Test: Short Transmission Line

Types of transmission lines:
Depending upon the length of the line, it is divided into three types:

Shunt capacitance is neglected in the analysis of short transmission lines. This is because:
The capacitive reactance is given by:

Since the value of length (l) is small in the short transmission lines, therefore the reactance becomes very high.
High reactance means an open circuit, hence effect of capacitance is neglected in the short transmission lines.

Concept:

The ABCD parameters of the above circuit diagram are 

The ABCD parameters of the above circuit diagram are 

When the networks are connected in cascade, their ABCD parameters will get multiplied.
Calculation:
A short transmission line can be modelled as a series impedance
z = R + jωL
The ABCD constants of this series impedance are

A = D = 1, B = z, C = 0

Classification of Overhead Transmission Lines:

• A transmission line has three constants R, L, and C distributed uniformly along the whole length of the line.
• The resistance and inductance form the series impedance.
• The capacitance existing between conductors for a 1-phase line or from a conductor to neutral for a 3-phase line forms a shunt path throughout the length of the line. Therefore, capacitance effects introduce complications in transmission line calculations.

 
Depending upon the manner in which capacitance is taken into account, the overhead transmission lines are classified as:
Short transmission lines:
When the length of an overhead transmission line is up to about 50 km to 80 km and the line voltage is comparatively low (< 20 kV), it is usually considered as a short transmission line. Due to smaller lengths and lower voltage, the capacitance effects are small and hence can be neglected. Therefore, while studying the performance of a short transmission line, only the resistance and inductance of the line are taken into account.

Medium transmission lines:
When the length of an overhead transmission line is about 80-150 km and the line voltage is moderately high (>20 kV < 100 kV), it is considered a medium transmission line. Due to the sufficient length and voltage of the line, the capacitance effects are taken into account. For purposes of calculations, the distributed capacitance of the line is divided and lumped in the form of condensers shunted across the line at one or more points.

Long transmission lines:
When the length of an overhead transmission line is more than 150 km and line voltage is very high (> 100 kV), it is considered a long transmission line. For the treatment of such a line, the line constants are considered uniformly distributed over the whole length of the line and rigorous methods are employed for the solution

Switching surges are the governing factor in the design of insulation for EHV and UHV transmission lines.

Origins of switching overvoltage:

• Energization of transmission lines and cables
• Switching of high-voltage reactors
• Switching of transformers that are loaded by a reactor on their tertiary winding
• Switching of a transformer at no load
• Switching at intermediate substations
• Fault initiation and clearing

Short transmission lines:

• The effective length of a short transmission line is less than 80 km.
• When the length of an overhead transmission line is up to about 50 km to 80 km and the line voltage is comparatively low (< 20 kV), it is usually considered as a short transmission line. 
• Due to smaller lengths and lower voltage, the capacitance effects are small and hence can be neglected.
• Therefore, while studying the performance of a short transmission line, only the resistance and inductance of the line are taken into account.

Concept:
Short Transmission Line:
When the length of the overhead transmission line is up to about 50 km and the line voltage is comparatively up to 20 kV.
Due to smaller lengths and lower voltage, the capacitance effect is small and may be neglected.
Considered a short transmission line of resistance R and Reactance XL over a length.

V_s is the sending end voltage
V_R is the receiving end voltage
I is the load current.
cos ϕ_R is receiving end power factor
and, cos ϕ_s is sending end power factor
The phasor diagram of the system can be drawn by taking load current as a reference,

From the phasor,
(OC)² = (OD)² + (DC)²
(OC)² = (OE + ED)² + (DB + BC)²
V²S = (V_R cos ϕ_R + IR)² + (V_R sin ϕ_R + IX_L)²  

Calculation:
Given, P = 5500 kW
V_R = 11 kV
R = 10 Ω
X_L = 10 Ω
P = V_R I cos ϕ_R
cos ϕ_R = cos (45°) = 0.707

From above concept,

sin ϕ_R = 0.707
(V_R cos ϕ_R + IR) = (11000 × 0.707) + (707.21 × 10) = 14849 volt
(V_R sin ϕ_R + IX_L) = (11000 × 0.707) + (707.21 × 10) = 14849 volt

⇒ V_S = 21 kV

Short Transmission Line:

• When the length of the overhead transmission line is up to about 50 km  - 80 km and the line voltage is comparatively up to 20 kV.
• Due to smaller lengths and lower voltage, the capacitance effect is small and may be neglected.

Considered a short transmission line of resistance R and Reactance XL over a length.

V_s is the sending end voltage
V_R is the receiving end voltage
I is the load current.
cos ϕ_R is receiving end power factor
and, cos ϕ_s is sending end power factor
The phasor diagram of the system can be drawn by taking load current as a reference,

From the phasor,
(OC)² = (OD)² + (DC)²
(OC)² = (OE + ED)² + (DB + BC)²
V²S = (V_R cos ϕ_R + IR)² + (V_R sin ϕ_R + IX_L)²   

Transmission efficiency (η):
The power obtained at the receiving end of a transmission line is generally less than the sending end power due to losses in the line resistance.

The ratio of receiving end power (Power delivered) to the sending end power (Power supplied) of a transmission line is known as the transmission efficiency of the line.

Hence, η = (Power delivered  / Power supplied × 100)

Concept:
In a single phase transmission line
V_S = V_R + I_L∠ϕ Z
Where,
V_S = Sending end voltage
V_R = Receiving end voltage
I_L = Current flowing through the transmission line
Z = Impedance of line
ϕ = Phase angle
Calculation:
Given-
S_L = 100 kVA
V_R = 2000 V
cos ϕ = 1, ϕ = 0°
Z = R + j X = 1.4 +j 0.8
|I_L|= 100 x 10³ / 2000 = 50 A
I_L ∠ϕ = 50 ∠0°
Now sending end voltage is
V_S = 2000 + 50 ∠0° (1.4 +j 0.8)
V_S = 2000 + 50(cos0° + sin0°)(1.4 +j 0.8)
V_S = 2000 + 70 + j40
V_S = 2070 + j40
V_S = 2070.40∠1.1°
|V­_S| = 2070.40 V

Short Transmission Line:
When the length of the overhead transmission line is up to about 50 km and the line voltage is comparatively up to 20 kV.
Due to smaller lengths and lower voltage, the capacitance effect is small and maybe neglected, and hence Shunt admittance too.
Considered a short transmission line of resistance R and Reactance XL over a length.

V_s is the sending end voltage
V_R is the receiving end voltage
I is the load current.
cos ϕ_R is receiving end power factor
and, cos ϕ_s is sending end power factor
The phasor diagram of the system can be drawn by taking load current as a reference,

From the phasor,
(OC)² = (OD)² + (DC)²
(OC)² = (OE + ED)² + (DB + BC)²
V²S = (V_R cos ϕ_R + IR)² + (V_R sin ϕ_R + IX_L)²  

Voltage regulation: The voltage regulation of transmission line is,
Percentage voltage regulation

Here,
Vs is the sending end voltage
Vr is the receiving end voltage
Voltage Regulation Curve at different Power Factor:

 

From The curve:

• Voltage regulation at the lagging power factor will be more than the voltage regulation at the unity power factor.
• Negative voltage regulation and zero voltage regulation occur at the leading power factor.
• Positive voltage regulation occurs at both unities as well as lagging power factors.