ac transmission kundur

8/7/2019 AC Transmission Kundur

1/42

AC TRANSMISSIONAC TRANSMISSION

1539pk

Copyright P. KundurThis material should not be used without the author's consent


2/42

Performance Equations and ParametersPerformance Equations and Parameters

of Transmission Linesof Transmission Lines

A transmission line is characterized by four

parameters:

series resistance (R) due to conductor resistivity

shunt conductance (G) due to currents along

insulator strings and corona; effect is small and

usually neglected

series inductance (L) due to magnetic field

surrounding the conductor

shunt capacitance (C) due to the electric field

1539pkACT - 1

between the conductors

These are distributed parameters.

The parameters and hence the characteristics of

cables differ significantly from those of overhead

lines because the conductors in a cable are

much closer to each other

surrounded by metallic bodies such as shields,

lead or aluminum sheets, and steel pipes

separated by insulating material such as

impregnated paper, oil, or inert gas


3/42

For balanced steady-state operation, the performance of

transmission lines may be analyzed in terms of single-

phase equivalents.

Fig. 6.1 Voltage and current relationship of a distributed

parameter line

1539pkACT - 2

The general solution for voltage and current at adistance x from the receiving end (see book: page 202)

is:

where

(6.8)

(6.9)

xRCRxRCR eIZV

eIZV

V

++

=2

~~

2

~~~

xR

C

R

xR

C

R

eI

ZV

eI

ZV

I

+

=2

~~

2

~~

~

jzy

yzZC

+==

=


4/42

The constant ZC is called the characteristic

impedance and is called the propagation constant.

The constants and ZCare complex quantities. The

real part of the propagation constant is called theattenuation constant , and the imaginary part the

phase constant .

If losses are completely neglected,

1539pkACT - 3

)resistance(pure

NumberReal== CLZC

numberImaginary== j


5/42

For a lossless line, Equations 6.8 and 6.9 simplify to

When dealing with lightening/switching surges, HV

lines are assumed to be lossless. Hence, ZC with

losses neglected is commonly referred to as the surge

impedance.

The power delivered by a line when terminated by its

(6.17)

(6.18)

xIjZxVV RCR sincos~~

+=

xZ

VjxII

C

RR sin

~cos

~~

+=

1539pkACT - 4

impedance load.

where V0 is the rated voltage

At SIL, Equations 6.17 and 6.18 further simplify to

wattsZ

VSIL

C

2

0=

x

R

x

R

eII

eVV

=

=

~

~~(6.20)

(6.21)


6/42

Hence, for a lossless line at SIL,

V and I have constant amplitude along the line

V and I are in phase throughout the length of the line

The line neither generates nor absorbs VARS

As we will see later, the SIL serves as a convenientreference quantityfor evaluating and expressing line

performance

Typical values of SIL for overhead lines:

1539pkACT - 5

SIL (MW): 140 420 1000 2300

Underground cables have higher shunt capacitance;

hence, ZC is much smaller and SIL is much higher than

those for overhead lines.

for example, the SIL of a 230 kV cable is about

1400 MW

generate VARs at all loads


7/42

Typical ParametersTypical Parameters

Table 6.1 Typical overhead transmission line parameters

Note: 1. Rated frequency is assumed to be 60 Hz

1539pkACT - 6

Table 6.2 Typical cable parameters

2. Bundled conductors used for all lines listed, except for the 230 kV line.

3. R, xL, and bC are per-phase values.4. SIL and charging MVA are three-phase values.

* direct buried paper insulated lead covered (PILC) and high pressure pipe

type (PIPE)


8/42

Voltage Profile of a Radial Line at NoVoltage Profile of a Radial Line at No--LoadLoad

With receiving end open, IR = 0. Assuming a

lossless line from Equations 6.17 and 6.18, we have

At the sending end (x = l),

( )

( ) ( )xsinZV~jI~xcosV

~V~

CR

R

=

=

=

=

cosV~

lcosV~

E~

R

RS

(6.31)

(6.32)

(6.33)

1539pkACT - 7

where = l. The angle is referred to as the

electrical length or the line angle, and is expressed

in radians.

From Equations 6.31, 6.32, and 6.33

(6.35)

(6.36)

=

=

cos

xsin

Z

EjI

cos

xcosE~

V~

C

S

S


9/42

As an example, consider a 300 km, 500 kV line with

= 0.0013 rads/km, ZC = 250 ohms, and ES = 1.0 pu:

Base current is equal to that corresponding to SIL.

Voltage and current profiles are shown in Figure 6.5.

The only line parameter, other than line length, thataffects the results of Figure 6.5 is . Since ispractically the same for overhead lines of all voltagelevels (see Table 6.1), the results are universally

pu411.0I

pu081.1V

3.22

rads39.00013.0x300

S

R

=

=

=

==o

1539pkACT - 8

The receiving end voltage for different line lengths:

- forl= 300 km, VR = 1.081 pu- forl= 600 km, VR = 1.407 pu- forl= 1200 km, VR = infinity

Rise in voltage at the receiving end is because of

capacitive charging current flowing through lineinductance.

known as the "Ferranti effect".


10/42

(a) Schematic Diagram

1539pkACT - 9

Figure 6.5 Voltage and current profiles for a 300 km lossless

line with receiving end open-circuited

(b) Voltage Profile

(c) Current Profile


11/42

VoltageVoltage -- Power CharacteristicsPower Characteristics

of aof a Radial LineRadial Line

Corresponding to a load ofPR+jQRat the receiving end, wehave

Assuming the line to be lossless, from Equation 6.17

with x = l

Fig. 6.7 shows the relationship between VRand PRfor a300 km line with different loads and power factors.

*~~

R

RRR

V

jQPI

=

+=

*~sincos~~

R

RRCRS

V

jQPjZVE

1539pkACT - 10

e oa s norma ze y v ng R y 0, e na ura

load (SIL), so that the results are applicable to overheadlines of all voltage ratings.

From Figure 6.7 the following fundamental properties of actransmission are evident:

a) There is an inherent maximum limit of power that can betransmitted at any load power factor. Obviously, therehas to be such a limit, since, with ESconstant, the only

way to increase power is by lowering the loadimpedance. This will result in increased current, butdecreased VRand large line losses. Up to a certain pointthe increase of current dominates the decrease ofVR,thereby resulting in an increased PR. Finally, thedecrease in VRis such that the trend reverses.


12/42

1539pkACT - 11

Figure 6.7 Voltage-power characteristics of a 300 km

lossless radial line


13/42

VoltageVoltage -- Power CharacteristicsPower Characteristics

of a Radial Lineof a Radial Line (cont'd)(cont'd)

b) Any value of power below the maximum can be

transmitted at two different values ofVR. The

normal operation is at the upper value, within

narrow limits around 1.0 pu. At the lower voltage,

the current is higher and may exceed thermallimits. The feasibility of operation at the lower

voltage also depends on load characteristics, and

may lead to voltage instability.

c) The load power factor has a significant influence

1539pkACT - 12

transmitted. This means that the receiving end

voltage can be regulated by the addition of shunt

capacitive compensation.

Fig. 6.8 depicts the effect of line length:

For longer lines, VR is very sensitive to variations

in PR.

For lines longer than 600 km ( > 45), VRatnatural load is the lower of the two values which

satisfies Equation 6.46. Such operation is likely

to be unstable.


14/42

1539pkACT - 13

Figure 6.8 Relationship between receiving end voltage,line length, and load of a lossless radial line


15/42

VoltageVoltage--Power Characteristic of a LinePower Characteristic of a Line

Connected to Sources at Both EndsConnected to Sources at Both Ends

With ESand ERassumed to be equal, the following

conditions exist:

the midpoint voltage is midway in phase between

ESand ER

the power factor at midpoint is unity

with PR>P0, both ends supply reactive power to the

line; with PR


16/42

Power Transfer and StabilityPower Transfer and Stability

ConsiderationsConsiderations

Assuming a lossless line, from Equation 6.17 with

x = l, we can show that

where = llll is the electrical length of line and is theangle by which ESleads ER, i.e. the load angle.

IfES= ER= rated voltage, then the natural load is

(6.51)sin

sinC

RSR

Z

EEP =

C

RSO

Z

EEP =

1539pkACT - 15

and Equation 6.51 becomes

The above is valid for synchronous as well as

asynchronous load at the receiving end.

Fig. 6.10(a) shows the ---- PRrelationship for a 400 km

line.

For comparison, the Vm - PRcharacteristic of the line is

shown in Fig. 6.10(b).

sin

sinO

R

PP =


17/42

1539pkACT - 16

Figure 6.10 PR- and Vm-PR characteristics of 400 km lossless

line transmitting power between two large systems


18/42

Reactive Power RequirementsReactive Power Requirements

From Equation 6.17, with x = land ES= ER= 1.0, we canshow that

Fig. 6.11 shows the terminal reactive powerrequirements of lines of different lengths as a functionofPR.

Adequate VAR sources must be available at the twoends to operate with varying load and nearly

( )

sin

coscos2

C

S

SR

Z

E

QQ

=

=

1539pkACT - 17

cons an vo age.

General Comments

Analysis of transmission line performancecharacteristics presented above represents a highlyidealized situation

useful in developing a conceptual understanding ofthe phenomenon

dynamics of the sending-end and receiving-endsystems need to be considered for accurateanalysis.


19/42

1539pkACT - 18

Figure 6.11 Terminal reactive power as a function of power

transmitted for different line lengths


20/42

Loadability CharacteristicsLoadability Characteristics

The concept of "line loadability" was introduced by

H.P. St. Clair in 1953

Fig. 6.13 shows the universal loadability curve for

overhead uncompensated lines applicable to allvoltage ratings

Three factors influence power transfer limits:

thermal limit (annealing and increased sag)

voltage drop limit (maximum 5% drop)

1539pkACT - 19

steady-state stability limit (steady-state stability

margin of 30% as shown in Fig. 6.14)

The "St. Clair Curve" provides a simple means of

visualizing power transfer capabilities of transmission

lines.

useful for developing conceptual guides to

preliminary planning of transmission systems

must be used with some caution

Large complex systems require detailed assessment

of their performance and consideration of additional

factors


21/42

"St. Clair Curve""St. Clair Curve"

1539pkACT - 20

Figure 6.13 Transmission line loadability curve


22/42

Stability Limit Calculation for LineStability Limit Calculation for Line

LoadabilityLoadability

1539pkACT - 21

Figure 6.14 Steady state stability margin calculation


23/42

Factors Influencing Transfer of ActiveFactors Influencing Transfer of Active

and Reactive Powerand Reactive Power

Consider two sources connected by an inductive

reactance as shown in Figure 6.21.

representation of two sections of a power system

interconnected by a transmission system

a purely inductive reactance is consideredbecause impedances of transmission elements

are predominately inductive

effects of shunt capacitances do not appear

explicitly

1539pkACT - 22

Figure 6.21 Power transfer between two sources

(a) Equivalent system diagram

(b) Phasor diagram

= load angle

= power factor angle


24/42

The complex power at the receiving end is

Hence,

+=

==+=

jX

EjEEE

jX

EEEIEjQPS

RSSR

RSRRRRR

sincos

~~~~~~ *

X

EEEQ

X

EEP

RRSR

RSR

2cos

sin

=

=

(6.79)

(6.80)

1539pkACT - 23

,

Equations 6.79 to 6.82 describe the way in which

active and reactive power are transferred

Let us examine the dependence ofPand Qtransfer

on the source voltages, by considering separately

the effects of differences in voltage magnitudes and

angles

X

EEEQ

X

EEP

RSSS

RSS

cos

sin

2 =

=(6.81)

(6.82)


25/42

From Equations 6.79 to 6.82, we have

With ES> ER, QSand QRare positive

With ES< ER, QSand QRare negative

As shown in Fig. 6.22,

transmission of lagging current through an

inductive reactance causes a drop in receiving

end voltage

(a) Condition with = 0:

0== SR PP

( ) ( )X

EEEQ

X

EEEQ RSSS

RSRR

=

= ,

1539pkACT - 24

transmission of leading current through an

inductive reactance causes a rise in receiving

end voltage

Reactive power "consumed" in each case is

Figure 6.22 Phasor diagrams with = 0

( ) 22

XIX

EEQQ RS

RS=

=

(a) ES>ER (b) ER>ES


26/42

From Equations 6.79 to 6.82, we now have

With positive, PSand PRare positive, i.e., active

power flows from sending to receiving end

(b) Condition with ES= ERand

0

( )

2

2

2

2

1

cos1

sin

IX

X

EQQ

X

EPP

RS

SR

=

==

==

1539pkACT - 25

In each case, there is no reactive power transferredfrom one end to the other; instead, each end

supplies half ofQconsumed by X.

Figure 6.23 Phasor diagram with ES = ER

(b) < 0(a) > 0


27/42

We now have

If, in addition to X, we consider series resistance R

of the network, then

(c) General case applicable to any condition:

( ) 22

22 cos2

sincos

XIX

XI

X

EEEEQQ

jX

EjEEI

RSRSRS

RSS

==

+=

+=

(6.83)

(6.84)

1539pkACT - 26

The reactive power "absorbed" by Xfor all

conditions is X I2. This leads to the concept of

"reactive power loss", a companion term to active

power loss.

An increase in reactive power transmitted increases

active as well as reactive power losses. This has an

impact on efficiency and voltage regulation.

2

222

2

222

R

RRloss

R

RR

loss

E

QPRIRP

E

QP

XIXQ

+==

+

== (6.85)

(6.86)


28/42

Conclusions Regarding Transfer of Active andConclusions Regarding Transfer of Active and

Reactive PowerReactive Power

The active power transferred (PR) is a function ofvoltage magnitudes and . However, for satisfactoryoperation of the power system, the voltage magnitudeat any bus cannot deviate significantly from thenominal value. Therefore, control of active power

transfer is achieved primarily through variations inangle .

Reactive power transfer depends mainly on voltagemagnitudes. It is transmitted from the side with highervoltage magnitude to the side with lower voltagemagnitude.

1539pkACT - 27

Reactive power cannot be transmitted over longdistances, since it would require a large voltagegradient to do so.

An increase in reactive power transfer causes anincrease in active as well as reactive power losses.

Although we have considered a simple system, the general

conclusions are applicable to any practical system, In fact, the basic

characteristics of ac transmission reflected in these conclusions

have a dominant effect on the way in which we operate and control

the power system.


29/42

Appendix to Section on AC TransmissionAppendix to Section on AC Transmission

1. Copy of Section 6.4 from the book Power System

Stability and Control

provides background information related topower flow analysis techniques

1539pkACT - 28


30/42

1539pkACT - 29


31/42

1539pkACT - 30


32/42

1539pkACT - 31


33/42

1539pkACT - 32


34/42

1539pkACT - 33


35/42

1539pkACT - 34


36/42


37/42

1539pkACT - 36


38/42

1539pkACT - 37


39/42

1539pkACT - 38


40/42

1539pkACT - 39


41/42

1539pkACT - 40


42/42

ac transmission kundur

Documents