02-4 AC Transmission

8/10/2019 02-4 AC Transmission

1/421539pk

AC TRANSMISSION

Copyright P. Kundur

This material should not be used without the author's consent


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Performance Equations and Parameters

of 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 andusually neglected

series inductance (L) due to magnetic field

surrounding the conductor

shunt capacitance (C) due to the electric field

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


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The constant ZCis called the character is t icimpedanceand is called the propagat ion constant.

The constants and ZCare complex quantities. The

real part of the propagation constant is called the

attenuation constant , and the imaginary part the

phase constant .

If losses are completely neglected,

)resistance(pure

NumberReal

C

LZC

numberImaginary j


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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, ZCwithlosses neglected is commonly referred to as the surge

impedance.

The power delivered by a line when terminated by its

surge impedance is known as the natural load or su rge

impedance load.

where V0 is the rated voltage

At SIL, Equations 6.17 and 6.18 further simplify to

(6.17)

(6.18)

xIjZxVVRCR

sincos~~

xZ

VjxII

C

RR sin

~cos

~~

wattsZ

VSIL

C

2

0

x

R

x

R

eII

eVV

~

~~

(6.20)

(6.21)


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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 convenient

reference q uant i tyfor evaluating and expressing line

performance

Typical values of SIL for overhead lines:

nominal (kV): 230 345 500 765

SIL (MW): 140 420 1000 2300

Underground cables have higher shunt capacitance;

hence, ZCis 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


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Typical Parameters

Table 6.1 Typical overhead transmission line parameters

Table 6.2 Typical cable parameters

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

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

3. R, xL, and bCare 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)


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Voltage Profile of a Radial Line at No-Load

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),

where = l. The angle is referred to as the

electr ic al lengthor the l ine angle, and is expressed

in radians.

From Equations 6.31, 6.32, and 6.33

xsinZV~jI~xcosV

~V~

CR

R

cosV~

lcosV~

E~

R

RS

(6.31)

(6.32)

(6.33)

(6.35)

(6.36)

cos

xsin

Z

EjI

cos

xcosE~

V~

C

S

S


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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. Sinceispractically the same for overhead lines of all voltagelevels (see Table 6.1),the results are universallyapplicable, not just for a 500 kV line.

The receiving end voltage for different line lengths:

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

Rise in voltage at the receiving end is because ofcapacitive charging current flowing through line

inductance.

known as the "Ferranti effect".

pu411.0I

pu081.1V

3.22

rads39.00013.0x300

S

R


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Figure 6.5 Voltage and current profiles for a 300 km lossless

line with receiving end open-circuited

(b) Voltage Profile

(a) Schematic Diagram

(c) Current Profile


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Voltage - Power Characteristics

of a Radial Line

Corresponding to a load of PR+jQRat the receiving end, wehave

Assuming the line to be lossless, from Equation 6.17with x = l

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

The load is normalized by dividing PRbyP0, the naturalload (SIL), so that the results are applicable to overhead

lines 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 onlyway 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 of VR,thereby resulting in an increased PR. Finally, thedecrease in VR is such that the trend reverses.

*~~

R

RRR

V

jQPI

*~sincos~~

R

RRCRS

V

jQPjZVE


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Figure 6.7 Voltage-power characteristics of a 300 km

lossless radial line


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Voltage - Power Characteristics

of a Radial Line (cont'd)

b) Any value of power below the maximum can be

transmitted at two different values of VR. 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 thermal

limits. The feasibility of operation at the lower

voltage also depends on load characteristics, andmay lead to voltage instability.

c) The load power factor has a significant influence

on VRand the maximum power that can be

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, VRis 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.


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Figure 6.8 Relationship between receiving end voltage,

line length, and load of a lossless radial line


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Voltage-Power Characteristic of a Line

Connected 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 theline; with PR


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Power Transfer and Stability

Considerations

Assuming a lossless line, from Equation 6.17 with

x = l,we can show that

where = lis the electrical length of line and is the

angle by which ESleads ER, i.e. the load angle.

If ES= ER=rated voltage, then the natural load is

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).

(6.51)sin

sinC

RSR

Z

EEP

C

RSO

Z

EEP

sin

sinO

R

PP


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

line transmitting power between two large systems


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Reactive 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 functionof PR.

Adequate VAR sources must be available at the twoends to operate with varying load and nearlyconstant voltage.

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.

sin

coscos2

C

S

SR

Z

E

QQ


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Figure 6.11 Terminal reactive power as a function of power

transmitted for different line lengths


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Loadability 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 all

voltage ratings

Three factors influence power transfer limits:

thermal limit (annealing and increased sag)

voltage drop limit (maximum 5% drop)

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


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Figure 6.13 Transmission line loadability curve

"St. Clair Curve"


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Figure 6.14 Steady state stability margin calculation

Stability Limit Calculation for Line

Loadability


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Factors Influencing Transfer of Active

and 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 considered

because impedances of transmission elements

are predominately inductive

effects of shunt capacitances do not appear

explicitly

Figure 6.21 Power transfer between two sources

(a) Equivalent system diagram

(b) Phasor diagram

= load angle

= power factor angle


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The complex power at the receiving end is

Hence,

Similarly,

Equations 6.79 to 6.82 describe the way in which

active and reactive power are transferred

Let us examine the dependence of Pand Qtransfer

on the source voltages, by considering separately

the effects of differences in voltage magnitudes and

angles

jX

EjEEE

jX

EEEIEjQPS

RSSR

RSRRRRR

sincos

~~~~~~

*

X

EEEQ

X

EEP

RRSR

RS

R

2cos

sin

(6.79)

(6.80)

X

EEEQ

XEEP

RSSS

RSS

cos

sin

2

(6.81)

(6.82)


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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

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

(a) Condition with = 0:

0 SR PP

X

EEEQ

X

EEEQ RSSS

RSRR

,

22

XIX

EEQQ RSRS

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


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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

In each case, there is no reactive power transferred

from one end to the other; instead, each endsupplies half of Qconsumed by X.

(b) Condition with ES= ERand 0

Figure 6.23 Phasor diagram with ES= ER

2

2

2

2

1

cos1

sin

IX

X

EQQ

X

EPP

RS

SR

(b) < 0(a) > 0


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We now have

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

of the network, then

The reactive power "absorbed" by Xfor all

conditions is XI 2. 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.

(c) General case applicable to any condition:

22

22 cos2

sincos

XI

X

XI

X

EEEEQQ

jX

EjEEI

RSRSRS

RSS

2

22

2

2

22

2

R

RRloss

R

RRloss

E

QPRIRP

E

QPXIXQ

(6.83)

(6.84)

(6.85)

(6.86)


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Appendix to Section on AC Transmission

1. Copy of Section 6.4 from the book Power System

Stability and Control

provides background information related to

power flow analysis techniques


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02-4 AC Transmission

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