Introduction to Power

Systems(ECEG-3176)

Addis Ababa University

Addis Ababa Institute of Technology (AAiT)
School of Electrical & Computer Engineering
Instructor: Awraris Getachew
 Chapter 5
Characteristics and performance of power
transmission
1 lines
Representation of Transmission Lines
● Transmission lines are normally operated with a
balanced three phase load.
● The analysis can therefore proceed on a per phase
basis.
𝐼𝑆 𝐼𝑅

Two Port Network

𝑉𝑆 𝐴 𝐵 𝑉𝑅
𝐶 𝐷

2
Cont’d…
3
 It is convenient to represent the single phase equivalent
of a transmission line by the two-port network, where in
the sending end voltage VS and current IS are related to
the receiving end voltage VR and current IR through A, B,
C and D parameters as:
𝑉𝑆 = 𝐴𝑉𝑅 + 𝐵𝐼𝑅 𝑉𝑜𝑙𝑡𝑠
𝐼𝑆 = 𝐶𝑉𝑅 + 𝐷𝐼𝑅 𝐴𝑚𝑝

● In matrix form:
𝑉𝑆 𝐴 𝐵 𝑉𝑅
=
𝐼𝑆 𝐶 𝐷 𝐼𝑅

3
Cont’d… 4

 A, B, C and D are the parameters that depend on the line

parameters R, L, C and G.
 The ABCD parameters are, in general complex numbers.
 A and D are dimensionless.
 B has units of Ohms and C has units of Siemens.
 The following identity holds true for ABCD constants:
𝐴𝐷 − 𝐵𝐶 = 1
 For symmetrical network A and D are equal

4
Cont’d…
5

 To avoid confusion between total series impedance and

series impedance per unit length, the following notation
is used:
 Series impedance per unit length: 𝑧 = 𝑟 + 𝑗𝜔𝐿 Ω/m
 Shunt admittance per unit length: 𝑦 = 𝐺 + 𝑗𝜔𝐶 𝑆/m
 Total series impedance: 𝑍 = 𝑧𝑙 Ω
 Total shunt admittance: 𝑌 = 𝑦𝑙 S
 Line length in meter: 𝑙 m
 The shunt conductance G is usually neglected for
overhead transmission lines
5
Cont’d… 6
 The parameters of transmission lines which are discussed
in chapter three are uniformly distributed along the lines.
 For lines of short and medium length we can use lumped
parameters with good accuracy.
 For long transmission lines the parameters must be taken
as distributed parameters.
 Because approximating the uniformly distributed
parameters of long lines to lumped parameters results
considerable error.

6
Short Transmission Line (< 80km)
7

 Capacitance may be ignored with out much error if the

lines are less than 80 km long or if the voltage is not over
66 kV.

𝑉𝑆 1 𝑍 𝑉𝑅
=
𝐼𝑆 0 1 𝐼𝑅

7
Cont’d… 8
 The phasor diagram for the short line is shown below for
lagging current.
𝑉𝑠 cos 𝛿𝑠 − 𝛿𝑅 = 𝐼 𝑅𝑐𝑜𝑠𝛿𝑅 + 𝐼 𝑋𝑠𝑖𝑛𝛿𝑅 + 𝑉𝑅
𝛿𝑠 − 𝛿𝑅 is very small
∴ 𝑐𝑜𝑠 𝛿𝑠 − 𝛿𝑅 =1.0
∴ 𝑉𝑠 = 𝑉𝑅 + 𝐼 (𝑅𝑐𝑜𝑠𝛿𝑅 +𝑋𝑠𝑖𝑛𝛿𝑅 )

8
Voltage Regulation
9

 Voltage regulation of transmission lines may be defined

as the percentage change in voltage at the receiving end
of the line expressed as percentage of full load voltage in
going from no-load to full-load
𝑉𝑅 𝑁𝐿 − 𝑉𝑅 𝐹𝐿
% 𝑣𝑜𝑙𝑡𝑎𝑔𝑒 𝑟𝑒𝑔𝑢𝑙𝑎𝑡𝑖𝑜𝑛 = 𝐹𝐿 × 100
𝑉𝑅
 Where:
𝑉𝑅 𝑁𝐿 = magnitude of no-load receiving end voltage

𝑉𝑅 𝐹𝐿 = magnitude of full-load receiving end voltage

9
Cont’d…
10

 For short lines ,at no load, 𝐼𝑅 = 0, 𝑉𝑅 𝑁𝐿 = 𝑉𝑆 and:

𝑉𝑅 𝐹𝐿 = 𝑉𝑅

𝑉𝑆 − 𝑉𝑅
% 𝑣𝑜𝑙𝑡𝑎𝑔𝑒 𝑟𝑒𝑔𝑢𝑙𝑎𝑡𝑖𝑜𝑛 = × 100
𝑉𝑅

10
Cont’d…
11

In the above derivation 𝛿𝑅 has been considered positive

for a lagging load. 𝛿𝑅 Will be negative for leading load
For leading power factor load

11
 Medium Transmission Line
(80 km < l < 250km)
12
 Transmission lines more than 80 km long and below 250
km in length are treated as medium lines, and the line
charging current becomes appreciable and the shunt
capacitance must be considered.
 Medium lines can be represented sufficiently well by R, L
and C as lumped parameters with:
 Half the capacitance to neutral of the line lumped at
each end of the equivalent circuit (π-model) or
 Half of the series impedance lumped at each side of
the line (T- model).

12
Cont’d… 13
 Nominal π-model

13
Cont’d…
13

14
Cont’d…
14

15
Cont’d…
15

 T-model

𝑌𝑍 𝑌𝑍
𝑉𝑆 1+ 𝑍 1+ 𝑉𝑅
= 2 4
𝐼𝑆 𝑌𝑍 𝐼𝑅
𝑌 1+
2
16
Long Transmission Line (> 250 km)
16

For short and medium length lines, accurate models were

obtained by assuming the line parameters to be lumped.
In case of long TLs, for accurate solutions the parameters
must be taken as distributed uniformly along the length

17
Cont’d…
17

18
Cont’d…
18
 If we differentiate again the above equation:

 The real part, 𝛼 is known as the attenuation constant, and

the imaginary part, 𝛽 is known as the phase constant.
19
Cont’d…
19
 For the current:

20
Cont’d…
20

 Next, the integration constants 𝐶1 and 𝐶2 are evaluated

from the boundary conditions.
 When 𝑥 = 0, 𝑉 𝑥 = 𝑉𝑅 and 𝐼 𝑥 = 𝐼𝑅 , from the above
voltage and current equations, we get:
𝑉𝑅 = 𝐶1 + 𝐶2
1
𝐼𝑅 = (𝐶1 − 𝐶2 )
𝑍𝐶
 Solving these equations, we obtain:
𝑉𝑅 + 𝑍𝐶 𝐼𝑅
𝐶1 =
2
𝑉𝑅 − 𝑍𝐶 𝐼𝑅
𝐶2 = 21
2
Cont’d…
21

 Substituting the values of 𝐴1 and 𝐴2 into the voltage and

current equations, we obtain:
(𝑉𝑅 + 𝑍𝐶 𝐼𝑅 ) 𝛾𝑥 (𝑉𝑅 − 𝑍𝐶 𝐼𝑅 ) −𝛾𝑥
𝑉 𝑥 = 𝑒 + 𝑒
2 2
(𝑉𝑅 + 𝑍𝐶 𝐼𝑅 ) 𝛾𝑥 (𝑉𝑅 − 𝑍𝐶 𝐼𝑅 ) −𝛾𝑥
𝐼 𝑥 = 𝑒 − 𝑒
2𝑍𝐶 2𝑍𝐶
 The equations can be rearranged as follows:
(𝑒 𝛾𝑥 + 𝑒 −𝛾𝑥 ) (𝑒 𝛾𝑥 − 𝑒 −𝛾𝑥 )
𝑉 𝑥 = 𝑉𝑅 + 𝑍𝐶 𝐼𝑅
2 2
(𝑒 𝛾𝑥 − 𝑒 −𝛾𝑥 ) (𝑒 𝛾𝑥 + 𝑒 −𝛾𝑥 )
𝐼 𝑥 = 𝑉𝑅 + 𝐼𝑅
2𝑍𝐶 2
22
Cont’d…
22

 Or 𝑉 𝑥 = cosh 𝛾𝑥 𝑉𝑅 + 𝑍𝐶 sinh(𝛾𝑥)𝐼𝑅
1
𝐼 𝑥 = sinh 𝛾𝑥 𝑉𝑅 + cosh(𝛾𝑥)𝐼𝑅
𝑍𝐶
 Our interest is in the relation between the sending end
and the receiving end of the line.
 Therefore, when 𝑥 = 𝑙, 𝑉 𝑙 = 𝑉𝑆 and 𝐼 𝑙 = 𝐼𝑆 . The
result is:
𝑉𝑆 = cosh 𝛾𝑙 𝑉𝑅 + 𝑍𝐶 sinh(𝛾𝑙)𝐼𝑅
1
𝐼𝑆 = sinh 𝛾𝑙 𝑉𝑅 + cosh(𝛾𝑙)𝐼𝑅
𝑍𝐶
23
Cont’d…
23
 Therefore, ABCD constants are:
𝐴 = cosh(𝛾𝑙)
𝐵 = 𝑍𝐶 sinh(𝛾𝑙)
1
𝐶= sinh(𝛾𝑙)
𝑍𝐶
𝐷 = cosh(𝛾𝑙)

24
The Equivalent Circuit of a Long TL
24
 The nominal-π circuit does not represent a
transmission line exactly because it does not account
for the parameters of the line being uniformly
distributed.
 The discrepancy between the nominal-π and the
actual line becomes larger as the length of line
increases.

25
Cont’d…
25
 It is possible, however, to find the equivalent circuit of a
long transmission line and to represent the line
accurately, in so far as ends of the line are concerned, by a
network of lumped parameters.

● Comparing the above equations with the equations in

slide 23 & 24 and making use of identity

26
Cont’d…
26
 The parameters of the equivalent 𝜋 model are obtained as

27
 Power Flow through Transmission Lines

Consider a simple power system as shown below

Equations for power can be derived in terms of ABCD

constants.
The equations apply to any network of two ports.
𝑉𝑠 = 𝐴𝑉𝑅 +𝐵𝐼𝑅
𝑉𝑆 −𝐴𝑉𝑅
𝐼𝑅 =
𝐵

28
 Cont’d…
The complex power leaving the sending end and entering to
the receiving end of the TL becomes(per-phase basis)
1 𝐴
𝑆𝑅 = 𝑃𝑅 + 𝑗𝑄𝑅 = 𝑉𝑅 𝐼𝑅 ∗ 𝐼𝑅 = 𝑉𝑆 − 𝑉𝑅
𝐵 𝐵
𝑆𝑆 = 𝑃𝑆 + 𝑗𝑄𝑆 = 𝑉𝑆 𝐼𝑆 ∗ 𝐷 1
𝐼𝑆 = 𝑉𝑆 − 𝑉𝑅
𝐵 𝐵

Let A,B,D , the transmission line constants be written as

𝐴 = 𝐴 ∠𝛼, 𝐵 = 𝐵 ∠𝛽, 𝐷 = 𝐷 ∠𝛼(𝑆𝑖𝑛𝑐𝑒 𝐴 = 𝐷)
Therefore, we can write
1 𝐴
𝐼𝑅 = 𝑉𝑆 ∠ 𝛿 − 𝛽 − 𝑉𝑅 ∠ 𝛼 − 𝛽
𝐵 𝐵
𝐷 1
𝐼𝑆 = 𝑉𝑆 ∠ 𝛼 + 𝛿 − 𝛽 − 𝑉𝑅 ∠ − 𝛽
𝐵 𝐵

29
 Cont’d…
Substituting 𝐼𝑅 in the equation of 𝑆𝑅
1 𝐴
𝑆𝑅 = 𝑉𝑅 ∠0 𝑉𝑆 ∠ 𝛽 − 𝛿 − 𝑉𝑅 ∠(𝛽 − 𝛼)
𝐵 𝐵
𝑉𝑆 𝑉𝑅 𝐴
= ∠ 𝛽−𝛿 − 𝑉𝑅 2 ∠(𝛽 − 𝛼)
𝐵 𝐵
Similarly
𝐷 2
𝑉𝑆 𝑉𝑅
𝑆𝑆 = 𝑉𝑆 ∠ 𝛽 − 𝛼 − ∠(𝛽 + 𝛿)
𝐵 𝐵
Then the three phase receiving end complex power is given by
𝑉𝑆 𝑉𝑅 × 106 𝐴 𝑉𝑅 2 × 106
𝑆𝑅 3 − 𝑝ℎ𝑎𝑠𝑒 𝑉𝐴 = 3 ∠ 𝛽−𝛿 − ∠(𝛽 − 𝑎)
3× 3 𝐵 𝐵 3

𝑉𝑆 𝑉𝑅 𝐴
𝑆𝑅 3 − 𝑝ℎ𝑎𝑠𝑒 𝑀𝑉𝐴 = ∠ 𝛽−𝛿 − 𝑉𝑅 2 ∠(𝛽 − 𝛼)
𝐵 𝐵
Please see the details from the book modern power system analysis by D.P. Kothari 30
 Cont’d…
From the previous equation the real and the reactive power
at the receiving end becomes
𝑉𝑆 𝑉𝑅 𝐴
𝑃𝑅 = cos(𝛽 − 𝛿) − 𝑉𝑅 2 cos(𝛽 − 𝛼)
𝐵 𝐵
𝑉𝑆 𝑉𝑅 𝐴
𝑄𝑅 = sin(𝛽 − 𝛿) − 𝑉𝑅 2 sin(𝛽 − 𝛼)
𝐵 𝐵

Similarly the real and reactive power at the sending end is

𝐷 2
𝑉𝑆 𝑉𝑅
𝑃𝑆 = 𝑉 cos( 𝛽 − 𝛼) − cos(𝛽 + 𝛿)
𝐵 𝑆 𝐵
𝐷 𝑉𝑆 𝑉𝑅
𝑄𝑆 = 𝑉𝑆 2 sin( 𝛽 − 𝛼) − sin(𝛽 + 𝛿)
𝐵 𝐵

31
 Cont’d…
The received power 𝑃𝑅 will be maximum at
𝛿=𝛽
Such that 𝑉𝑆 𝑉𝑅 𝐴 𝑉𝑅 2
𝑃𝑅 𝑚𝑎𝑥 = − cos(𝛽 − 𝛼)
𝐵 𝐵

The corresponding 𝑄𝑅 (at max 𝑃𝑅 ) is

𝐴 𝑉𝑅 2
𝑄𝑅 = − sin(𝛽 − 𝛼)
𝐵

That means the load must draw this much leading MVAR
in order to receive the maximum real power
 Please try to do Example 5.8 on page 162-165 (Modern
power system analysis by D.P.Kothari)

32
Reactive power Compensation of
Transmission Lines
 Reactive power (VAR) compensation is defined as the
management of reactive power to improve the performance
of ac systems.
 The performance of transmission lines, especially those of
medium length and longer, can be improved by reactive
compensation of a series or parallel type.
 Series compensation consists of a capacitor bank placed in
series with each phase conductor of the line.
 Shunt compensation refers to the placement of inductors
from each line to neutral to reduce partially or completely
the shunt susceptance of a high-voltage line, which is
particularly important at light loads when the voltage at the
receiving end may otherwise become very high. 33
Cont’d…
.

Reactive power Analogue

34
Cont’d…
. Series compensation reduces the series impedance of
the line, which is the principal cause of voltage drop
and the most important factor in determining the
maximum power which the line can transmit.
 The desired reactance of the capacitor bank can be
determined by compensating for a specific amount of
the total inductive reactance of the line.
 This leads to the term "compensation factor," which is
𝑋𝐶
defined by 𝑋𝐿 , where 𝑋𝐶 is the capacitive reactance
of the series capacitor bank per phase and 𝑋𝐿 is the
total inductive reactance of the line per phase.

35
Cont’d…
 When the nominal-π circuit is used to represent the line
and capacitor bank and if only the sending- and receiving-
end conditions of the line are of interest, the physical
location of the capacitor bank along the line is not taken in
to account.
 However, when the operating conditions along the line are
of interest, the physical location of the capacitor bank must
be taken into account.

36
Cont’d…
 This can be accomplished most easily by determining ABCD
constants of the portions of line on each side of the
capacitor bank and by representing the capacitor bank by
its ABCD constants.
 The equivalent constants of the combination (actually
referred to as a cascaded connection) of line – capacitor -
line can then be determined.

37
Ferranti Effect

 During light load or no-load condition, receiving end

voltage is greater than sending end voltage in long
transmission line or cable.
 This happens due to very high line charging current.
 This phenomenon is known as ferranti effect.
 A charged open circuit line draws significant amount of
current due to capacitive effect of the line.
 This is more in high voltage long transmission lines.

38