R. Fuerte-Esquivel

Abstract-Advanced load flow models for the static VAR cum-

pensator (SVC) are presented in this paper. The models are incor-

porate d into existing load flow

(LF)

and optimal power flow (OPF)

Newton algorithms. Unlike SVC models available in open litera-

ture, the new models dep art from the generator representation of

the SVC an d are based instead on the variable shunt susceptance

concept. In particular, a SVC model which uses the firing angle

as the state variable provides key information for cases when the

load flow solution is used t o initialize other powe r system applied-

tions, e.g., harm onic analysis. The SVC state variables a re com-

bined with the nodal voltage magnitudes and angles

of

the net-

work in

a

single frame-of-reference fur

U

unified, iterative solution

through Newton methods. Both algorithms, the L F and the

OPF

exhibit very strong convergence characteristics, regardless of net-

work size and the nu mb er of controllable devices. Results ar e pre-

sented which demonstrate the prowess of the new SVC models.

Index Terms-FACTS, Newton method, OP F, SVC, voltage con-

trol.

I.

INTRODUCTION

N ELECTRIC power systems, nodal voltages are signifi-

I antly affected by load variations and by network topology

changes. Voltages can drop considerably and even collapse

when the network is operating under heavy loading. This

may trigger the operation of under-voltage relays and other

voltage sensitive controls, leading to extensive disconnection

of loads and thus adversely affecting consum ers and company

revenue.

On

the other hand, when the load level in the system

is low, over-voltages can arise due to Ferranti effect. Capac-

itive over-compcnsation and over-excitation of synchronous

machines can also occur [ I ] . Over-voltages cause equipment

failures due to insulation breakdown and produce magnetic

saturation in transformers, rcsulting in harmonic generation.

Hence, voltage magnitudc throughout the network cannot

deviate significantly from its nominal value if an efficient and

reliable operation of the power system is to bc achieved.

Voltage regulation is achieve d by controlling the production,

absorption and flow of reactive power throughout the network.

Reactive power flows are minimized so as

to

reduce system

losses. Sources and sinks of reactive power, such as shunt ca-

pacitors, shunt reactors, rotating synchronous condensers and

SVC’s are used for this purpose. Shunt capacitors and shunt

Manuscript received

December

8, 1997; revised Septem ber

30,1998.H. Am-

briz-P6rez and C. R. Fue rte-Esqoivel were financially supported hy the Consejo

Nacional de Ciencia y Tecnologiil,MCxica.

H. Ambriz-PCrw and E. Acha

are

with the Department of Electronics and

Electrical Engineering, University of Glasgow, Scotland, UK.

C .

R. Fuerte-Esquivel is with the Departamentu de Ingenieria EICctrica y

Electrhica,

Institute

Tccnol6gico

de

M ar ch , Mexico.

Publishcr

Item Identifier

S

0885-8950 110)01862-9.

reactors are either permanently connected to the network, or

switched on and off according to operative conditions. They

only provide passive compensation since their productionlab-

sorption of reactive power depe nds on their ratings and local bus

voltage level. Conversely, the reactive power sup plieda bsorb ed

by rotating synchronous condensers and SVC’s is automatically

adjusted, attempting to maintain fixed voltage m agnitude at the

connection points.

This paper focuses on the development of new SV C models

and their implementation in Newton-Raphson load flow and

op-

timal power flow algorithms. The SVC is taken to be a con-

tinuous, variable-shunt susceptance, which is adjusted in order

to achieve a specified voltage magnitude while satisfying con-

straint conditions.

Two models are presented in this paper:

I SVC total susceptance model. A changing susceptance

R,,, represents the fundamental frequency equivalent

susceptance of all shunt modules making up the SVC.

This model is an improved version of SVC models

currently ava ilable in ope n literature.

2) SV C firing angle model. The equivalent susceptance B e ,

which is function of a changing firing angle (Y, is made

up of the parallel comb ination of a thyristor controlled re-

actor (TCR ) equivalent admittance and a fixcd capacitive

susceptance. This is a new and more advance d SVC rep-

resentation than those that are currently available in open

literature. This model provides information on the SVC

firing angle required to achieve a given level of co mpen-

sation.

The SV C models have been tested in a wide range of power

networks of varying sizes.

Io

this paper, an actual power network

consisting of

26

generators, 166 buses, 108 transmission lines

and 128 transformers is used as the test network.

11. STATIC

AR

COMPENSATOR‘S

QUIVALENT SUSCEPTANCB

Advances in power electronics technology together with so-

phisticated control methods m ade possible the development of

fast SVC‘s in the early 1970’s. The SVC consists

of

a group of

shunt-connected capacitors and reactors bank s with fast control

action by means of thyristor switching. From the operational

point of view, the SVC can be secn

as

a variable shunt reac-

tance that adjusts automatically in response to changing system

operative conditions. Depending on the nature of the equivalent

SVC ’s reactance, i.e., capacitive or inductive, the S VC draw s

either capacitive or inductive current from the network. Snit-

able control of this equivalent reactance allows voltage magni-

tude regulation at the SV C point

of

connection. SVC’s achieve

0885-8950/00 10.00 2000 IEEE


2/8

IBSS 1IlANSACTIONSO N

POWER

S Y S W M S ,

VOL.

I S , NO.

I ,

FEBRUARY 20110

: 0 0

\

XC

f i

-

'

A . ,

90 100 110 120 1 140 150 160 170 180

Firing angle

(degrees)

Fig.

I . SVC

stmctiirc.

60

40

E

6

-

20

4

o -

r

+

-20

.f -40

c

n

0

c

-

LT

-

-

-

-

-

-

Reactive region

J

-60 1 ,

,

,

,

,

,

,

,

90 100 110 120 130

Capacitive

region

40 150 160 170 180

Firing angle (degrees)

Fig.

2.

their main op erating Characteristic at the expen se of generating

harmonic cu rrents and filters are employed w ith this kind of de-

vices.

SVC 's normally include a combination of mechanically con-

trolled and thyristor controlled shunt capacitors andreactors

[U ,

[Z].

he most pop ular configuration for continuously controlled

S VC' s is the combination of either

fix

capacitor and thyristor

controlled reactor or thyristor sw itched capacitor and thyristor

contro lled reactor

[3], [41. As

far as steady-stale analysis is con-

cerned, both co nfigurations can be modeled along similar lines.

The SVC structure shown in Fig.

I

is used

to

derive a SVC

model that considers the TCR firing angle

Y

as state variablc.

This is a new and more advanced SV C representation than those

currently available in open literature.

The variable TCR equivalent reactance,

X.ce,,

at fundamental

frequency, is given by [31,

S V C equivalent

reactancc

as function of firing angle

wherc Y is the thyristor's firing angle.

allel combination

of X


3/8

A M BRIZP kREZ

c t n l . :

ADVANCE0 SVC

MOUE1.S

FOR

NEWTON-RAPHSON L O A D FLOW

AND NEWTON

OPTIMAL POWER FLOW STUDIES

131

v 4 112r

1 1 0 -

108 -

? 106-

'

04

-

e

g

100

1 0 2 -

System

reactive

characteristics

I

enerator

model /

--- S us c ep t onc e

model

//

/

,

-

-

Capacitive

I

ratine

r

Inductive

rating

b

s

IMlN

0

IM4X

Fig. 4.

characteristic.

Comparison between actual

a n d

idcalized

SVC voltage c~irrcnt

changing the SVC representation to a fixed reactive suscep-

tance. This combined model yields accurate results. However,

both representations require a different number of nodes.

The generator uses two or three nodes [4] whereas the fixed

susceptance uses only one node [ I ] . In Newton-Raphson load

flow implementations this may require Jacobian reordering and

redimensioning during the iterative solution. Also, extensive

checking becomes necessary in order to verify whether or not

the SVC can return to operation inside limits.

It mu st be remarked that for operation outside limits, it is im-

portant to model the SVC as a susceptance and not as a gener-

ator set at its violated limit, Qv i o l i L t c c ~ . Ignoring this point will

lead to inaccurate results. The reason is that the amount

of

reac-

tive power drawn by the S VC is given by the product

of

the fixed

susceptance, nf iXed and the nodal voltage magnitude, V ,Since

V,

is

a

function of network operating conditions, the am ount of

reactive power drawn by the fixed suscepta nce model may differ

from the reactive power drawn by the generator model, i.e.

Qviolalcd

f - Rfixcdvb2

(4)

This point is exemplified in Fig. 5 where the reactive power

output of the generator was set at 100MVAR. This value is con-

stant as it is voltage independent. Th e result given by the con-

stant susceptanc e model varies with nodal voltage magnitude. A

voltage range of 0.95-1 .OS was considered. A susce ptance value

of

1

p a . o n a 100MVA base was used.

The SVC m odel presented above, based on the generator and

fixed susceptan ce representations, is better handle d as

a

suscep-

tance model only. It takes the form

of a

variable susceptance

when the SVC is operating within reactive limits and it takes

the form

of

a fixed susceptance otherwise.

Moreover, a new and more advanced SVC representation then

becomes possible, wherc the thyristor firing angle me chanism is

represented explicitly as a function of network opera ting condi-

tions. In contrast

to

all SVC models proposed heretofore, this

model assesses ranges of firing angle operation leading to in-

trinsic, internal SV C resonances. Both representations, the total

susceptance and the firing angle models, are presented below.

-

Fig. 5.1.

Variable shunt susceptance.

IV. NEW SVC LOAD

FLOWMODELS

In practice the SVC can be seen

as an

adjustable reactance

with either firing angle limits or reactance limits. The circuit

shown

in

Fig. 5.1 is used

to

derive the SV C's nonlinear power

equations and the linearized equations required by Newton's

method.

In general, the transfer admittance equation for the variable-

shunt compensator is,

/ =

jov, (4)

and the reactive power equation is,

Q , = v;u.

3

Linearized, positive sequence SVC models for

Newton-Raphson load flows are presen tcd in this section

whereas SVC models optimal power flows are presented in

Sc ctio n V.

A . SVC Total Suscnptance Model

H = BSVC;)

total susceptance BSVC s taken to be the sta te variable,

The l inearized equation ofth e SV Cis given by (6),wherc the


4/8

I12 IEEE TRANSACTIONS ON POWhK SYSTEMS, VOL IS.

NO

I , rEBRUARY 2000

At the end of iteration i the variable shunt susceptance Hsvt:

i q updated according to (7),

This changing susceptance represents the tolal SV C susceptance

necessary to iiiaintsin the nodal voltage magnitude at the spec-

ified value.

Once the level of compensation has been determined, the

firing angle requircd to achievc such compensation level can

be calculated. This assumes that the SV C is represented by the

slriicture shown in Fig. 1. Since the S VC susceptance given hy

(3) is a transcendental equation, tlie coinpulation of the tiring

angle valirc is dctcrmincd by iteration.

R. SVC Firing Ang le Model U

=

De, )

Provided the SV C can be represented by the structure shown

in Fig. 1, it is possihle to consider the firing angle to be the state

variable. In this casc, the lincarized SVC equa tion is given as,

where

At the end oC iteration i lhc variahle firing angle is updated

according to

I

0),

(10)

t l

~

i t i

4-

ACV‘

and the new SVC susceptancc 1Lq is calculated from (3).

It should be remarked that both models, the total susceptance

model and the tiring anglc tnodcl obscrve good numerical prop-

erties. However, the Cortncr model requires of an additional it-

crativc procedure, after the load flow solution has converged, to

determine the tiring angle. These models were developed to sat-

isly similar requirements, hut different parametcrs are adjusted

during the iterative process. Hence, heir inathcniatical formu-

lalions arc quite different. Accura te information about the SV C

tiring angle, as given by the load tlow solution, is of paramount

important in harmonics and clcctromagnetic transients studies

~51 .

C. Nodal Voltage Magnitude Controlled

by

SVC

Th e implemeiitation of the variahlc shunt susceptance models

in a Newlon-Raphson load

flow

program has required the in-

corporation of a nonstandard type of bus, namcly PVB This

is a controlled bus where the nodal voltage inagnitiide and ac-

tive and rcactive powers are specified whilst either the SVC’s

firing anglc (I or the SVC ’s total susceptance Usvc are handled

as s tate variables. If a or l l s ~ ~ ~ ~rc within limits, the specified

voltagc magnitude is atlained and the coiitrolled bus remains

I’VD-type. However, i i

CY

or Dsv,: go out of limits, they are

fixed at the violatcd lim it and tlie bus becom es’ PCJ-type. Th is

is, of course, i n the absence

of

other FACTS devices capable o l

providing reactive powe r support.

D. Revision @SVC Limits

The revision criterion of SVC limits is based mainly on the

rcactive power misma tch values at controlled buses. SVC limits

revision starts just after the reactive power at the controlled node

is lcss than a specified tolerance. Hcrc this value wa s set at

p.u.

If the SV C violates limits, the SV C’s state variable is fixed at

the offending limit. In this case the node is changed from

PVB

to PQ type. In this situation the SVC will act as an unregu-

lated voltage compensator whose productionhibsorption reac-

tive power capabilities will be a function of the nodal voltage at

the SVC point of connection.

At the end of each iteration, after the network and SVC’s state

variables have been updated, the voltage magnitudes of all nodes

transformed from I’VU to I’ type are revised. The purpose

of this exercise is to check whether or not it is still possible

to maintain the SVC’ s firing angle or S VC’ s total susceptance

fixed and to check whether 01’not the node has changed back to

thc original P V R type without exceeding a or

Bsvc:

limits. The

nodal voltagc magnitudes of the converted nodes are compa red

against the specified nodal voltage ma gnitudes.

Th e node is recoiivcrted to

P V R

type if any of the following

conditions are satisfied:

I) The SV C violates its upper n

or

Dsvc limit and the ac-

tual nodal voltage magnitude is larger than the specified

voltage.

2) The SVC violates its lower (Y or Usvc limit and the ac-

tual nodal voltage magnitude is lower than ‘the specified

voltage.

After the node is returned to

P V B

type, the voltage magni-

tude is fixed at the original target value.

E.

Control Coordination Between Reactive Sources

In cases when different kinds of reactive power sources are

set to control voltage magnitude at the sam e node, they have to

be prioritized in order to have a single control criterion. Syn-

chronous reactive sources have been chosen to be the regulating

components with the highest priority order, holding all other re-

active power so urces fixed at their initial condition

so

long as the

synchronous Renerators and condensers operate within limits.

After these rotating sources start to violate reactive limits, they

arc fixed at the violated limit and auother kind

of

reactive powe r

source, e.g., SV C’s, are activated. In this case, the node is trans-

formed from PV type to PVH type.

v. L O A D FLOW

EST ASE

An ohject oriented, Newton-Raphson load flow program

[61 has been extended in order to incorporate the models and

methods presented above. A real life power network, consisting

of

166 buses,

108

transmission lines and

128

transformers [7],

has been used to show the capab es of the proposcd SVC

models.

Three SV C’s have been embedded in key locations of th e net-

‘work, resulting in significant improvement of the voltage m ag-

nitude profile. The SV C’s were set to control nodal voltage mag-

nitude at

I p.u.

The relevant SVC’s parameters are given in

Section 11.


5/8

133

A M R K I ZPER EZ eta .: ADVANCE0 SVC MODELS FOR NF.WTON~RAP1iSON

OAD

Fl.OW AND NEWTON OPTIMAL POWER PLOW STUDIFS

static

VAR

susceptance

Compensator Model

Firing angle

model

SVCl

s v c 2

s v c 3

L

1 2

3

4

lteroiions

Fig. 6 .

Power mismatches as function of number

of

iterationb.

In order

to

compare the virtues and limitations

of

the new

SV C total susceptance model and the S VC firing anglc model

load flow solutions were carried out using both models. The

number of iterations required by th e load flow to converge war

5

iterations in all cases. The same final values of

SVC's

total

reactance required to achieve the specified nodal voltage m a g

nitude contr ol were arrived at. Th ese values are shown in Tablc

I

Bsvc @.U,)

a

degrees) Bdp.u.)

0.2378 136.038 0.2378

0.0848 136.016 0.0848

0.1709 136.02 8 0.1709

state variables are combined with the

AC

nctwork nodal state.

variables for a unified, optimal solution via Newton's method.

The S VC state variables are adjusted antoinatically

so

as

to

sat-

isfy specified power flows, voltage magnitudes and optimality

conditions

as

given by Kuhn and Tuckcr

[ I l l .

The first stcp in finding thc optimal power flow solution is

to

build a Lagrangian function corresponding to the active and

reactive power flow mismatch equations at every nodc, SVC

nodes included. The contribution of the SVC

to

the Lagrangian

functio n is cxplicitly modeled in

OPF

Newton's method

as

an

equality constraint given by the following equation:

h h ,

A)

= XqrBr ( 1

1

where

e

is the reactive power injected by the SV C at nodc

le

as defined in

5 ) . q e

is the Ltigrange multiplier at node

le

The SVC linearized systcm

of

equations for minimizing the

Lagrangian function via Newton's method is given by

[SI

and

PI,

[1/1

WI[Azl

(12)

where matrix

[ W ]

contains second partial derivatives

of

the Lagrangian function / , h ( V h ,

13, X q k )

with respect to

statc variables

V,, H ,

and

Xq r .

The gradicnt vector

[g]

is

[VVk

VH VAJt .

It consists of first partial derivative

tcrms. [A z] is the vector of corrcction terms, given by

[Al lh AR

Also, [ A z ]

= [ I i -

z] ', where

i is

the

itcration counter.

Once

(12)

has been assembled and combined with matrix

[ W ]

and gradient vector [g] of the entirc network then a sparsity-

oriented soltition

i s

carried out. This process

is

repeated until

a

small, prespecified tolerance is reached.

The Dartial derivativcs, with resoect to the variable shu nt sus-

The behavior of the lnaximum absolute power mismatches, as

function

of

the number

of

iterations,

is

plotted in Fig.

6

for both

cases. straints.

q t :

ice

/?, are function

of

the

svc

State variable to be ad-

justed, i.c.,

Ilsvc:

o r o , in order to satisfy the network con-

VI. N E W S V C MODEL FOR O P TI M A L P O WER FLOW

A . SVC ?btu1Susceptance Model H

=

I ;

J

In this casc, (12) takes the following form, as shown in

13)

at the bottom of the page. The partial derivativcs corresponding

to the SV C are derivcd from

( 5 )

and

I I).

erms written in bold

Linearized, positive sequ ence SVC models suitable for OPF

solutions are described in this section, Similarly to the S VC load

flow model implcmentations described in Section IV, the SV C


6/8

134

Base

C a S e

Losses M W )

13.48

co s t Sh) 264.137

I E E E

TRANSACTIONS

ON POWBK SYSTEMS,

VOL.

15, NO. I. PCDRUARY znnn

Susceptance

Firing

angle

Model model

264.135 264.136

13.47 13.475

- aL - I

aI,

a

BL

i lXi

d L

-~

asi,

-

-

--

a

aL

-_

- a c t -

(14)

Compensator

svc

1

s v c 2

svc3

The relevant partial derivative terms are derived from

5 )

and ( I I). T he derivative terms corrcsponding to ineqtiality

constraints are not required

at

the beginning of the iterative

solution, they are introduced into matrix

(I

)

or

(14)

only after

limits arc enforced. Further explanation is given below.

C. Handling Limits of SVC Controllable Variable

In the O PF forinnlation, voltage m agnitude and active power

limits are included in the inequality constraints set. The inulti-

plier method

1101,

[ I

I ]

s

used to han dle this set. Here, a penalty

term is added

to

the Lagrangian function, which then becomes

the augme nted Lagrang ian function. Variables inside boun ds are

ignore d whilst binding ineqtiality constraints beco me part of the

augmented Lagrangian function and, hence, hecome enforced.

The handling of SVC inequality constraints can be carried out

by using the following generic function 1101,

di (S i (Xj ,

l J i j

3;) i f p i + c g i z ) B ~ )

_

+ , ( S i ( x j - G ) ~

f p i

+

c s; z)

9 . )

2

c

e Model mo del

Bsvc

(P.U.)

a degrees)

Blb

P.U.)

0.1905

136.28 0.1908

0.0754

136.11 0.0751

0.1194 136.18 0.1242

TAULE If

OWIMAI.

GENERATION

OST AND SYSrF.M LOSSES

TABLE Ill

COMPARSION

OFOPTIMAI.

SUSCEPTANCBS FOR BOT?[

SVC MOIX1 S

I Staticvm

~usceptanc

I

Firing angle

I

1

8)

wheregi(x) is thes ta tevariablevalue, i . e. ,

Hsvc

orru , a t theend

of each iteration. 4 and

g

are the maximum and minimum limits

of the SV C state

variable.

p

is

a

multiplier term and

c

is

a pcnalty

term that adapts itself in order to force inequality constraints

within limits while minim ising the objective function.

D. Lugrange Multiplier

The Lagrange multiplier for active and reactive powcr flow

mismatch equations are initialized at

1

and 0, respectively. For

the S VC Lagr angc mu ltiplier the initial valiic of

A

is set equal

to 0.

E. SVC Control of Voltage Magnitu de u t

a

Fixed Value

In O PF studies i t is normal to asstime that voltage mag nitudes

arc controlled within certain limits, e.g.,

0.95-1.10

p.u. How-

ever, if the voltage magn itude is to h e contro lled at a Fixed value,

then matrix [ W ] is suitably modified to reflect this opcrational

constraint. This is done by adding to the second derivative term

of the Lagrangian function with respect

to

the voltage inagni-

tude

Vi

he seco nd derivative term

of

a large, quadratic penalty

factor.

Also,

the first derivative term of the quadratic pcnally

function is evaluated and added to the corresponding gradient

element. Hence, the d iagonal element corresponding to voltage

magnitude

V

will have a very large value, resulting i n a null

voltage increment, A

l/k.

This is equivalent to deactivating the

equation of partial derivatives of the Lagrangian function with

respect to

K:

rom matrix

[I&'].

VI1. OPF

TRST

ASES

The SVC models described above have been implemented

in an OPE To test the robustness of the algorithm, the electric

system described in Section IV-F was used

[7].

The objective

function to be minim ized is mainly active power gen eration cost.

A .

Variable Susceptanc e and Firing Angle M ode ls

The op timal generation cost for the base and mo dified cases

(SVC upgraded) are given

in

Table

11,

together with system

losses. The optimal susceptance values of SVC's are given in

Table

111.

The SV C's w ere embedded in nodes with low voltage


7/8

AMBRIZ-PBRBZ ADVANCED svc

MODEIS

FOR NEWTON-RAPHSON

M A D

FLOW A N D NEWTON

OPTIMAL

POWER FLOW

STUDIES

135

3

3 0 -

YI 2 8 -

3

2 4

a

2 0 .

%

2 6 -

g 2 2 -

1 8 -

'

6 -

1 4

300 -

2

.-

290

-

.

s 280

-

?

2

+

D

t

270

-

260 -

SVC toto1 s us c ep t an r e m ode l

$ 2 5 0 . SVCfir ing

ongle

model

?

-

-

1

2

3 4 5

6

4

2 4 C L ' ' '

0

l terot ions

Fig. 7 .

Active power

generiltion

cos1 p r ~ f i l ~ s

34 1

Base

case

N C otal susceptonce model

. SVC i ir ing

ong le

model

\

---

3 4 5

6

Iterations

1 2 5 r - 7 '

Fig.

X.

magnitudes and,

as

expected, the voltage profiles improved in

such nodes (not shown). SVC voltage magnitudes were sub-

jected to inequality constraints within 0.95-1.10 p a .

In all cases, solutions were achieved in 6 iterations. Fig. 7

shows active power generation cost as a function of the iter-

ation number whilst Fig.

8

shows the active power losses as a

function of the iteration number. Oscillations can be observed in

the cost and losses profiles in the early iterations. This is due to

large variations in both the active set constraint and the penalty

weighting factors during the early iterations.

The SVC control specifications used above werc modified

in order to assess the SVC inodcl capability to control voltage

magnitude at fixed values. In this case the SVC's were set to

control voltage magnitudes at

I

p.u.

The optimal SVC state variable values for both models, i.c.

susceptance a nd firing angle models, are shown in Table IV.

As

expected, these values difler from those given in Table 111.

It is interesting to note that both SVC operating control

modes, i.e. fixed and free, produce similar active power gener-

ation cost and active power losses.

B.

Transmission Lusses M inimization

The algorithm is now applied

to

investigate the problem of

transmission loss tniniinimtion. The active power cost opti-

mization and the transmission losses minimization procedures

Active puwer lasses profiles.

TABLE 1V

COMPARISON

OmlMAL

SUSCEPPANCE FOR BOTH

SVC

MODELS

Firing angle

0.2150

136.32 0.2151

0.0741 136.10 0.0741

0.1376

136.20 0.1377

TABLE

V

OI'TIMAL GENERATION COST A N U SYSTEM LOSSES

Base

Susceptance Firingangle

case Model

model

cost

S i b ) 264.136 264.134 264.134

I 7 475 1 1 A K l

1.018L-12.23 1.021L-12.23

40'00 6 1 4 1 . 7 3 6

+

Fig. 9.

the case when u p p ~imits have

bcen reached.

Comparison

between the g e w m o v and

susceptiince

SVC ,nod for

are carried out sequentially. Transmission

loss

minimization is

amenable to

a

redistribution of the reactive power throughout

the network, which in turn induces changes in the active power

generated by the slack generator. This may then result in an

additional reduction of the active power generation cost. Table

V presents results obtained with the combined optimization

algorithm for the case when the SVC's are not set to main-

tain nodal voltage magnitude at a specified value. It can

be

observed that the second optimization exercise, i.e., network

loss minimization, has only a minor effect on the active power

generation cost. The network losses were reduced in only

0.15%, but a more uniform voltage profile was observed in all

nodes (not shown). The reason for such a small variation in

the network losses is due to the fact that the first optimization

exercise, active power cost optimization, indirectly reduces the

transmission network losses, i.e., large network losses means

more active generation. T he network loss minimization cases

converged in 2 iterations

C

Limitations of the Generator Model

In order to show the limitations of the

generator

representa-

tion

of

an SV C, compared to the two SV C models introduced

in this paper, a cas e is presented below where the SV C hits its

upper reactive limit. SVC 1 was assumed to have a

40

MVAR

upperlimit. Fig.9 shows thereac tive powers contributed by both

models, where the network losses objective function w as m ini-

mized. It can be observed that the reactive powers drawn by bath

SV C models differ. The reason is that the reactive power is not

really constant, as taken to be by the

generator

representation,

but afunction

of

nodal voltage mag nitude, as correctly indicated


8/8

116

IEEE

TRANSACTIONS ON POWER SYSTEMS, VOL

15.

NO. I.

PEBRUARY 2000

by the variable susceptance representation.

As

expected, both

representations have a minimum impact on network

losses,

but

REFERHNCES

111 CIGRB.workine

r o u o

8-01,

Task

F~~~~ ~ .un SVC. -stitticVAR

at the point of SVC connection differences can be observed in

the voltage profile and in the reactive power contributed by both

models.

As discussed in Section 111, the compound generator-con-

stunt

susceptance representation provides an ac curate load flow

represetitation of SVC's for the complete range of operation.

However, inefficiencies may be observed owing to the need to

reorder and to redimension the linearized systems of equations.

In the

OPF

formulation, further complications may arise with

this compound representation. The multiplier method, which

has shown to be very effective in enforcing variables outside

limits, may have difficulties in checking whether or not a S VC

is operating within limits when represented as a constant sus-

ceptancc.

VIII.

CONCLUSIONS

Comprehensive SVC models suitable for conventional

and optimal power flow analysis have been presented in

this paper, namely

SVC total suscepiance

model and

SVC

firing angle model. In contrast to SVC models reported in

the open literature, the proposed models do not make use

of

the

generafor

concept normally employed for the SVC

representation. Instead, they use the variable shu nt susceptance

concept. Arguably, this has the advantage of representing actual

SV C operation more realistically. Moreover, since S VC shun t

susceptance models only make use of one node

to

represent

SVC's operating inside and outside ranges then Newton based

pow er flow solutions becom e more efficient, comp ared to cases

when

generafor

based models of SVC's are used in Newton

algorithms. An S VC mo del which uses the thyristor firing angle

as the state variable has shown to provide fuller information

that existing SVC mo dels. The firing angle required

to

achieve

a specified level of compensation becomes readily available

from the power flow solution, as well as the fundamental

frequency, internal SVC resonant points. A Newton-Raphson

load flow and a Newton's

OPF

algorithms have been upgraded

to incorporate the new S VC models. A real-life, bulk transmis-

sion system has been used as the test case. Conventional and

optimal solutions were obtained in less than 6 iterations.

ACKNOWLEDGMENT

H. Ambriz-PBrez would like to thank Comisi6n Federal de

Electricidad, MBxico for granting him study leave to ca rry out

Ph.D. studies at the University of Glasgow, Scotland, UK.

- .

comnensutors. . I. A. Enrimez.

EL.

1986.

sciencc, 1982.

141

IEEE Special Stability Controls Working Glaup, Working Group

38-01,

Task

Force

No.

2

an

SVC, Static VAR compensator models

for

power

f low

and dynamic

performance

simulation, IEEE T,una.

on

Po owrr

Syr-

tems,

vol.

9, no.

I ,

pp.

229-240, Feb. 1995.

151

J .

1

Rico, E. Acha,

and 1

J H. Miller, 'Harmunic dainain modelling of

three phme

thyristor-cuntmlled ceilctors by means of switching vectors

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July 1996.

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jcct orienled power systems saftwarc for the antilysis of lsrge~scalc et-

works

containing FACTS-controlled branches,

I B m

7mn s .

on Power

S y . ~ l o n s , ol. 13, no. 2,

pp.

4 6 4 4 7 2 , May 1998.

171 E

Aboytes

and

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gitudinal power systems, lEEE

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on P w r r S y s t c m s , vol. PWRS-I,

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May 1986.

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D.

I .

Sun, B.

Ashley. B.

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Hughes,

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W. E

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by

Newton approach, IEEE Tmns. on owerAppapnmtusnnd

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IO

pp.

286G2880, Oct.

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G.

S.

in,

C. 1.

Lin,

and

C. H. Chen, Expericnces

wilh implementing optimal power flow for ceactive scheduling

i n

the

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S),,rremu,

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pp,

1193-1200, Aug. 1988.

I101

D. P.

Bettsekas, Multiplier methods: A

wrvcy,

in

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amonPress,

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vo l . 12, pp. 133-145.

[I 11

D.

G .

Luenberger,

Intmducfirm o Linear and Nonl ineur Pmgramming,

2nd e Addisun-Wesley Publishing Ca., 1984.

It .

Aeha

was

horn

in

Mexico.

He

graduated

from

University

of

Michoilcdn

in

1979 and

rcceiverl the Ph.D.

degrcc from

the University of Canterbury,

Christchurch, New Zealand, in 1988,

He

was a postdoctoral Fellow at the

University of Toronto, Cantidti and the Univcrsity of Durham,

England.

He

is currently

n Senior

Lecturer

ut the

University

01

Glasgow,

Scotland, wherc

hc lectures and conducts mseilrc l i

on

power systems analysis i ~ ndpower

electronics applications. His

research

interests

arc

in

the aceus of

FACTS,

custom

power,

and real-time modeling and iinalysis.

C.

R.Focrte-Esqaivel

was barii i n Mexico in 1964. He received the

B.Eng.

de

gree (Hans) from

Institute

Tecnol6gicn de M a r c h , M e x ic o

i n 1990,

the M S c .

degree

1wm

lnstiluto Politecnico Naciunal,

Mexico i n 1993,

and thc

Ph.D

de-

gree from the U niversity dGlasgaw, Scotland,

UK,

in lY97. He is currently :NI

Assistant

Professorat

thc

Institute

TecnolOgico de M a r c h His main

research

interests lie

oil

the dynamic and steady-state analysis of FACTS, custom

power,

and

real-time modeling and analysis.

Documents

[B] Advanced SVC models for NewtonRaphson.pdf