AC TRANSMISSIONAC TRANSMISSION

1539pk

Copyright © P. Kundur

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

Performance Equations and Parameters Performance 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

� 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

� 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 a

distance x from the receiving end (see book: page 202)

is:

where

(6.8)

(6.9)

xRCRxRCRe

IZVe

IZVV γγ −−

++

=2

~~

2

~~~

xR

C

R

xR

C

R

eI

ZV

eI

ZV

I γγ −−

−+

=2

~~

2

~~

~

βαγ jzy

yzZC

+==

=

The constant ZC is called the characteristic

impedance and γγγγ is called the propagation constant.

� The constants γγγγ and ZC are complex quantities. The

real part of the propagation constant γγγγ is called the

attenuation constant α, and the imaginary part the

phase constant β.

� If losses are completely neglected,

1539pkACT - 3

)resistance (pure

Number Real==C

LZC

numberImaginary== βγ j

� 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

surge impedance is known as the natural load or surge

(6.17)

(6.18)

xIjZxVV RCR ββ sincos~~

+=

xZ

VjxII

C

RR ββ sin

~cos

~~

+=

1539pkACT - 4

surge impedance is known as the natural load or surge

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

γ

γ

=

=

~

~~(6.20)

(6.21)

� 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 quantity for evaluating and expressing line

performance

� Typical values of SIL for overhead lines:

nominal (kV): 230 345 500 765

1539pkACT - 5

nominal (kV): 230 345 500 765

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

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)

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

β=

β=

θ=

β=

cosV~

lcosV~

E~

R

RS

(6.31)

(6.32)

(6.33)

1539pkACT - 7

where θ = βl. The angle θ 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)θβ

=

θβ

=

cos

xsin

Z

EjI

cos

xcosE~

V~

C

S

S

� As an example, consider a 300 km, 500 kV line with β = 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, that affects the results of Figure 6.5 is β. Since β is practically the same for overhead lines of all voltage levels (see Table 6.1), the results are universally applicable, not just for a 500 kV line.

pu411.0I

pu081.1V

3.22

rads39.00013.0x300

S

R

=

=

=

==θo

1539pkACT - 8

applicable, 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 of capacitive charging current flowing through line inductance.

� known as the "Ferranti effect".

(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

Voltage Voltage -- Power Characteristics Power Characteristics

of a of a Radial LineRadial Line

� Corresponding to a load of PR+jQR at the receiving end, we have

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

� Fig. 6.7 shows the relationship between VR and PR for a 300 km line with different loads and power factors.

The load is normalized by dividing P by P , the natural

*~~

R

RRR

V

jQPI

−=

−+=

*~sincos~~

R

RRCRS

V

jQPjZVE θθ

1539pkACT - 10

The load is normalized by dividing PR by P0, the natural load (SIL), so that the results are applicable to overhead lines of all voltage ratings.

� From Figure 6.7 the following fundamental properties of ac transmission are evident:

a) There is an inherent maximum limit of power that can be transmitted at any load power factor. Obviously, there has to be such a limit, since, with ES constant, the only way to increase power is by lowering the load impedance. This will result in increased current, but decreased VR and large line losses. Up to a certain point the increase of current dominates the decrease of VR, thereby resulting in an increased PR. Finally, the decrease in VR is such that the trend reverses.

1539pkACT - 11

Figure 6.7 Voltage-power characteristics of a 300 km

lossless radial line

Voltage Voltage -- Power Characteristics Power Characteristics

of a Radial Line of a Radial Line (cont'd)(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, and

may lead to voltage instability.

c) The load power factor has a significant influence

on VR and the maximum power that can be

1539pkACT - 12

on VR and 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, VR is very sensitive to variations

in PR.

� For lines longer than 600 km (θ > 45°), VR at

natural load is the lower of the two values which

satisfies Equation 6.46. Such operation is likely

to be unstable.

1539pkACT - 13

Figure 6.8 Relationship between receiving end voltage,

line length, and load of a lossless radial line

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

Connected to Sources at Both EndsConnected to Sources at Both Ends

� With ES and ER assumed to be equal, the following

conditions exist:

� the midpoint voltage is midway in phase between

ES and ER

� the power factor at midpoint is unity

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

line; with PR<P0, both ends absorb reactive power

from the line.

1539pkACT - 14

� Fig. 6.8 (developed for a radial line) may be used to

analyze how Vm varies with PR.

� with the length equal to half that of the actual line,

plots of VR shown in Figure 6.8 give Vm.

Fig. 6.9 Voltage and current phase relationships with ESequal to ER, and PR less than Po

Power Transfer and Stability Power Transfer and Stability

ConsiderationsConsiderations

� Assuming a lossless line, from Equation 6.17 with

x = l, we can show that

where θ = βllll is the electrical length of line and is the

angle by which ES leads ER, i.e. the load angle.

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

(6.51)δθ

sinsinC

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 δ ---- PR relationship for a 400 km

line.

For comparison, the Vm - PR characteristic of the line is

shown in Fig. 6.10(b).

δθ

sinsin

OR

PP =

1539pkACT - 16

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

line transmitting power between two large systems

Reactive Power RequirementsReactive Power Requirements

� From Equation 6.17, with x = l and ES = ER = 1.0, we can show that

� Fig. 6.11 shows the terminal reactive power requirements of lines of different lengths as a function of PR.

� Adequate VAR sources must be available at the two ends to operate with varying load and nearly constant voltage.

( )θ

θδsin

coscos2

C

S

SR

Z

E

QQ

−=

−=

1539pkACT - 17

constant voltage.

General Comments

Analysis of transmission line performance characteristics presented above represents a highly idealized situation

� useful in developing a conceptual understanding of the phenomenon

� dynamics of the sending-end and receiving-end systems need to be considered for accurate analysis.

1539pkACT - 18

Figure 6.11 Terminal reactive power as a function of power

transmitted for different line lengths

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 all

voltage 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

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

1539pkACT - 20

Figure 6.13 Transmission line loadability curve

Stability Limit Calculation for Line Stability Limit Calculation for Line

LoadabilityLoadability

1539pkACT - 21

Figure 6.14 Steady state stability margin calculation

Factors Influencing Transfer of Active Factors 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 considered

because 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

δ = load angle

Φ = power factor angle

The complex power at the receiving end is

Hence,

Similarly,

−+=

−==+=

jX

EjEEE

jX

EEEIEjQPS

RSSR

RSRRRRR

δδ sincos

~~~~~~ *

X

EEEQ

X

EEP

RRSR

RSR

2cos

sin

−=

=

δ

δ (6.79)

(6.80)

1539pkACT - 23

Similarly,

� Equations 6.79 to 6.82 describe the way in which

active and reactive power are transferred

� Let us examine the dependence of P and Q transfer

on the source voltages, by considering separately

the effects of differences in voltage magnitudes and

angles

X

EEEQ

X

EEP

RSSS

RSS

δ

δ

cos

sin

2 −=

=(6.81)

(6.82)

� From Equations 6.79 to 6.82, we have

� With ES > ER, QS and QR are positive

With ES < ER, QS and QR are 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 δ = 0:

0== SR PP

( ) ( )X

EEEQ

X

EEEQ RSS

SRSR

R

−=

−= ,

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 δ = 0

( ) 2

2

XIX

EEQQ RS

RS=

−=−

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

� From Equations 6.79 to 6.82, we now have

� With δ positive, PS and PR are positive, i.e., active

power flows from sending to receiving end

(b) Condition with ES = ER and δ ≠≠≠≠ 0

( )

2

2

2

2

1

cos1

sin

IX

X

EQQ

X

EPP

RS

SR

=

−=−=

==

δ

δ

1539pkACT - 25

� In each case, there is no reactive power transferred

from one end to the other; instead, each end

supplies half of Q consumed by X.

Figure 6.23 Phasor diagram with ES = ER

(b) δ < 0(a) δ > 0

� We now have

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

of the network, then

(c) General case applicable to any condition:

( ) 2

2

22 cos2

sincos

XIX

XI

X

EEEEQQ

jX

EjEEI

RSRSRS

RSS

==

−+=−

−+=

δ

δδ (6.83)

(6.84)

1539pkACT - 26

� The reactive power "absorbed" by X for all

conditions is X I 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.

2

222

2

222

R

RRloss

R

RRloss

E

QPRIRP

E

QPXIXQ

+==

+== (6.85)

(6.86)

Conclusions Regarding Transfer of Active and Conclusions Regarding Transfer of Active and

Reactive PowerReactive Power

� The active power transferred (PR) is a function of voltage magnitudes and δ. However, for satisfactory operation of the power system, the voltage magnitude at any bus cannot deviate significantly from the nominal value. Therefore, control of active power transfer is achieved primarily through variations in angle δ.

� Reactive power transfer depends mainly on voltage magnitudes. It is transmitted from the side with higher voltage magnitude to the side with lower voltage magnitude.

1539pkACT - 27

� Reactive power cannot be transmitted over long distances, since it would require a large voltage gradient to do so.

� An increase in reactive power transfer causes an increase 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.

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 to

power flow analysis techniques

1539pkACT - 28

1539pkACT - 29

1539pkACT - 30

1539pkACT - 31

1539pkACT - 32

1539pkACT - 33

1539pkACT - 34

1539pkACT - 35

1539pkACT - 36

1539pkACT - 37

1539pkACT - 38

1539pkACT - 39

1539pkACT - 40

1539pkACT - 41