Performance of Transmission Lines

Table of Contents
1. Classification of Transmission Lines
2. Short Transmission Line
3. Medium Transmission Line
4. Long Transmission Line
5. Power Flow On Transmission Line
6. Important Conclusions
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Classification of Transmission Lines

On the basis of the length of the transmission line, transmission lines are classified into three categories:

• Short transmission lines
• Medium transmission lines
• Long transmission lines

1. Short Transmission Line

A transmission line is classed as short when its length is less than about 80 km.

Assumptions

• The shunt capacitance effect is negligible.
• The line parameters can be modelled as lumped parameters (single series impedance per phase).

Equivalent circuit

Under balanced conditions a short transmission line is represented by a single-phase series impedance consisting of resistance R and inductive reactance X_L (or series impedance Z).

Sending-end relations

The sending-end current equals the receiving-end current under series model:

I_s = I_r

The sending-end voltage is the sum of the receiving-end voltage and the voltage drop across the line impedance:

V_s = V_r + I_r Z

Voltage regulation

Voltage regulation of a line is defined as the rise in the receiving-end voltage when the full load at a specified power factor is removed while the sending-end voltage is kept constant.

% voltage regulation =

Where V_s is the sending voltage (or no-load receiving voltage) and V_r is the full-load rated receiving voltage.

From the phasor diagram of the short transmission line, the sending-end voltage can be obtained as the phasor sum of receiving voltage and impedance drop.

Where φ_r is the phase angle at the receiving end.

Voltage regulation =

Notes

• In the above equation φ_r is positive for a lagging power factor load and negative for a leading power factor load.
• The power factor strongly affects voltage regulation: for lagging power-factor loads the regulation is positive; for leading power-factor loads the regulation may be negative.
• Condition for zero voltage regulation: I_r R cos φ_r = I_r X_L sin φ_r or tan φ_r = R / X.

Efficiency

The efficiency of the transmission line is the ratio of power at the receiving end to the power at the sending end.

% efficiency = (Power delivered at the receiving end / Power sent from the sending end) × 100

ABCD parameters (short line)

The sending-end quantities V_s, I_s can be expressed in terms of the receiving-end quantities V_r, I_r by the two-port parameter form:

V_s = A V_r + B I_r

I_s = C V_r + D I_r

• A: Voltage ratio when the receiving end is open (dimensionless).

• B: Transfer impedance; voltage at sending end to produce 1 A at short-circuited receiving end.

• C: Transfer admittance; current at sending end per volt on open-circuited receiving end.

• D: Current ratio for 1 A at short-circuited receiving end.

In matrix form:

• ABCD parameters of short transmission line: A = 1, B = Z, C = 0, D = 1
• Efficiency of short transmission line:

where P is the per-phase power received and R is the resistance per phase of the line.

2. Medium Transmission Line

A transmission line is classed as medium when its length lies approximately between 80 km and 240 km. The single-phase equivalent is modelled by either the nominal-T or nominal-π circuit.

Assumption

• Line parameters are still treated as lumped parameters, but shunt capacitance is no longer negligible.

Nominal-T circuit

In the nominal-T model the shunt admittance is assumed concentrated at the midpoint and the series impedance is split into two equal halves placed on either side.

• The sending-end parameters in terms of the receiving-end parameters are derived from the T network relations.

• In matrix form:

• A, B, C, D parameters for the nominal-T representation are obtained from the T-network element relationships.

Therefore:

• Current in the shunt branch:

• Voltage across the shunt branch:

Nominal-π circuit

In the nominal-π model one half of the total line capacitance of each conductor is placed at each end; the series impedance remains in the middle.

• The sending-end parameters in terms of the receiving-end parameters follow from the π network relations.

V_S = A V_r + Z* I_r

• In matrix form:

• A, B, C, D parameters (for π) are obtained and are used for power flow and regulation studies.

3. Long Transmission Line

A transmission line is classed as long when its length is greater than about 240 km.

• Line parameters are distributed uniformly over the entire length (R, L, G, C per unit length).

Assumptions

• The line operates under sinusoidal, balanced, steady-state conditions.
• The line is transposed to make each phase electrically identical over the length.

• In the figure, V(x) and I(x) denote the phasor voltage and current at distance x measured from the receiving end.
• The series impedance per unit length is z = R + jωL, where R and L are series resistance and inductance per unit length.
• The shunt admittance per unit length is y = G + jωC, where G and C are shunt conductance and capacitance to neutral per unit length.

Consider an element of infinitesimal length Δx at a distance x from the receiving end. Let:

• V = voltage just before entering the element Δx
• I = current just before entering the element Δx
• V + ΔV = voltage leaving the element Δx
• I + ΔI = current leaving the element Δx
• ΔV = voltage drop across element Δx
• z Δx = series impedance of element Δx
• y Δx = shunt admittance of element Δx

Total line impedance Z = z ℓ and total shunt admittance Y = y ℓ, where ℓ is the total line length.

The voltage drop across the infinitesimal element Δx is given by:

Apply Kirchhoff's current law at the node to determine ΔI:

Since the term ΔV · y · Δx is product of two infinitesimals it may be neglected; therefore:

Differentiate the equation (1) with respect to x:

Substitute z and y relations to obtain the second-order differential equations:

The solution of the above second-order differential equation is:

Differentiate the voltage solution with respect to x to obtain the current expression:

Compare with the earlier expressions to identify constants and relationships:

Define the characteristic impedance Z_c and propagation constant γ of the long line as:

Then the voltage and current along the line can be written in terms of Z_c and γ:

At x = 0 (receiving end) V = V_R and I = I_R. Substituting these conditions into the general solutions gives:

Solving for the integration constants A₁ and A₂ yields:

Now apply the condition at x = ℓ (sending end): V = V_S, I = I_S. Substituting x = ℓ and the found constants gives expressions for V_S and I_S:

Using hyperbolic/trigonometric relations:

The above expressions can be re-written in compact hyperbolic form:

Thus, compared with the general ABCD two-port form, the ABCD parameters of a long transmission line are:

Ferranti effect

Ferranti effect: When a long line operates under no-load or very light-load conditions, the receiving-end voltage may be higher than the sending-end voltage. This phenomenon is called the Ferranti effect.

Reasons and remarks

• Usually the capacitive susceptance of the line is significant relative to its inductive susceptance for long lines; charging current causes leading reactive current in the line under light load.
• Under no-load or light load the line current is largely capacitive and leads the voltage; the capacitive charging current produces a voltage rise across the series reactance, increasing the receiving-end voltage.

• If reactive power generated at a point exceeds reactive power absorbed, the voltage at that point rises above normal and vice versa.
• Inductive reactance of the line absorbs (sinks) reactive power, while shunt capacitances generate reactive power.
• If the line loading equals the surge impedance loading (SIL), the reactive power generated by the line equals the reactive power absorbed and the voltage is uniform along the line.
• If loading is less than SIL, reactive power generated exceeds absorbed reactive power and voltages tend to rise toward the receiving end (Ferranti effect).

Power Flow On Transmission Line

• In the typical two-bus model the sending-end bus is fed by a generator and the receiving-end supplies the load. S_r and S_s denote complex power at receiving and sending ends respectively.
• Let receiving end voltage be V_r = |V_r| ∠0° and sending end voltage be V_s = |V_s| ∠δ. Let ABCD parameters be A = D = |A| ∠α, B = |B| ∠β.
• Receiving-end current in terms of V_s, V_r and ABCD parameters:

• The complex power per phase at the receiving end:

• Sending-end current in terms of V_s, V_r, and ABCD parameters:

• The complex power per phase at the sending end (power injected by the generator):

Important observations

• The equations give per-phase power if phase voltages are used for V_s and V_r. Total three-phase power equals three times the per-phase power.
• If V_s and V_r are line-to-line voltages, the equations directly yield three-phase values.

Real and reactive power expressions

Receiving end real and reactive power expressions are:

Sending end real and reactive power expressions are:

• For fixed magnitudes |V_s| and |V_r|, the receiving end real power is maximum when the power-angle δ = β.

• The load must draw leading vars (i.e., supply negative reactive power) to achieve the condition of maximum real power transfer at the receiving end in some cases.

For a short transmission line, where resistance is small compared to inductive reactance:

The transmission line generally has small resistance compared to reactance.

Important Conclusions

• For fixed |V_s|, |V_r| and line reactance X, the real power transferred depends on the power angle δ, which is the phase angle difference between V_s and V_r. Power can be transferred even when |V_s| = |V_r|.
• A decrease in line inductance increases the line transfer capacity.
• Inductance can be reduced by using double-circuit lines or bundled conductors, which increases transfer capability and reduces reactance-limited power transfer.