04 Loadflow Theory

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Loadflow Calculations

Basic Principles and Models

Training Course Documents


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Version: 25. Juli 2003


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Table of Contents

1 Application areas for the loadflow calculations...................................................................

3

2 Calculation Methods............................................................................................................

4

3 Models of Network Elements for Load Flow Calculation.....................................................

3.1 Syncronous !enerator........................................................................................................

"

3.# $%ternal !rid........................................................................................................................

&

3.3 !eneral Load.........................................................................................................................

'

3.( Motor............................................................................................................................. ........

'

3." )*eread+ Transmission Lines.............................................................................................

,

3.- Transformer

............................................................................................................................................ ..

11

4 Loadflow calculation in the transmission s!stem...............................................................

12

(.1 Power Transfer

............................................................................................................................................ ..

1#

(.# cti*e Power Distribution

..............................................................................................................................................

1#

(.3 Loadflow )ptimi/ation

..............................................................................................................................................

1(

(.( 0eacti*e Power Control

..............................................................................................................................................

1(

Loadflow Calculations in Medium "olta#e $!stems %&istri'ution(......................

1

) Loadflow Calculations in Low*+olta#e $!stems...................................................................


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1)

, -eferences..........................................................................................................................

1,


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1

.pplication areas for te loadflow calculations

Loadflow calculations are used to analyse the power system under steady state and unfaulted (short-circuit-free) conditions. he load flow calculates the acti!e and reacti!e power flows for all "ranches# and

the !olta$es for all nodes# in ma$nitude and phase an$le.

he usual parameters $i!en for a loadflow calculation are $enerator dispatch or set points (acti!e andreacti!e power# or acti!e power and !olta$e)# the acti!e and reacti!e power of loads and the impedances

of all "ranches. %f more comple& acti!e and reacti!e power control mechanisms (secondary control#

re$ulated static compensators or '*) are to "e simulated in the loadflow calculation# the setpoints

and control characteristics of these elements also ha!e to "e pro!ided.

+e$ardin$ the areas of application of the loadflow calculation one should differentiate "etween

simulations under ,normal, and ,a"normal, conditions.

his distinction affects the modellin$ of the operatin$ plant. nder normal operating conditions the

$enerator dispatches as well as the loads are nown. %t is sufficient for the calculation to represent these

$enerator dispatches and to pro!ide the acti!e and reacti!e power of all loads. he resultin$ loadflowcalculation should "e a system condition in which none of the "ranch or $enerator limitations are

e&ceeded.

he second area of application# representin$ the abnormal conditions# re/uires a hi$her de$ree of

accuracy from the models. ere it can no lon$er "e assumed that all system plant is operatin$ within

limits. he models must "e a"le to correctly simulate conditions that are de!iatin$ from the normaloperatin$ conditions. 'or e&ample# the reacti!e power limits of $enerators or the !olta$e-dependency

of loads ha!e to "e modelled. 'urther# in many applications# the power "alance cannot "e esta"lishedwith a sin$le slac "us (or machine). %nstead# a more realistic representation of the acti!e and reacti!e

power control mechanisms ha!e to "e considered to de termine a correct sharin$ of the acti!e and reacti!e

$eneration.

he main areas of application for loadflow calculations are:

alculation of "ranch loadin$# system losses and !olta$e profiles for system plannin$ and

operation

(normal and a"normal conditions)

ontin$ency analysis# networ security assessment (a"normal conditions)

1ptimisation tass (minimiin$ system losses# minimiin$ $eneration costs# open tie

optimisation in

distri"uted networs# etc. normal or a"normal condition)

Verification of system conditions durin$ the relia"ility calculations. utomatic determination ofopti-

mal system re-supplyin$ strate$ies. 1ptimisation of load sheddin$ (a"normal conditions)

alculation of steady state initial conditions for sta"ility simulations or short-circuit calculationsusin$

the complete superposition method (usually normal conditions).


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# Calculation Metods

he most important calculation methods for the solution of loadflow pro"lems are descri"ed in detail inthe re-

ferred literature (e.$. 6 2). %t is important for the application that the implemented procedure

determines thesolution fast and relia"ly. *ince the loadflow pro"lem (in contrast to the short-circuit calculation)

represents a

non-linear pro"lem# the solution can only "e determined with an iterati!e process# which may# in some

cases# notlead to a solution.

he main difference "etween the many methods# which are used for loadflow calculations# lies in the

specifica-tion of the nodal e/uations. hese can "e set up usin$ either the ener$y conser!ation law or 7irchhoff8s

law

(current e/uation).

9ependin$ on the application# "oth nodal specification methods offer ad!anta$es# which e&plains why

PowerFactory supports "oth. lthou$h "oth methods normally con!er$e without any pro"lems#

e&perience has shown that the applications of the two methods are "est used as follows:

ower e/uation: Lar$e transmission networs with lar$e an$le de!iations.

urrent e/uation: *ystems# in which the !olta$e and5 or reacti!e power control is pro"lematic# aswell

as systems with power electronics.

o sol!e the system of e/uations# PowerFactory$enerally uses methods that are "ased on a ;ewton+aphson iteration# independently of whether the nodal e/uations are specified as power or currente/uations.

'urther# ower'actory allows for the calculation of "alanced and un"alanced load flows. %n theclassical loadflow calculation system# the un"alances "etween phases are completely ne$lected. 'or theanalysis of transmission networs this assumption is $enerally admissi"le. 'or distri"ution networsthis assumption may "e inappropriate dependin$ on the characteristics of the system6. ll "alancedsystem elements can "e represented "y a sin$le-phase e/ui!alent circuit# which only represents the

positi!e se/uence component of a networ (refer topaper on


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69istri"ution networs in merica or frica $enerally utilise sin$le- or two-phase technolo$ies. 'orsuch systems it is important that the full un"alanced ,three-phase loadflow calculation, is used# whichcorrectly models all phases includin$ (mutual) couplin$s. 1n the other hand# in some central europeandistri"ution systems the "alanced calculation perfectly applies.


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3 Models of etwor2 $lements for Load low Calculation

%n this section the most important aspects of the ;etwor >lement ?odels for loadflow calculations will"e descri"ed. omplete descriptions of the models can "e o"tained from thePowerFactorymanual 3.

hese elements can "e classified accordin$ to their function in the system: *upply# Load andtransmission or 9istri"ution >lements.

he most important power supply elements are:

(synchronous-) $enerators

e&ternal infeeds5 $rids

?ost types of loads are represented with the Load elements:

$eneral loads

motors

'inally# the transmission or distri"ution networ elements consist mainly of

o!erhead lines

transformers

3.1 Syncronous !enerator

3.1.1 $4ui*alent Circuit

s shown in 'i$ure 3-6# a synchronous machine can "e modelled in steady state "y an e/ui!alent !olta$e

source with the synchronous reactance as internal impedance. 'or load flow calculations howe!er# it is

incon!enient to specify the internal !olta$e and phase an$le. he $enerated acti!e and reacti!e power#

or the $enerated acti!e power and the !olta$e at the terminals of the $enerator# are rather specified.

%n 'i$ure 3-6 a re$ulated synchronous machine is shown modelled for a loadflow calculation. %n reality

the acti!e power output of a machine is controlled "y a tur"ine $o!ernor# which ad@usts the tur"ine

power output accordin$ to the speed# electrical power output or some other prime mo!er si$nal (i.e.

e&haust temperature in a $as tur"ine# steam pressure in a steam tur"ine# etc.)

'or the reacti!e power control# it is possi"le to distin$uish "etween !olta$e control and power factorcontrol.

Lar$e machines in power stations# connected to a transmission system# are usually re/uired to supply

reacti!e

power support to the system. he reacti!e power of the machines# in these cases# is usually modified in

order to

control the !olta$e at the terminals of the $enerator (or the hi$h !olta$e side of the $enerator

transformer) to a

preset !alue (primary !olta$e control). *uch $enerators can "e modelled "y a ,V, control characteristic


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(power

and !olta$e control).

Aenerators of small power stations# which feed into a medium !olta$e system# are usually operated with

a constant power factor. he power factor should "e as close as possi"le to unity# so that for a specific

tur"ine power the stator current of the machine is as small as possi"le and thus the $enerator runs at its

optimum point. *uch $enerators can "e represented as


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ondary power control as the "alance machine. *ince this application is only used in transmissionnetwors# it is descri"ed in more detail in section 4.

x I#

U

0

#

d I

U 0 U# U

Qma&

P#

co

s(

)

P#

U

U#U

Qmin

igure 351: *ynchronous $enerator model for the loadflow calculation

3.1.# Capability Cur*e of Syncronous Macines

he permissi"le operatin$ ran$e of synchronous machines is usually defined "y the followin$ limitations:

6. ?a&imum tur"ine power output

2. ?a&imum stator current

3. ?a&imum rotor (e&citation) current

4. *teady state sta"ility limit

5. nder-e&citation limit

o descri"e the operatin$ ran$e of a synchronous machine on a B dia$ram# the output power is defined:

S= P+jQ =3UI=3UI(cos( U

I

)+jsin(

U

I

))

t rated !olta$e and current the output power can "e represented "y a circle in the B dia$ram. he radiusof

this circle is e/ual to the specific output power of the machine:


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S = 3U Ir r r

9ependin$ on the rotor induced !olta$e (L-L) and the load an$le the output power of a synchronous

$enerator can "e determined:

Ur

P= U0

sin(

U 0

U)

XdUr

Q=Xd

(U0

cos(

U 0

U)U

r

)

*ince the rotor induced !olta$e in steady state is proportional to the e&citation current# a third condition

can "e added to the B dia$ram# which represents the tra@ectory of ma&imum rotor induced !olta$e.his tra@ectory is represented "y a circle with the followin$ centre co-ordinates:

P

=

0

Q

=

U

X

2

r

d

D


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and radius:U

r=

U0r r

Xd

he two circles# ma&imum stator current and ma&imum e&citation current cut each other at the nominaloperatin$ point of the machine.

igure 35#6 ypical B dia$ram of a synchronous machine (in relation to rated power output)

'or under e&cited operation it is more difficult to determine the operation limits. 1n the one hand the"order of static sta"ility specifies the limitation# which depends# howe!er# on "oth the parameters of themachine# as well as the surroundin$ system. 1n the other hand# and usually more restricti!e# the under-e&cited operation is limited due to heatin$ of the end re$ion of the armature "y eddy-current losses.

;e!ertheless# in an o!er-e&cited operatin$ condition parasitic effects also e&ist# which result in the actualB

dia$ram de!iatin$ from the analytically deri!ed dia$ram. *aturation effects also play a role# as thesynchronous reactance is then not constant# "ut depends on the operatin$ point. 'or salient polemachines the influence of ma$netic anisotropy# which results in a difference "etween the d- and /-a&isreactances# may not "e ne$lected. his results in the circle characteristics descri"ed a"o!e appearin$ as aspiral.

1win$ to these parasitic effects it is customary to specify the operatin$ limits "y e&plicit parameters ofma&i-

mum and minimum reacti!e power# which are determined from measurements and a detailed analysis

of the

desi$n.

3.# $%ternal !rid

n e&ternal $rid represents a reduced# o!erlyin$ system. %n principle it can "e treated lie a $enerator. %t

should"e noted# howe!er# that for the modellin$ of hi$h and medium !olta$e systems# the acti!e and

reacti!e power"alance usually taes place on the transmission le!el# which e&plains why an e&ternal $rid

is usually defined as the


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system#one of the $rids is usually defined as


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3.3 !eneral Load

ll non-motor loads# as well as $roups of non-motor loads or whole su"-systems# for e&ample# medium!olta$e systems as !iewed from a 632V system or a low-!olta$e system !iewed from a medium !olta$e

system# are modelled as a ,$eneral load,.

nder ,normal conditions, it is permissi"le to represent such loads as constant B loads.

nder ,a"normal conditions,# for e&ample durin$ !olta$e collapse conditions the !olta$e-dependencyof the loads should "e considered.

PowerFactory uses an e&ponential model to simulate the !olta$e-dependency:

P+ jQ=

UP

kpu

+Q

U

kqu

0

U0

0

U0

he special cases ,constant power,# ,constant current, and ,constant impedance, can "e descri"ed "y thefollow-in$ choice of the coefficients pu and /u:

6. kpukqu0: Constantacti!e and reacti!epower "oltage independence

P+jQ = P+jQ0 0

2. kpukqu#: Constantacti!e and reacti!e current $inear !oltage dependence

P Q0 0

P+ jQ = U+ j U =3(IjIUU

0

P0 Q0

U0

3. kpukqu%:Constantmpedance&or 'dmittance( Quadratic !oltage dependence

P Q0 2 0 2 2P+jQ = U +j U = 3() j* ) U

U

3.( Motor

20 U2 00

0

+s Xs Xr +r,s

U Xm

igure 3536 >/ui!alent circuit of an asynchronous machine in steady state

%n comparison to synchronous machines# asynchronous machines do not possess an e&citation windin$.

Volta$e or reacti!e power control is thus not possi"le 2. 'or an asynchronous machine it is therefore not

possi"le to specify acti!e and reacti!e power independently# or to control the terminal !olta$e.

1nly for analysis under normal conditions is it sometimes permissi"le to model asynchronous machines


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as constant B loads. he nominal power factor of the machine can then "e used as power factor for theload.

Aenerally howe!er# an asynchronous machine model as shown in 'i$ure 3-3 should "e used. 'rom thee/ui!alent circuit it follows that# in steady state conditions# an asynchronous machine can "e modelledwith a slipdependent impedance. he acti!e power output depends on a specific !alue of the slip. heresultin$ acti!epower flow throu$h the reactance of the machine also influences the reacti!e power needs.

2 n e&ception is the slip controlled asynchronous machine# as used for wind $enerators or small

hydroelectric

power plants. hese machines allow reacti!e power control usin$ a current con!erter# which is connected

to therotor circuit.


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nfortunately the resistance and reactance !alues of machines are usually unnown. ;ormally only theratedpower# power factor# !olta$e# num"er of pole pairs and may"e the ma&imum tor/ue are a!aila"le.

PowerFactoryallows the user to enter @ust these operatin$ !alues and is then a"le to appro&imate thee/ui!alent electrical circuit parameters from this limited information.

3." )*eread+ Transmission Lines

3.".1 $4ui*alent Circuit

+-l X-l

U# *-l,% *-l,% U%

igure 35(6 >/ui!alent circuit for an o!erhead line and ca"le

he sin$le-phase e/ui!alent circuit for a ca"le or o!erhead line is displayed in 'i$ure 3-4. he lineparameters are entered per len$th. he shunt capacitances are distri"uted e!enly at either end of the line.

PowerFactory permits lines to "e di!ided into routes and sections. Line routes are used if further

elements are connected within this route# e.$. loads connected directly to the line. >ach indi!idual line

route is modelled accordin$ to 'i$ure 3-4.

Line sections are used for lines or ca"les# which consist of se!eral portions# each represented "y a

different impedance (for e&ample# a ca"le connected to an o!erhead line). t the transition point from

one portion to the ne&t no other plant is connected# which allows the indi!idual portions to "e lumped into

one model as per 'i$ure 3-4 for the whole line ca"le. he parameters of the e/ui!alent circuit in thiscase result "y switchin$ the indi!idual impedances in series# and the indi!idual capacitances in parallel.

3.".# )*eread line inductances

L6

r

d36

d62

d23

L2

igur

e 35"6

1!erh

ead


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line

L3

'or the calculation of line inductances# "oth the field outside of the conductors# and the field inside theconductors (internal inductance) must "e considered.

he influence of the field outside of the conductors can "e calculated with the help of mpere8s circuital

law:


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.ds=I

%n the case of an infinitely lon$ conductor with circular cross section is this e/ui!alent to:

2x.=I

he !aria"lexdescri"es the radial distance from the centre of the conductor. 'rom this the ma$netic

flu& density alon$ a circle around the conductor can "e calculated:

* =

0. =

0I

2x

+esultin$ from a current in L6# the surface "etween two conductors (arran$ed accordin$ to 'i$ure 3-5) is

in-duced "y the followin$ flu&:

d62

d62 dx d62

=62r*(x)dx=

0

2 r I6=x

0

2 ln r I6

'or the calculation of the total flu the currents in conductor 2 and 3 must also "e considered. %n the

case of a

completely symmetrical arran$ement all partial flu&es of conductor 3 eliminate themsel!es# whiche&plains why

the current in conductor 3 does not influence the flu& "etween L6 and L2. he total flu& therefore is

e/ual to:

0 d62 0 r 0 d62 = ln I+ ln I = ln (II )

62

6

2 r 2

2 d626 2

2 r he inductance related the field e&ternal to the conductors is therefore:

d62$ext=

0

2ln

r

he influence of the ma$netic field inside the conductors can "e considered "y the internal inductance. he

in-

ternal inductance is fre/uency dependent. %n case of a cylindrical conductor# the followin$ low-fre/uency

ap-pro&imation is !alid:

$

=int

0 r

2 4

he total inductance of the symmetrical three-phase line is:

d62$= $int+$ext=

0

2

r

4+ln r

*ometimes# the internal inductance is e&pressed "y the e/ui!alent $eometric mean radius (A?+):

)/+

=re he inductance can now "e e&pressed "y:


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r

4

d62 $=

0

2ln )/+

'or the calculation of the other o!erhead line parameters (resistance# capacitance) reference is made to the

literature (e.$.2).


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3.- Transformer

t(:#

u.

r. x. x$ r$

bm gm u$

igure 35-: >/ui!alent circuit of a two-windin$ transformer

'i$ure 3-C shows the transformer model in the positi!e phase-se/uence system. he e/ui!alent circuit is

descri"ed usin$ the per unit system# which e&plains why no ideal transformer is used to model the!olta$e ratio from the hi$h !olta$e to the low !olta$e le!el.

he sum of the series impedances corresponds to the short-circuit impedance of the transformer. he

series

impedances r.and r$are proportional to the copper losses. he ma$netiin$ current is modelled with

bm# the

iron losses are modelled with conductance gm. %n PowerFactory the !alues of the resistances and

reactances are

determined automatically from the measured short-circuit !olta$e# the copper losses# the no-load current

and the

iron losses.

he ideal transformer on the hi$h !olta$e side represents the tap chan$er# which can "e ad@usted "oth in

ma$nitude and phase an$le. Hy ad@ustin$ the real part of tthe !olta$e or the reacti!e power can "econtrolled. Hy alterin$ the ima$inary part of tthe acti!e power can "e controlled (phase shifter).

'or the tap chan$er the num"er of steps# the !olta$e per step# as well as the an$le of tare defined in the

transformer type parameters. he control mode# as well as the tap chan$er setpoints# can "e indi!idually

specified for each transformer element. 9urin$ a loadflow simulation# the automatic ad@ustment of all

transformer tap chan$ers can $lo"ally "e switched 1; or 1'' (in the loadflow command dialo$ue).


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( Loadflow calculation in te transmission system

(.1 Power Transfer

6#B6 + X 2#B2

U# U%

igure (516 ower transmission o!er throu$h an impedance

he fundamental pro"lem of the transfer of power usin$ alternatin$ current is represented in 'i$ure 4-6. %t

deals with the transfer of acti!e and reacti!e power throu$h an impedance# which is defined "y a resistanceand a reactance. he power on side 2 is calculated as:

S =P+jQ =

where:

3(U

1

U2 6

U

2

2)

6

=6

=++jX

2 2= ) j*

+ +X

9escri"in$ the !olta$es in polar coordinates:

6

U =U e662

U

he power flow on side 2 is: 2

=U

2e

2

P2= 3( )U

U6 2

cos

62 *U U6 2

sin

62 )U2)

2

Q =3( )U Usin *U U cos +*U)2 6 2 62 6 2 62 2

articularly interestin$ are the appro&imated power e/uations for the case of small !olta$e drops and

small an$lede!iations:

2

P 2

3 )U U *U262(

62)

2

2

Q 3()U62 2 2 *U U )

62 2

%n transmission networs the relationship of +I (and thus also of AH) for o!erhead lines and ca"les liesin aran$e of "etween 0.6 and 0.2. his e&plains why# for transmission lines# the !olta$e an$les across a line


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primar-ily depends on the acti!e power flow# and the !olta$e drop across a line depends primarily on the reacti!e

power

flow.

(.# cti*e Power Distribution

'or the analysis of transmission networs# in particular where ,a"normal condition, operations are

re$arded# it is

important to represent the acti!e power sharin$ amon$ $eneratin$ units as close as possi"le to the real or

actualnetwor.

%t is therefore necessary to cate$orie the control processes that# for instance# follow the loss of a lar$epowerstation.

he followin$ description is an e&tensi!e simplification that# in many cases# $i!es reasona"leappro&imations ofthe system "eha!iour usin$ loadflow calculation capa"ilities. he actual system "eha!iour is more

comple in

D

D

1 + jXD


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- 63 -

particular where"y the primary and secondary fre/uency control o!erlap each other. he description

ser!es thepurpose of a "etter cate$oriation and allows a $ood appro&imation of the acti!e power sharin$ at

differentpoints in time.

(.#.1 7nertial Loadflow

%mmediately followin$ a distur"ance# the missin$ power is deli!ered from the inetic ener$y stored in the

rotatin$ mass of the tur"ines. his leads to a deceleration and thus to a decrease of the fre/uency. he

contri"ution of each indi!idual $enerator towards the total additional power re/uired is proportional to its%nertia.

hese relations can "e descri"ed mathematically as follows. 'or each $enerator the followin$ swin$

e/uation

applies:

2

=

PtP

el

n

ssumin$ a completely synchronied system# in which the fre/uency of all $enerators is the same# the

time deri!ati!e of fre/uency# which is due to the total missin$ power# can "e calculated as follows:

=

Ptot

2n

%nsertin$ the fre/uency deri!ati!e into the swin$ e/uation of each $enerator# its acti!e power

contri"ution towards the total re/uired power can "e determined:

2i

P= Pi

(.#.# Primary Controlled

Loadflow

2itot

fter a short time (within a few seconds)# the $o!ernors of units tain$ part of the primary fre/uency

control increase the tur"ine power and dri!e the fre/uency "ac to a !alue close to nominal fre/uency.

sin$ a pure droop control each $o!ernor increases the tur"ine power proportionally to the fre/uency

de!iation. he resultin$ system fre/uency de!iation (') will then "e associated to the system droop. heacti!e power $eneration in the system is di!ided amon$ units accordin$ to the $ain (7) of the indi!idual

primary controllers.

'or each tur"ine# the de!iation of the power output from the dispatch power () is$i!en "y:

P= 3F

ddin$ the power contri"ution or de!iations from dispatch power from all power stations (J tot)#

assumin$ a uniform fre/uency in the entire system# the fre/uency de!iation remainin$ after the

primary control operation can "e descri"ed as follows:

F

=


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Ptot

3i

%nsertin$ this fre/uency de!iation into the e/uation of each indi!idual tur"ine# its acti!e power

contri"ution in a state of primary control is o"tained:

3i

P= Pi

(.#.3 Secondary Controlled Loadflow

3itot

he secondary controlled loadflow is of special importance# since it deals with the power system in an

almost stationary condition. lthou$h a stationary condition scarcely e!er e&ists in a power system# as

the loads are chan$in$ continuously# the secondary controlled condition (system condition after the

operation of a secondary fre/uency control or A) is still /uite o"ser!a"le.

he $oal of the secondary fre/uency control is to reduce the remainin$ fre/uency de!iation that has

resulted from the first few minutes of primary control operation. %n an interconnected power system the

defined power e&chan$es "etween different areas is $enerally also re-esta"lished "y this secondarycontrol.

i


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- 64 -

sually with properly set power e&chan$es "etween areas and enou$h secondary spinnin$ reser!e in allareas#the fre/uency de!iation after the operation of the secondary control should e/ual ero. %n this case# theoutput

power of the primary controlled $enerators should e/ual the output power @ust "efore the distur"ance.1n thecontrary# when there is a deficit of secondary reser!e in an area of the system# the fre/uency will not returnto itsnominal !alue and the machines under primary fre/uency control will still de!iate from their dispatch

power.

he few power stations in each networ one that participate in the secondary control adapt their outputpower in such a way that the acti!e power un"alance is compensated. he missin$ power due to thedistur"ance is thus distri"uted amon$ the secondary controlled $enerators accordin$ to the participationfactors# which are indi!idually set for the secondary controller of each machine.

(.3 Loadflow )ptimi/ation

'rom an economical point of !iew# the power station dispatch resultin$ from the secondary control actionis not necessarily optimal. %t @ust $uarantees that a distur"ance (e.$. a power station loss) will "ecompensated for. %n the lon$er term howe!er# the most fa!oura"le $enerator economic dispatch has to

"e stri!ed for. he re/uired output power has to "e shared amon$ the indi!idual power stations in themost cost effecti!e way.

%n a li"eralised power maret# howe!er# contractually a$reements need to "e adhered to. Aeneratordispatch then o"eys maret rules and not the total system optimisation.

(.( 0eacti*e Power Control

;ot only the acti!e# "ut also the reacti!e power control must "e considered in a loadflow analysis of atransmis-sion networ. he reacti!e power reser!es of synchronous $enerators in transmission networs are used tocon-trol the !olta$es at specific nodes in the system and or to control the reacti!e power e&chan$e withnei$h"ourin$networ ones.

>ach $enerator is e/uipped with a !olta$e re$ulator that controls the $enerator !olta$e. his !olta$e

re$ulator has a !olta$e setpoint# which is set manually# or from the secondary reacti!e power control# sothat the desired !olta$es and reacti!e power flows are attained in the system.

%f the !olta$e of a specific node is controlled from se!eral $enerators# the contri"ution of reacti!e

power fromeach of the $enerators must "e defined. his definition can "e made inPowerFactoryusin$ the o"@ect

,auto-

matic station control, where the reacti!e power participation factors can "e specified for each$enerator.

9urin$ the secondary reacti!e power control# not only the $enerators play an important role# "ut also the

transformer tap chan$ers and switcha"le shunts or re$ulated static compensators. %t is difficult toillustrate the interaction of these control processes in a loadflow calculation.


26/30

he optimal operation of all secondary !olta$e and reacti!e power controllers can "e "est determined

usin$ an optimal loadflow analysis tool.

'or the realistic consideration of secondary reacti!e power controllers in an e&ceptional loadflow

calculation# a method has "een de!eloped that calculates the time constants of the controllers in such a

way that realistic final states of the controllers result.


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- 65 -

" Loadflow Calculations in Medium 8oltage Systems

9Distribution:

he +5I ratio of medium-!olta$e ca"les and o!erhead lines is si$nificantly hi$her compared to those

used in hi$h and e&tra-hi$h !olta$e systems. his ratio normally ran$es "etween 6 and 4. onsiderin$ thewhole distri"ution system# includin$ the transformers in su"stations# the o!erall +I ratio normally is stillless than 6 howe!er. %n medium !olta$e systems the reacti!e power flow and !olta$e drop are thereforestill closely related. he de-couplin$ of the reacti!e power !olta$e drop and acti!e power !olta$e an$lerelationship is howe!er "y far not as pronounced as in hi$h !olta$e systems.

'or the analysis of transmission networs the acti!e powerflow distri"ution and the related $eneratordispatch is of prime concern. 'or distri"ution networs# this is not the case. ere it can "e assumed thatthe main portion ofpower is drawn from a hi$h-!olta$e transmission system. *hould power stations#which feed directly into the distri"ution networ e&ist# these normally run at ma&imum output power anddo not perform primary fre/uency control. %t can "e assumed that# followin$ a distur"ance# the missin$

power is compensated almost completely !ia the connected transmission system.he main focus for loadflow analysis in distri"ution networs is centred on the determination of systemlosses as well as the assessment of "ranch utilisation and !olta$e profiles.

+e$ardin$ the modellin$# the main challen$e lies in a realistic representation of the loads.hronolo$icalchan$es are usually represented "y load profiles# which descri"e the actual load at any time durin$ a day.oallow for weeday or seasonal !ariations# PowerFactoryallows the option of definin$ two-dimensionalload

profiles# where each column of the daily load profile represents the load for a specific weeday or season.

s soon as the load profiles are defined# it is possi"le to calculate a loadflow for any time of the dayweeend

season. Hy runnin$ se/uential loadflows# any !aria"le of interest can "e plotted a$ainst the time of day.hisind of loadflow analysis allows for a !ery accurate estimate of line loadin$s and or !olta$e le!els atspecific

points in the networ.

%n radially operated distri"ution systems the pro"lem often arises that !ery little is nown a"out the actualload-in$s of the loads connected at each mini su"station. he only information sometimes a!aila"le is the total

powerflowin$ into a radial feeder. o "e a"le to still estimate the !olta$e profile alon$ the feeder a load scalin$tool isused. %n the simplest case the distri"ution loads are scaled accordin$ to the nominal power ratin$s ofthe trans-

formers in the mini su"stations. 1f course such scalin$ leads to inaccurate results. Hetter results areo"tained "yusin$ the a!era$e annual load# which could "e determined from the total annual ener$y consumed and

"illed.


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- 6C -

- Loadflow Calculations in Low5*oltage Systems

;ormally the focus for loadflow calculations in low-!olta$e systems is to determine ma&imum "ranchcurrents as well as ma&imum !olta$e drop.

%n low-!olta$e systems the +I ratio is considera"ly lar$er than 6. he !olta$e drop therefore depends

mainly on the acti!e power flow. +eacti!e power flow in low-!olta$e systems are of lesser interest.

he modellin$ of the loads# as for distri"ution systems# represents a ma@or challen$e. %n addition to thetime dependence# a stochastic component is introduced in the low-!olta$e systems# which is usuallye&pressed in the form of a coincidence factor. his considers# that for two connections it is !ery

impro"a"le that "oth are drawin$ ma&imum load at the same time. Kith three connections this is e!en

more unliely etc. he ma&imum load current thus depends on the num"er of connections supplied from

a ca"le.

%n PowerFactorya special low-!olta$e load model is implemented# which is defined "y the num"er ofconnections supplied. %n addition# the load is assi$ned to a load cate$ory# which defines for each

cate$ory the ma&imum load of the connection# as well as the coincidence factor# for an infinite num"erof connections (thus the relationship of a!era$e and ma&imum load).

onsiderin$ the stochastic independent component of low-!olta$e loads# the ma&imum load (dependanton the num"er of connections) can "e descri"ed "y the followin$ formula:

S ma& (n) =ng(n)S ma& (6)

he functiong&n(descri"es the ma&imum coincidence# dependant on the num"er of connections# n. %f aAaussian (normal) distri"ution is assumed for the coincidence# the coincidence functions reads:

g(n)

=g

6

g+

n

his function depends only on the coincidence of an infinite num"er of connections.

1,2

1

0,8

0,6

0,4

0,2


29/30

0

0 10 20 30 40 50 60 70 80 90

100

igure -516 oincidence as a function of the num"er of connections ($inf0.6)

'or the calculation of ma&imum "ranch utiliation as well as the ma&imum !olta$e drop# PowerFactoryuses a

pro"a"ilistic loadflow calculation# which is a"le to calculate "oth ma&imum and a!era$e currents aswell as the


30/30

- 6E -

a!era$e losses. he pro"a"ilistic loadflow calculation used "y PowerFactorycan "e applied to anysystem topolo$y# thus also to meshed low-!olta$e systems.

;ot only can these low-!olta$e loads "e attached to networ nodes# "ut also ar"itrarily to line sections#

which thus reduces the clutter of the sin$le line dia$ram.

& 0eferences

6 H. 1swald. *erec4nung station5rer und quasistation5rer *etriebs6ust5nde in7lektroenergie!ersorgungs8

net6en9 V9>-Verla$# 6GG2

2 J. J. Arain$er und K. 9. *te!enson. Power System 'nalysis?c. Araw

ill# 6GG4 3 9%$*%L>; Am".PowerFactory /anual "#; 2002